mintOC - User contributions [en]

Exponential OED

2026-09-07T10:44:01Z

RobertLampel: /* Miscellaneous and Further Reading */

{{Dimensions
|nd = 1
|nx = 2
|nw = 1
}}

The '''Exponential OED problem''' was formulated as minimal design problem that to highlight one important difference between single and multiple shooting. We are interested in finding an optimal experimental design to determine the parameter in a one-dimensional [[:Category:ODE model|ODE model]], where can directly measure the single state.

The optimal integer control functions shows [[:Category:Chattering|bang bang]] behavior.

== Mathematical formulation ==
For a single parameter <math>p</math> the original initial value problem is given by
<math>
\dot{x}(t) = x(t) \cdot (x(t) + p), \quad t \in [0, t_f], \quad x(0) = x_0.
</math>

We furthermore restrict the state to be in the interval <math>[0, 200]</math>. We assume that we have one measurement at the end time point <math>t_f</math>. This allows to eliminate sampling function directly from the control problem and to use
the objective function

<math>
\text{trace} \ C(t_f) = \frac{1}{D(t_f)} = \frac{1}{H(0) + G(t_f)^2 \cdot w_1} = \frac{1}{G(t_f)^2}
</math>


Applying our transformation, we obtain the following experimental design control
problem:


<math>
\begin{array}{lll}
\displaystyle \min_{q} && 1 / G(t_f)^2 \\
\text{subject to} \\
\quad \dot{x}(t) & = & x(t) \cdot (x(t) + p),\\
\quad \dot{G}(t) & = & (p + 2\cdot x(t)) \cdot G(t) + x(t), \\
\quad \dot{F}(t) & = & w(t) \cdot G(t)^2, \\
\quad \dot{z}(t) & = & w(t), \\
\quad x(0) &=& q, \\
\quad G(0) &=& \varepsilon, \\
\quad x(t) &\leq& 200, \\
\quad q &\in& \mathcal{D}
\end{array}
</math>


== Parameters ==
These fixed values are used within the model:

<math>
p = 1; \quad t_f = 0.6; \quad \varepsilon = 10^{-3}.
</math>


== Reference Solutions ==

As can be seen, the optimal solution will maximize the value of <math>G(t_f)</math> and hence also of <math>x(t_f)</math>.
For <math>t_f = 0.6</math> the optimal initial value is given by <math>q^* = 1.203</math> leading to state
values of <math>x(t_f) = 200</math> and <math>G(t_f) = 6088</math> and an objective value of <math>\varphi^* = 2.7 \cdot 10^{-8}</math>.
The main problem with direct single shooting is that a large part of the feasible
domain <math>\mathcal{D}</math> of <math>q</math> will cause the integrator to run into a singularity before <math>t_f</math>. Hence only initial guesses for the optimization variable <math>q</math> that are below a critical value
of <math>\approx 1.23</math> will give rise to a successful optimization. For multiple shooting the
situation is different, due to the decoupling of the integration

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Exponential_OED.png| Left: State variable x(·). Right: Sensitivity G(·). Top row shows initialization of multiple shooting variables at constant values <math>x(t_i) = 2, \ G(t_i) = 10^{-3}</math>. Note that the solution to the initial value problem on <math>[0, 0.6]</math> with <math>x(t_0) = q = 2</math> does not exist, hence the single shooting approach will fail. The bottom row shows the converged solution.
</gallery>

== Miscellaneous and Further Reading ==
The Exponential OED problem was introduced by Körkel et al. in [[#Koerkel | [1]]], which contains further details.

== References ==
[1] "A Multiple Shooting Formulation for Optimum Experimental Design" by S. Körkel, A. Potschka, H.G. Bock, and Sebastian Sager 


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]

Tumor Anti Angiogenesis

2026-09-04T11:30:42Z

RobertLampel:

{{Dimensions
|nd = 1
|nx = 3
|nw = 1
}}

The '''tumor anti-angiogenesis problem''' is a classical benchmark in biomedical optimal control.

It is based on the mathematical model of Hahnfeldt et al. [[#Hahnfeldt|[1]]] for tumor growth under angiogenic signalling, as studied from an optimal control perspective by Ledzewicz and Schättler [[#Ledzewicz|[2]]].
Angiogenesis is the process by which tumors recruit a network of blood vessels to sustain their growth; anti-angiogenic therapy targets this process by administering inhibitor agents.
The two biological states are the tumor volume and the carrying capacity of the vascular endothelium (which sets the maximum supportable tumor size).
A third state tracks the cumulative inhibitor dose, enforcing a total-dose constraint.
The control is the instantaneous inhibitor dose rate.
The final time is free and is itself an optimization variable, bounded within a physiologically reasonable range.
The objective is to minimise the tumor volume at the final time.
The problem is known to exhibit a rich optimal control structure, including bang–bang arcs, singular arcs, and boundary arcs corresponding to the dose constraint.

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{u,\, t_\mathrm{f}} && p(t_\mathrm{f}) \\[6pt]
\text{subject to} \\
\quad \dot{p}(t) &=& -\xi\, p(t)\,\ln\!\left(\dfrac{p(t)}{q(t)}\right), \\[6pt]
\quad \dot{q}(t) &=& \bigl(b - \mu - d\, p(t)^{2/3} - G\, u(t)\bigr)\, q(t), \\[6pt]
\quad \dot{y}(t) &=& u(t), \\
\quad (p(0),\, q(0),\, y(0)) &=& (p_0,\, q_0,\, 0), \\
\quad y(t_\mathrm{f}) &\leq& A, \\
\quad u(t) &\in& [0,\, u_{\max}] \quad \forall\, t \in [0, t_\mathrm{f}], \\
\quad t_\mathrm{f} &\in& [0.5,\, 2.0],
\end{array}
</math>


where <math>p(t)</math> is the tumor volume (mm3), <math>q(t)</math> is the endothelial carrying capacity (mm3), and <math>y(t)</math> is the cumulative inhibitor dose (mg/kg) administered up to time <math>t</math>.
The tumor dynamics follow the Gompertz law <math>\dot{p} = -\xi\,p\ln(p/q)</math>, which models logistic-like growth bounded by the carrying capacity <math>q</math>.
The carrying capacity evolves through a balance of self-stimulation (<math>b\,q</math>), natural endothelial cell loss (<math>\mu\,q</math>), inhibition proportional to tumor surface area (<math>d\,p^{2/3}\,q</math>), and drug-induced endothelial loss (<math>G\,u\,q</math>).
The state <math>y(t)</math> enforces the total-dose constraint <math>y(t_\mathrm{f}) \leq A</math> as an isoperimetric inequality.
The terminal time <math>t_\mathrm{f}</math> is a free optimization variable.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>\xi</math> || align=right | 0.084 || Tumor growth rate (day−1)
|-
| align=center | <math>b</math> || align=right | 5.85 || Endothelial self-stimulation rate (day−1)
|-
| align=center | <math>\mu</math> || align=right | 0 || Natural endothelial cell loss rate (day−1)
|-
| align=center | <math>d</math> || align=right | 0.00873 || Angiogenic inhibition coefficient (mm−2 day−1)
|-
| align=center | <math>G</math> || align=right | 0.15 || Drug efficacy on endothelium (kg mg−1 day−1)
|-
| align=center | <math>u_{\max}</math> || align=right | 75 || Maximum inhibitor dose rate (mg kg−1 day−1)
|-
| align=center | <math>A</math> || align=right | 75 || Maximum cumulative inhibitor dose (mg kg−1)
|-
| align=center | <math>p_0</math> || align=right | 12000 || Initial tumor volume (mm3)
|-
| align=center | <math>q_0</math> || align=right | 15000 || Initial endothelial carrying capacity (mm3)
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | free || Terminal time, optimized in [0.5, 2.0] (days)
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Tumor_Anti_Angiogenesis.png| Tumor volume, carrying capacity, cumulative dose, and optimal inhibitor schedule for a reference solution.
</gallery>

== Miscellaneous and Further Reading ==
The equilibrium <math>p^* = q^*</math> corresponds to a tumor at maximum sustainable size.
Without treatment, <math>p(t)</math> converges to this steady state.
Ledzewicz and Schättler [[#Ledzewicz|[2]]] characterise the optimal control using Pontryagin's minimum principle and show that the optimal policy consists of a sequence of full-dose (<math>u = u_{\max}</math>), no-dose (<math>u = 0</math>), and singular arcs, depending on the initial conditions and remaining dose budget.
The problem is included in the GPOPS-II benchmark suite [[#Patterson|[3]]].

== References ==
[1] Hahnfeldt, P., Panigrahy, D., Folkman, J., & Hlatky, L. (1999). Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Research, 59(19), 4770–4775. 
[2] Ledzewicz, U., & Schättler, H. (2007). Antiangiogenic therapy in cancer treatment as an optimal control problem. SIAM Journal on Control and Optimization, 46(3), 1052–1079. 
[3] Patterson, M. A., & Rao, A. V. (2014). GPOPS-II: A MATLAB software for solving multiple-phase optimal control problems using hp-adaptive Gaussian quadrature collocation methods and sparse nonlinear programming. ACM Transactions on Mathematical Software, 41(1), 1. 

[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Singular arc]]
[[Category:Free terminal time]]

Ramsey Growth

2026-09-04T11:29:39Z

RobertLampel:

{{Dimensions
|nd = 1
|nx = 1
|nw = 1
}}

The '''Ramsey growth problem''' is a foundational benchmark in optimal control and macroeconomic theory.

First formulated by Frank Ramsey in 1928 [[#Ramsey|[1]]] and later extended by Cass [[#Cass|[2]]] and Koopmans [[#Koopmans|[3]]], the Ramsey–Cass–Koopmans model determines the socially optimal path of capital accumulation and consumption for a representative household economy.
The single state is the capital stock per effective worker, and the single control is consumption per effective worker.
Capital accumulates through Cobb–Douglas production, net of depreciation, population growth, technological progress, and consumption.
The objective is to maximise the present value of a stream of discounted CRRA utility over a finite horizon, plus a bequest value at the terminal time, which is equivalent to minimising its negative.
The resulting optimal trajectories converge towards a steady-state balanced growth path.

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{c} && -\int_0^{T} e^{-(\rho - n)\, t}\, \frac{c(t)^{1-\sigma}}{1 - \sigma}\, \mathrm{d}t \;-\; V(k(T)) \\[6pt]
\text{subject to} \\
\quad \dot{k}(t) &=& k(t)^\alpha - c(t) - (n + g + \delta)\, k(t), \\
\quad k(0) &=& k_0, \\
\quad k(t) &>& 0 \quad \forall\, t \in [0, T], \\
\quad c(t) &>& 0 \quad \forall\, t \in [0, T],
\end{array}
</math>


where <math>k(t)</math> is capital per effective worker, <math>c(t)</math> is consumption per effective worker,
<math>k^\alpha</math> is the Cobb–Douglas production function (with total factor productivity normalised to one),
and <math>c^{1-\sigma}/(1-\sigma)</math> is the constant-relative-risk-aversion (CRRA) utility function.
The effective discount rate in the objective is <math>\rho - n</math>, reflecting that utility is evaluated per capita while the economy grows at population rate <math>n</math>.
As <math>\sigma \to 1</math>, the CRRA utility converges to <math>\ln c</math> (logarithmic utility).

The terminal bequest value <math>V(k(T))</math> approximates the discounted steady-state continuation:

<math>
V(k(T)) = e^{-(\rho-n)\,T}\,\frac{u(c^*_{\mathrm{ss}})}{\rho - n}\,\min\!\left(\frac{k(T)}{k^*_{\mathrm{ss}}},\, 1\right),
</math>

where <math>k^*_{\mathrm{ss}}</math> and <math>c^*_{\mathrm{ss}}</math> are the steady-state capital and consumption (see below).

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>\alpha</math> || align=right | 0.33 || Capital share in production (Cobb–Douglas exponent)
|-
| align=center | <math>\delta</math> || align=right | 0.05 || Capital depreciation rate (year−1)
|-
| align=center | <math>n</math> || align=right | 0.01 || Population (labour) growth rate (year−1)
|-
| align=center | <math>g</math> || align=right | 0.02 || Technological progress rate (year−1)
|-
| align=center | <math>\rho</math> || align=right | 0.04 || Pure rate of time preference (year−1)
|-
| align=center | <math>\sigma</math> || align=right | 1.5 || Inverse of the intertemporal elasticity of substitution
|-
| align=center | <math>k_0</math> || align=right | 3.0 || Initial capital per effective worker
|-
| align=center | <math>T</math> || align=right | 100 || Planning horizon (years)
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Ramsey_Growth.png| Optimal capital and consumption paths for a reference solution.
</gallery>

== Miscellaneous and Further Reading ==
The steady-state capital stock satisfies the modified golden rule
<math>\alpha\,(k^*_{\mathrm{ss}})^{\alpha-1} = \rho + \delta + n + g</math>,
giving <math>k^*_{\mathrm{ss}} = \bigl(\alpha/(\rho+\delta+n+g)\bigr)^{1/(1-\alpha)}</math> and
<math>c^*_{\mathrm{ss}} = (k^*_{\mathrm{ss}})^\alpha - (n+g+\delta)\,k^*_{\mathrm{ss}}</math>.
The optimal consumption path satisfies the Keynes–Ramsey rule
<math>\dot{c}/c = \bigl(\alpha\,k^{\alpha-1} - \delta - \rho - g\bigr)/\sigma</math>.
The problem is solved analytically in the infinite-horizon limit and numerically for finite horizons using shooting or direct methods; the finite-horizon version with a bequest value is a standard test for indirect shooting and continuation techniques.

== References ==
[1] Ramsey, F. P. (1928). A mathematical theory of saving. The Economic Journal, 38(152), 543–559. 
[2] Cass, D. (1965). Optimum growth in an aggregative model of capital accumulation. The Review of Economic Studies, 32(3), 233–240. 
[3] Koopmans, T. C. (1965). On the concept of optimal economic growth. The Econometric Approach to Development Planning. North-Holland. 

[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Economic model]]

SEIR Epidemic

2026-09-04T11:28:46Z

RobertLampel:

{{Dimensions
|nd = 1
|nx = 4
|nw = 2
}}

The '''SEIR epidemic optimal control problem''' models the spread of an infectious disease through a homogeneous population divided into four compartments: susceptible (<math>S</math>), exposed (<math>E</math>), infectious (<math>I</math>), and recovered (<math>R</math>).
Two controls are applied simultaneously: a vaccination rate that removes susceptible individuals from the transmission chain, and a social-distancing intensity that reduces the effective contact rate.
The objective is to minimise a weighted combination of the cumulative infectious burden and the quadratic costs of both interventions over a finite horizon, yielding a trade-off between disease suppression and resource expenditure.
The problem is nonlinear and non-convex and is calibrated to a COVID-19-like baseline scenario.

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{u_1,\,u_2} && \displaystyle\int_0^{T} \Bigl[ A\, I(t) + B\, u_1(t)^2 + C\, u_2(t)^2 \Bigr]\, \mathrm{d}t \\[8pt]
\text{subject to} \\[4pt]
\quad \dot{S}(t) &=& -\beta\,(1 - u_2(t))\,S(t)\,I(t) - u_1(t)\,S(t), \\[4pt]
\quad \dot{E}(t) &=& \beta\,(1 - u_2(t))\,S(t)\,I(t) - \sigma\,E(t), \\[4pt]
\quad \dot{I}(t) &=& \sigma\,E(t) - \gamma\,I(t), \\[4pt]
\quad \dot{R}(t) &=& \gamma\,I(t) + u_1(t)\,S(t), \\[4pt]
\quad x(0) &=& (S_0,\; E_0,\; I_0,\; R_0)^T, \\
\quad u_1(t) &\in& [0,\, u_{1,\max}] \quad \forall\, t \in [0, T], \\
\quad u_2(t) &\in& [0,\, u_{2,\max}] \quad \forall\, t \in [0, T],
\end{array}
</math>


where <math>x = (S, E, I, R)^T</math>.
All compartments are fractions of the total population, so <math>S(t) + E(t) + I(t) + R(t) = 1</math> for all <math>t</math>.

The first control <math>u_1(t) \in [0, u_{1,\max}]</math> is the vaccination rate: the term <math>-u_1(t)\,S(t)</math> removes susceptible individuals through vaccination, moving them directly to the recovered compartment at rate <math>u_1(t)\,S(t)</math>.

The second control <math>u_2(t) \in [0, 1]</math> is the social-distancing intensity: <math>u_2 = 0</math> corresponds to no intervention and <math>u_2 = 1</math> to full isolation.
It enters multiplicatively as <math>(1 - u_2(t))</math> in the force of infection <math>\beta\,(1-u_2)\,S\,I</math>, reducing effective contacts without directly removing individuals from any compartment.

The quadratic penalties <math>B\,u_1^2</math> and <math>C\,u_2^2</math> model the increasing marginal cost of each intervention.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>\beta</math> || align=right | 0.5 || Contact (transmission) rate (day−1)
|-
| align=center | <math>\sigma</math> || align=right | 0.2 || Progression rate from exposed to infectious (day−1)
|-
| align=center | <math>\gamma</math> || align=right | 0.1 || Recovery rate (day−1)
|-
| align=center | <math>A</math> || align=right | 1 || Weight on infectious burden
|-
| align=center | <math>B</math> || align=right | 1 || Quadratic cost weight on vaccination
|-
| align=center | <math>C</math> || align=right | 1 || Quadratic cost weight on social distancing
|-
| align=center | <math>u_{1,\max}</math> || align=right | 0.10 || Maximum vaccination rate (day−1)
|-
| align=center | <math>u_{2,\max}</math> || align=right | 1.00 || Maximum social-distancing intensity
|-
| align=center | <math>S_0</math> || align=right | 0.990 || Initial susceptible fraction
|-
| align=center | <math>E_0</math> || align=right | 0.005 || Initial exposed fraction
|-
| align=center | <math>I_0</math> || align=right | 0.005 || Initial infectious fraction
|-
| align=center | <math>R_0</math> || align=right | 0 || Initial recovered fraction
|-
| align=center | <math>T</math> || align=right | 100 || Planning horizon (days)
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:SEIR_Epidemic.png| Compartment trajectories and optimal vaccination and social-distancing schedules for a reference solution.
</gallery>

== Miscellaneous and Further Reading ==
The basic reproduction number for the uncontrolled SEIR model is <math>\mathcal{R}_0 = \beta/\gamma</math>.
With the parameters above, <math>\mathcal{R}_0 = 5</math>, corresponding to a highly transmissible pathogen.
The two-control formulation combines pharmaceutical intervention (vaccination) with non-pharmaceutical intervention (social distancing), reflecting the dual-measure response used during the COVID-19 pandemic.
Necessary optimality conditions via Pontryagin's minimum principle lead to a coupled system of state and adjoint ODEs; the optimal control typically exhibits a bang–singular–bang structure for the vaccination arc and a smooth interior arc for social distancing.

== References ==
[1] Behncke, H. (2000). Optimal control of deterministic epidemics. Optimal Control Applications and Methods, 21(6), 269–285. 
[2] Perkins, T. A., & España, G. (2020). Optimal control of the COVID-19 pandemic with non-pharmaceutical interventions. Bulletin of Mathematical Biology, 82, 118. 

[[Category:MIOCP]]
[[Category:ODE model]]

Brachistochrone

2026-09-04T11:27:52Z

RobertLampel:

{{Dimensions
|nd = 1
|nx = 3
|nw = 1
}}

The '''brachistochrone problem''' (from Greek ''brachístos chrónos'', "shortest time") is one of the founding problems of the calculus of variations, and serves as a classical benchmark in optimal control.

First posed by Johann Bernoulli in 1696 as a public challenge, the problem asks for the shape of a frictionless curve connecting two fixed points in a vertical plane such that a particle released from rest at the upper point reaches the lower point in minimum time under gravity.
As an optimal control problem, the system state consists of the horizontal position, the downward vertical position, and the speed of the particle, while the control is the instantaneous angle of the velocity vector measured below the horizontal.
The terminal time is free and is the quantity to be minimised; it is handled by augmenting it as a global time-scaling variable and optimising over a normalised unit horizon.
The analytical solution is a cycloid, obtainable through Pontryagin's minimum principle or the classical Euler–Lagrange equations.

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{\theta,\, t_\mathrm{f}} && t_\mathrm{f} \\[6pt]
\text{subject to} \\
\quad \dot{x}(t) &=& v(t)\,\cos(\theta(t)), \\
\quad \dot{y}(t) &=& v(t)\,\sin(\theta(t)), \\
\quad \dot{v}(t) &=& g\,\sin(\theta(t)), \\
\quad (x(0),\, y(0),\, v(0)) &=& (0,\; 0,\; 0), \\
\quad x(t_\mathrm{f}) &=& x_\mathrm{f}, \\
\quad y(t_\mathrm{f}) &=& y_\mathrm{f}, \\
\quad \theta(t) &\in& [0,\, \pi] \quad \forall\, t \in [0, t_\mathrm{f}],
\end{array}
</math>


where <math>x(t)</math> is the horizontal position (m), <math>y(t)</math> is the downward vertical position (m, positive downward), and <math>v(t)</math> is the speed of the particle (m/s).
The control <math>\theta(t) \in [0, \pi]</math> is the angle of the velocity vector below the horizontal: <math>\theta = 0</math> corresponds to purely horizontal motion and <math>\theta = \pi/2</math> to straight downward motion.
The speed dynamics <math>\dot{v} = g\sin\theta</math> follow from the component of gravitational acceleration along the direction of motion.
Since <math>v(0) = 0</math>, the particle starts from rest.
The constraint <math>\theta \in [0, \pi]</math> restricts the particle to descent arcs (no upward motion).

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>g</math> || align=right | 9.81 || Gravitational acceleration (m s−2)
|-
| align=center | <math>x_\mathrm{f}</math> || align=right | 1.0 || Target horizontal position (m)
|-
| align=center | <math>y_\mathrm{f}</math> || align=right | 0.5 || Target downward vertical position (m)
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Brachistochrone.png| Optimal trajectory and control for the brachistochrone problem (cycloid solution).
</gallery>

== Miscellaneous and Further Reading ==
The analytical solution is a cycloid: the locus traced by a point on the rim of a circle rolling along a flat surface.
It can be parametrised as
<math>x(\phi) = r(\phi - \sin\phi)</math>, <math>y(\phi) = r(1 - \cos\phi)</math>,
where the radius <math>r</math> is determined by the boundary conditions.
The brachistochrone problem is a standard example in textbooks on optimal control and the calculus of variations, and is used as a test case for direct transcription methods due to its free terminal time and the mild singularity at <math>t = 0</math> where the speed vanishes.
A detailed formulation and further discussion can be found in [[#Bryson|[1]]].

== References ==
[1] Bryson, A. E., & Ho, Y.-C. (1975). Applied Optimal Control: Optimization, Estimation, and Control. Taylor & Francis. 

[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Free terminal time]]

Battery Storage

2026-09-04T11:26:53Z

RobertLampel: /* Reference Solutions */

{{Dimensions
|nd = 1
|nx = 1
|nw = 2
}}

The '''battery storage problem''' models the optimal operation of a grid-scale battery energy storage system (BESS) over a 24-hour scheduling horizon.
The operator minimises total electricity cost given a fixed, time-varying demand profile and a time-of-use electricity price signal.
Both signals are piecewise-constant, stepping every four hours through six daily blocks.
The two non-negative controls represent charging power and discharging power separately, which avoids the need for integer variables while keeping the physics transparent.
The optimal strategy is to charge during cheap night-time periods and discharge during expensive morning and evening peak periods, yielding an energy-arbitrage policy.

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{P_c,\,P_d} && \displaystyle\int_0^{T} p(t)\,\bigl(P_{\mathrm{load}}(t) - P_d(t) + P_c(t)\bigr)\,\mathrm{d}t \\[8pt]
\text{subject to} \\[4pt]
\quad \dot{x}(t) &=& \dfrac{\eta_c\,P_c(t) - P_d(t)/\eta_d}{E_{\mathrm{cap}}}, \\[8pt]
\quad x(0) &=& x_0, \\
\quad x(T) &=& x_0, \\
\quad x(t) &\in& [x_{\min},\, x_{\max}] \quad \forall\, t \in [0, T], \\
\quad P_c(t) &\in& [0,\, P_{\max}] \quad \forall\, t \in [0, T], \\
\quad P_d(t) &\in& [0,\, P_{\max}] \quad \forall\, t \in [0, T],
\end{array}
</math>


where <math>x(t)</math> is the state of charge (SoC, dimensionless fraction of <math>E_{\mathrm{cap}}</math>), <math>P_c(t) \geq 0</math> is the charging power (kW), <math>P_d(t) \geq 0</math> is the discharging power (kW), and <math>P_{\mathrm{grid}}(t) = P_{\mathrm{load}}(t) - P_d(t) + P_c(t)</math> is the net power drawn from the grid.
The cyclic boundary condition <math>x(T) = x_0</math> enforces a daily cycling constraint so the battery ends each day at the same charge level it started.

=== Price and demand profiles ===

Both <math>p(t)</math> and <math>P_{\mathrm{load}}(t)</math> are piecewise-constant step functions defined on six consecutive 4-hour blocks:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Block !! Time interval (h) !! <math>p(t)</math> (€/kWh) !! <math>P_{\mathrm{load}}(t)</math> (kW)
|-
| align=center | 1 || align=center | [0, 4) || align=right | 0.08 || align=right | 30
|-
| align=center | 2 || align=center | [4, 8) || align=right | 0.20 || align=right | 60
|-
| align=center | 3 || align=center | [8, 12) || align=right | 0.30 || align=right | 80
|-
| align=center | 4 || align=center | [12, 16)|| align=right | 0.15 || align=right | 50
|-
| align=center | 5 || align=center | [16, 20)|| align=right | 0.28 || align=right | 70
|-
| align=center | 6 || align=center | [20, 24]|| align=right | 0.10 || align=right | 35
|}

The price pattern is cheap at night (blocks 1 and 6), expensive at the morning peak (block 3), and moderately expensive at the evening peak (block 5).
The demand profile follows a corresponding load curve with low overnight consumption and a mid-morning peak.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>E_{\mathrm{cap}}</math> || align=right | 200 kWh || Battery energy capacity
|-
| align=center | <math>P_{\max}</math> || align=right | 50 kW || Maximum charge / discharge power
|-
| align=center | <math>\eta_c</math> || align=right | 0.95 || Charging efficiency
|-
| align=center | <math>\eta_d</math> || align=right | 0.95 || Discharging efficiency
|-
| align=center | <math>x_{\min}</math> || align=right | 0.10 || Minimum state of charge
|-
| align=center | <math>x_{\max}</math> || align=right | 0.90 || Maximum state of charge
|-
| align=center | <math>x_0</math> || align=right | 0.50 || Initial (and final) state of charge
|-
| align=center | <math>T</math> || align=right | 24 h || Scheduling horizon
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Battery_Storage.png| State of charge and optimal power schedule for a reference solution.
</gallery>

== Miscellaneous and Further Reading ==
The battery storage optimal control problem arises in the operation of grid-scale energy storage, demand-response programs, and behind-the-meter storage systems.
Formulating charging and discharging as two separate non-negative controls avoids the need for complementarity constraints or integer variables; the NLP relaxation is typically tight for cost profiles of this type because the optimizer has no incentive to charge and discharge simultaneously when round-trip efficiency is less than one.
Numerical solutions generally exhibit a bang–bang structure in each control: the battery charges at full power during the cheapest blocks and discharges at full power during the most expensive blocks, with smooth transitions determined by the SoC bounds.

== References ==
[1] Rahbar, N., Chai, C. C., & Zhang, R. (2015). Energy cooperation optimization in microgrids with renewable energy integration. IEEE Trans. Smart Grid, 6(3), 1482–1493. 
[2] Nottrott, A., Kleissl, J., & Washom, B. (2013). Energy dispatch schedule optimization and cost benefit analysis for grid-connected, photovoltaic-battery storage systems. Renewable Energy, 55, 230–240. 

[[Category:ODE model]]
[[Category:MIOCP]]

Battery Storage

2026-09-04T11:26:32Z

RobertLampel:

{{Dimensions
|nd = 1
|nx = 1
|nw = 2
}}

The '''battery storage problem''' models the optimal operation of a grid-scale battery energy storage system (BESS) over a 24-hour scheduling horizon.
The operator minimises total electricity cost given a fixed, time-varying demand profile and a time-of-use electricity price signal.
Both signals are piecewise-constant, stepping every four hours through six daily blocks.
The two non-negative controls represent charging power and discharging power separately, which avoids the need for integer variables while keeping the physics transparent.
The optimal strategy is to charge during cheap night-time periods and discharge during expensive morning and evening peak periods, yielding an energy-arbitrage policy.

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{P_c,\,P_d} && \displaystyle\int_0^{T} p(t)\,\bigl(P_{\mathrm{load}}(t) - P_d(t) + P_c(t)\bigr)\,\mathrm{d}t \\[8pt]
\text{subject to} \\[4pt]
\quad \dot{x}(t) &=& \dfrac{\eta_c\,P_c(t) - P_d(t)/\eta_d}{E_{\mathrm{cap}}}, \\[8pt]
\quad x(0) &=& x_0, \\
\quad x(T) &=& x_0, \\
\quad x(t) &\in& [x_{\min},\, x_{\max}] \quad \forall\, t \in [0, T], \\
\quad P_c(t) &\in& [0,\, P_{\max}] \quad \forall\, t \in [0, T], \\
\quad P_d(t) &\in& [0,\, P_{\max}] \quad \forall\, t \in [0, T],
\end{array}
</math>


where <math>x(t)</math> is the state of charge (SoC, dimensionless fraction of <math>E_{\mathrm{cap}}</math>), <math>P_c(t) \geq 0</math> is the charging power (kW), <math>P_d(t) \geq 0</math> is the discharging power (kW), and <math>P_{\mathrm{grid}}(t) = P_{\mathrm{load}}(t) - P_d(t) + P_c(t)</math> is the net power drawn from the grid.
The cyclic boundary condition <math>x(T) = x_0</math> enforces a daily cycling constraint so the battery ends each day at the same charge level it started.

=== Price and demand profiles ===

Both <math>p(t)</math> and <math>P_{\mathrm{load}}(t)</math> are piecewise-constant step functions defined on six consecutive 4-hour blocks:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Block !! Time interval (h) !! <math>p(t)</math> (€/kWh) !! <math>P_{\mathrm{load}}(t)</math> (kW)
|-
| align=center | 1 || align=center | [0, 4) || align=right | 0.08 || align=right | 30
|-
| align=center | 2 || align=center | [4, 8) || align=right | 0.20 || align=right | 60
|-
| align=center | 3 || align=center | [8, 12) || align=right | 0.30 || align=right | 80
|-
| align=center | 4 || align=center | [12, 16)|| align=right | 0.15 || align=right | 50
|-
| align=center | 5 || align=center | [16, 20)|| align=right | 0.28 || align=right | 70
|-
| align=center | 6 || align=center | [20, 24]|| align=right | 0.10 || align=right | 35
|}

The price pattern is cheap at night (blocks 1 and 6), expensive at the morning peak (block 3), and moderately expensive at the evening peak (block 5).
The demand profile follows a corresponding load curve with low overnight consumption and a mid-morning peak.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>E_{\mathrm{cap}}</math> || align=right | 200 kWh || Battery energy capacity
|-
| align=center | <math>P_{\max}</math> || align=right | 50 kW || Maximum charge / discharge power
|-
| align=center | <math>\eta_c</math> || align=right | 0.95 || Charging efficiency
|-
| align=center | <math>\eta_d</math> || align=right | 0.95 || Discharging efficiency
|-
| align=center | <math>x_{\min}</math> || align=right | 0.10 || Minimum state of charge
|-
| align=center | <math>x_{\max}</math> || align=right | 0.90 || Maximum state of charge
|-
| align=center | <math>x_0</math> || align=right | 0.50 || Initial (and final) state of charge
|-
| align=center | <math>T</math> || align=right | 24 h || Scheduling horizon
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:BatteryStorage.png| State of charge and optimal power schedule for a reference solution.
</gallery>

== Miscellaneous and Further Reading ==
The battery storage optimal control problem arises in the operation of grid-scale energy storage, demand-response programs, and behind-the-meter storage systems.
Formulating charging and discharging as two separate non-negative controls avoids the need for complementarity constraints or integer variables; the NLP relaxation is typically tight for cost profiles of this type because the optimizer has no incentive to charge and discharge simultaneously when round-trip efficiency is less than one.
Numerical solutions generally exhibit a bang–bang structure in each control: the battery charges at full power during the cheapest blocks and discharges at full power during the most expensive blocks, with smooth transitions determined by the SoC bounds.

== References ==
[1] Rahbar, N., Chai, C. C., & Zhang, R. (2015). Energy cooperation optimization in microgrids with renewable energy integration. IEEE Trans. Smart Grid, 6(3), 1482–1493. 
[2] Nottrott, A., Kleissl, J., & Washom, B. (2013). Energy dispatch schedule optimization and cost benefit analysis for grid-connected, photovoltaic-battery storage systems. Renewable Energy, 55, 230–240. 

[[Category:ODE model]]
[[Category:MIOCP]]

User:RobertLampel

2026-09-03T13:50:04Z

RobertLampel:

Working on
<ul>
<li> https://mintoc.de/index.php/DOW_Experimental_Design </li>
<li> https://mintoc.de/index.php/Urethane </li>
<li> https://mintoc.de/index.php/Toy_OED </li>
<li> https://mintoc.de/index.php/Rao_Mease </li>
<li> https://mintoc.de/index.php/Bryson_Denham </li>
<li> https://mintoc.de/index.php/Tubular_Reactor </li>
<li> https://mintoc.de/index.php/Mountain_Car </li>
<li> https://mintoc.de/index.php/Denbigh_Reaction </li>
<li> https://mintoc.de/index.php/Linear_Quadratic_Regulator </li>
<li> https://mintoc.de/index.php/Ocean </li>
<li> https://mintoc.de/index.php/Fermenter </li>
<li> https://mintoc.de/index.php/Van_der_Pol_OED </li>
<li> https://mintoc.de/index.php/Exponential_OED </li>
<li> https://mintoc.de/index.php/LV_Shared_Resource </li>
<li> https://mintoc.de/index.php/LV_Competitive </li>
<li> https://mintoc.de/index.php/Oscillating_OED </li>
<li> https://mintoc.de/index.php/Compartmental_OED </li>
<li> https://mintoc.de/index.php/Dielectrophoretic_Particle</li>
<li> https://mintoc.de/index.php/Double_Oscillator</li>
<li> https://mintoc.de/index.php/Ducted_Fan</li>
<li> https://mintoc.de/index.php/Jackson</li>
<li> https://mintoc.de/index.php/Robbins</li>
<li> https://mintoc.de/index.php/Hang_Glider</li>
<li> https://mintoc.de/index.php/Dielectrophoretic_Particle_OED</li>
<li> https://mintoc.de/index.php/Jackson_OED</li>
<li> https://mintoc.de/index.php/Moon_Landing</li>
<li> https://mintoc.de/index.php/Cart_Pendulum</li>
<li> https://mintoc.de/index.php/Lotka_Shared_OED</li>
<li> https://mintoc.de/index.php/Three_Tank_OED</li>
<li> https://mintoc.de/index.php/Battery_Storage</li>
<li> https://mintoc.de/index.php/Brachistochrone</li>
<li> https://mintoc.de/index.php/SEIR_Epidemic</li>
<li> https://mintoc.de/index.php/Ramsey_Growth</li>
<li> https://mintoc.de/index.php/Tumor_Anti_Angiogenesis</li>
</ul>

[[Category:Community]]

Tumor Anti Angiogenesis

2026-09-03T11:50:00Z

RobertLampel: /* References */

{{Dimensions
|nd = 1
|nx = 3
|nw = 1
}}

The '''tumor anti-angiogenesis problem''' is a classical benchmark in biomedical optimal control. This description is taken from [[#OCPjl | [1]]].

It is based on the mathematical model of Hahnfeldt et al. [[#Hahnfeldt|[2]]] for tumor growth under angiogenic signalling, as studied from an optimal control perspective by Ledzewicz and Schättler [[#Ledzewicz|[3]]].
Angiogenesis is the process by which tumors recruit a network of blood vessels to sustain their growth; anti-angiogenic therapy targets this process by administering inhibitor agents.
The two biological states are the tumor volume and the carrying capacity of the vascular endothelium (which sets the maximum supportable tumor size).
A third state tracks the cumulative inhibitor dose, enforcing a total-dose constraint.
The control is the instantaneous inhibitor dose rate.
The objective is to minimise the tumor volume at the final time.
The problem is known to exhibit a rich optimal control structure, including bang–bang arcs, singular arcs, and boundary arcs corresponding to the dose constraint.

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{u} && p(t_\mathrm{f}) \\[6pt]
\text{subject to} \\
\quad \dot{p}(t) &=& -\xi\, p(t)\,\ln\!\left(\dfrac{p(t)}{q(t)}\right), \\[6pt]
\quad \dot{q}(t) &=& b\, p(t) - \mu\, q(t) - \left(d\, p(t)^{2/3} + e\, u(t)\right) q(t), \\[6pt]
\quad \dot{y}(t) &=& u(t), \\
\quad (p(0),\, q(0),\, y(0)) &=& (p_0,\, q_0,\, 0), \\
\quad y(t_\mathrm{f}) &\leq& A, \\
\quad u(t) &\in& [0,\, a] \quad \forall\, t \in [0, t_\mathrm{f}],
\end{array}
</math>


where <math>p(t)</math> is the tumor volume (mm3), <math>q(t)</math> is the endothelial carrying capacity (mm3), and <math>y(t)</math> is the cumulative inhibitor dose (mg/kg) administered up to time <math>t</math>.
The tumor dynamics follow the Gompertz law <math>\dot{p} = -\xi\,p\ln(p/q)</math>, which models logistic-like growth bounded by the carrying capacity <math>q</math>.
The carrying capacity decreases due to natural endothelial cell loss (<math>\mu\,q</math>), inhibitor-induced endothelial cell loss (<math>e\,u\,q</math>), and stimulated shedding proportional to tumor surface area (<math>d\,p^{2/3}\,q</math>), while increasing due to endothelial recruitment by the tumor (<math>b\,p</math>).
The state <math>y(t)</math> enforces the total-dose constraint <math>y(t_\mathrm{f}) \leq A</math> as an isoperimetric inequality.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>\xi</math> || align=right | 0.084 || Tumor growth rate (day−1)
|-
| align=center | <math>b</math> || align=right | 5.85 || Endothelial birth rate coefficient (day−1)
|-
| align=center | <math>\mu</math> || align=right | 0.02 || Endothelial natural loss rate (day−1)
|-
| align=center | <math>d</math> || align=right | 0.00873 || Inhibition coefficient for endothelial shedding (mm−2 day−1)
|-
| align=center | <math>e</math> || align=right | 0.66 || Inhibitor effectiveness on endothelium (kg mg−1 day−1)
|-
| align=center | <math>a</math> || align=right | 75 || Maximum inhibitor dose rate (mg kg−1 day−1)
|-
| align=center | <math>A</math> || align=right | 15 || Maximum cumulative inhibitor dose (mg kg−1)
|-
| align=center | <math>p_0</math> || align=right | 12000 || Initial tumor volume (mm3)
|-
| align=center | <math>q_0</math> || align=right | 15000 || Initial endothelial carrying capacity (mm3)
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 1.2 || Horizon of the control problem (days)
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image: Tumor_Anti_Angiogenesis.png | Tumor volume, carrying capacity, cumulative dose, and optimal inhibitor schedule for a reference solution.
</gallery>

== Miscellaneous and Further Reading ==
The equilibrium <math>p^* = q^*</math> corresponds to a tumor at maximum sustainable size.
Without treatment, <math>p(t)</math> converges to this steady state.
Ledzewicz and Schättler [[#Ledzewicz|[3]]] characterise the optimal control using Pontryagin's minimum principle and show that the optimal policy consists of a sequence of full-dose (<math>u = a</math>), no-dose (<math>u = 0</math>), and singular arcs, depending on the initial conditions and remaining dose budget.
This formulation and detailed analysis can be found in [[#OCPjl|[1]]], [[#Hahnfeldt|[2]]], and [[#Ledzewicz|[3]]].

== References ==
[1] Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl 
[2] Hahnfeldt, P., Panigrahy, D., Folkman, J., & Hlatky, L. (1999). Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Research, 59(19), 4770–4775. 
[3] Ledzewicz, U., & Schättler, H. (2007). Antiangiogenic therapy in cancer treatment as an optimal control problem. SIAM Journal on Control and Optimization, 46(3), 1052–1079. 

[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Biomedical model]]
[[Category:Singular arc]]

Ramsey Growth

2026-09-03T11:49:47Z

RobertLampel: /* References */

{{Dimensions
|nd = 1
|nx = 1
|nw = 1
}}

The '''Ramsey growth problem''' is a foundational benchmark in optimal control and macroeconomic theory. This description is taken from [[#OCPjl | [1]]].

First formulated by Frank Ramsey in 1928 [[#Ramsey|[2]]] and later extended by Cass [[#Cass|[3]]] and Koopmans [[#Koopmans|[4]]], the problem determines the socially optimal path of capital accumulation and consumption for a representative household economy.
The single state is the capital stock per worker, and the single control is consumption per worker.
Capital accumulates through production (described by a Cobb–Douglas function), net of depreciation and consumption.
The objective is to maximise the present value of a stream of discounted utility of consumption over a finite horizon, which is equivalent to minimising its negative.
The resulting optimal trajectories converge towards a steady-state balanced growth path, and the problem serves as a standard test for indirect shooting methods and continuation techniques in optimal control.

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{c} && -\int_0^{t_\mathrm{f}} e^{-\rho\, t}\, \frac{c(t)^{1-\sigma} - 1}{1 - \sigma}\, \mathrm{d}t \\[6pt]
\text{subject to} \\
\quad \dot{k}(t) &=& A\, k(t)^\alpha - (\delta + n)\, k(t) - c(t), \\
\quad k(0) &=& k_0, \\
\quad k(t) &\geq& 0 \quad \forall\, t \in [0, t_\mathrm{f}], \\
\quad c(t) &>& 0 \quad \forall\, t \in [0, t_\mathrm{f}],
\end{array}
</math>


where <math>k(t)</math> is capital per worker, <math>c(t)</math> is consumption per worker,
<math>A\,k^\alpha</math> is the Cobb–Douglas production function, and
<math>\frac{c^{1-\sigma}-1}{1-\sigma}</math> is the constant-relative-risk-aversion (CRRA) utility function.
As <math>\sigma \to 1</math>, the CRRA utility converges to <math>\ln c</math> (logarithmic utility).
The transversality condition at the terminal time is <math>e^{-\rho\,t_\mathrm{f}}\,k(t_\mathrm{f}) \geq 0</math>.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>\rho</math> || align=right | 0.05 || Pure rate of time preference (discount rate)
|-
| align=center | <math>\sigma</math> || align=right | 1.5 || Inverse of the intertemporal elasticity of substitution
|-
| align=center | <math>\alpha</math> || align=right | 0.33 || Capital share in production (Cobb–Douglas exponent)
|-
| align=center | <math>\delta</math> || align=right | 0.05 || Capital depreciation rate
|-
| align=center | <math>n</math> || align=right | 0.01 || Population (labour) growth rate
|-
| align=center | <math>A</math> || align=right | 1 || Total factor productivity
|-
| align=center | <math>k_0</math> || align=right | 0.5 || Initial capital per worker
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 100 || Horizon of the control problem
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image: Ramsey_Growth.png | Optimal capital and consumption paths for a reference solution.
</gallery>

== Miscellaneous and Further Reading ==
The steady-state capital stock <math>k^*</math> satisfies <math>A\,\alpha\,(k^*)^{\alpha-1} = \rho + \sigma\,n + \delta</math> (the modified golden rule).
The optimal consumption path satisfies the Euler equation
<math>\dot{c}/c = (\alpha A k^{\alpha-1} - \delta - \rho - \sigma n)/\sigma</math>,
known as the Keynes–Ramsey rule.
The problem is solved analytically in the infinite-horizon limit and numerically for finite horizons using shooting or direct methods.
This formulation and further discussion can be found in [[#OCPjl|[1]]] and [[#Ramsey|[2]]].

== References ==
[1] Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl 
[2] Ramsey, F. P. (1928). A mathematical theory of saving. The Economic Journal, 38(152), 543–559. 
[3] Cass, D. (1965). Optimum growth in an aggregative model of capital accumulation. The Review of Economic Studies, 32(3), 233–240. 
[4] Koopmans, T. C. (1965). On the concept of optimal economic growth. The Econometric Approach to Development Planning. North-Holland. 

[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Economic model]]

SEIR Epidemic

2026-09-03T11:49:35Z

RobertLampel: /* References */

{{Dimensions
|nd = 1
|nx = 4
|nw = 1
}}

The '''SEIR epidemic optimal control problem''' is a classical benchmark in mathematical epidemiology and optimal control of biological systems. This description is taken from [[#OCPjl | [1]]].

It models the spread of an infectious disease through a homogeneous population divided into four compartments: susceptible (<math>S</math>), exposed (<math>E</math>), infectious (<math>I</math>), and recovered (<math>R</math>).
The control represents a time-varying vaccination (or isolation) rate applied to the susceptible compartment.
The objective is to minimise a weighted combination of the cumulative infectious burden and the cost of the intervention over a finite horizon, yielding a trade-off between disease suppression and resource expenditure.
The problem is nonlinear and non-convex, and typically admits a singular arc structure in the optimal control.

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{u} && \int_0^{t_\mathrm{f}} \Bigl[ A\, I(t) + \tfrac{1-A}{2}\, u(t)^2 \Bigr]\, \mathrm{d}t \\[6pt]
\text{subject to} \\
\quad \dot{S}(t) &=& -\beta\, S(t)\, I(t) - u(t)\, S(t), \\
\quad \dot{E}(t) &=& \beta\, S(t)\, I(t) - \sigma\, E(t), \\
\quad \dot{I}(t) &=& \sigma\, E(t) - \gamma\, I(t), \\
\quad \dot{R}(t) &=& \gamma\, I(t) + u(t)\, S(t), \\
\quad x(0) &=& (S_0,\; E_0,\; I_0,\; R_0)^T, \\
\quad u(t) &\in& [0,\, u_{\max}] \quad \forall\, t \in [0, t_\mathrm{f}],
\end{array}
</math>


where <math>x = (S, E, I, R)^T</math>.
All compartments are fractions of the total population, so <math>S(t) + E(t) + I(t) + R(t) = 1</math> for all <math>t</math>.
The term <math>-u(t)\,S(t)</math> removes susceptible individuals through vaccination (or effective isolation), moving them directly to the recovered compartment at rate <math>u(t)\,S(t)</math>.
The quadratic penalty <math>\tfrac{1-A}{2}\,u^2</math> models the increasing marginal cost of intervention.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>\beta</math> || align=right | 0.5 || Transmission rate (day−1)
|-
| align=center | <math>\sigma</math> || align=right | 0.2 || Progression rate from exposed to infectious (day−1)
|-
| align=center | <math>\gamma</math> || align=right | 0.1 || Recovery rate (day−1)
|-
| align=center | <math>A</math> || align=right | 0.5 || Weight on infection burden vs. control cost
|-
| align=center | <math>u_{\max}</math> || align=right | 0.5 || Maximum vaccination/isolation rate (day−1)
|-
| align=center | <math>S_0</math> || align=right | 0.99 || Initial susceptible fraction
|-
| align=center | <math>E_0</math> || align=right | 0.01 || Initial exposed fraction
|-
| align=center | <math>I_0</math> || align=right | 0 || Initial infectious fraction
|-
| align=center | <math>R_0</math> || align=right | 0 || Initial recovered fraction
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 100 || Horizon of the control problem (days)
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image: SEIR_Epidemic.png | Compartment trajectories and optimal vaccination rate for a reference solution.
</gallery>

== Miscellaneous and Further Reading ==
The basic reproduction number for the uncontrolled SEIR model is <math>\mathcal{R}_0 = \beta/\gamma</math>.
With the parameters above, <math>\mathcal{R}_0 = 5</math>, corresponding to a highly transmissible pathogen.
The optimal control problem was studied in detail by Lenhart and Workman [[#Lenhart|[2]]], where necessary conditions via Pontryagin's minimum principle are derived and adjoint equations are solved numerically.
This formulation and further discussion can be found in [[#OCPjl|[1]]] and [[#Lenhart|[2]]].

== References ==
[1] Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl 
[2] Lenhart, S., & Workman, J. T. (2007). Optimal Control Applied to Biological Models. CRC Press. 

[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Epidemiological model]]
[[Category:Biological systems]]

Brachistochrone

2026-09-03T11:49:20Z

RobertLampel: /* References */

{{Dimensions
|nd = 1
|nx = 3
|nw = 1
}}

The '''brachistochrone problem''' (from Greek ''brachístos chrónos'', "shortest time") is one of the founding problems of the calculus of variations, and serves as a classical benchmark in optimal control. This description is taken from [[#OCPjl | [1]]].

First posed by Johann Bernoulli in 1696 as a public challenge, the problem asks for the shape of a frictionless curve connecting two fixed points in a vertical plane such that a particle released from rest at the upper point reaches the lower point in minimum time under gravity.
As an optimal control problem, the system state consists of the horizontal position, vertical position, and speed of the particle, while the control is the instantaneous angle of the velocity vector with respect to the vertical.
The terminal time is free and is the quantity to be minimised.
The analytical solution is a cycloid, obtainable through Pontryagin's minimum principle or the classical Euler–Lagrange equations.

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{u,\, t_\mathrm{f}} && t_\mathrm{f} \\[6pt]
\text{subject to} \\
\quad \dot{x}_1(t) &=& x_3(t)\,\sin(u(t)), \\
\quad \dot{x}_2(t) &=& x_3(t)\,\cos(u(t)), \\
\quad \dot{x}_3(t) &=& g\,\cos(u(t)), \\
\quad x(0) &=& (0,\; 0,\; 0)^T, \\
\quad x_1(t_\mathrm{f}) &=& x_{1,\mathrm{f}}, \\
\quad x_2(t_\mathrm{f}) &=& x_{2,\mathrm{f}}, \\
\quad u(t) &\in& \left[-\tfrac{\pi}{2},\, \tfrac{\pi}{2}\right] \quad \forall\, t \in [0, t_\mathrm{f}],
\end{array}
</math>


where <math>x_1(t)</math> is the horizontal position, <math>x_2(t)</math> is the downward vertical position, <math>x_3(t)</math> is the speed of the particle, and <math>u(t)</math> is the angle of the velocity vector measured from the downward vertical.
The speed dynamics follow from conservation of energy under gravity: <math>\dot{x}_3 = g\cos(u)</math>.
Note that <math>x_3(0) = 0</math>, so the particle starts from rest.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>g</math> || align=right | 9.81 || Gravitational acceleration (m s−2)
|-
| align=center | <math>x_{1,\mathrm{f}}</math> || align=right | 1 || Target horizontal position (m)
|-
| align=center | <math>x_{2,\mathrm{f}}</math> || align=right | 1 || Target downward vertical position (m)
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image: Brachistochrone.png | Optimal trajectory and control for the brachistochrone problem (cycloid solution).
</gallery>

== Miscellaneous and Further Reading ==
The analytical solution is a cycloid: the locus traced by a point on the rim of a circle rolling along a flat surface.
It can be parametrised as
<math>x_1(\theta) = r(\theta - \sin\theta)</math>, <math>x_2(\theta) = r(1 - \cos\theta)</math>,
where the radius <math>r</math> is determined by the boundary conditions.
The brachistochrone problem is a standard example in textbooks on optimal control and the calculus of variations, and is used as a test case for direct transcription methods due to its free terminal time and the mild singularity at <math>t = 0</math> where the speed vanishes.
This formulation and further discussion can be found in [[#OCPjl|[1]]] and [[#Bryson|[2]]].

== References ==
[1] Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl 
[2] Bryson, A. E., & Ho, Y.-C. (1975). Applied Optimal Control: Optimization, Estimation, and Control. Taylor & Francis. 

[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Free terminal time]]
[[Category:Classical mechanics]]

Battery Storage

2026-09-03T11:48:57Z

RobertLampel: /* References */

{{Dimensions
|nd = 1
|nx = 1
|nw = 1
}}

The '''battery storage problem''' is a classical benchmark in optimal control for energy systems. This description is taken from [[#OCPjl | [1]]].

It models the optimal operation of a grid-connected battery over a finite time horizon.
The single state is the energy stored in the battery (state of charge), and the single control is the net power flow.
Positive control corresponds to charging (energy flows into the battery from the grid), while negative control corresponds to discharging (energy flows from the battery to the grid).
The objective is to minimise the total electricity cost, given by the integral of the product of a time-varying price signal and the net power drawn from the grid, subject to capacity constraints on the stored energy and power limits on the control.
The optimal strategy is to charge when prices are low and discharge when prices are high, yielding an energy-arbitrage policy that exhibits bang–bang structure when there is no regularisation term.

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{u} && \int_0^{t_\mathrm{f}} p(t)\, u(t)\, \mathrm{d}t \\[6pt]
\text{subject to} \\
\quad \dot{x}(t) &=& u(t), \\
\quad x(0) &=& x_0, \\
\quad x(t_\mathrm{f}) &=& x_0, \\
\quad x(t) &\in& [0,\, E_{\max}] \quad \forall\, t \in [0, t_\mathrm{f}], \\
\quad u(t) &\in& [-P_{\max},\, P_{\max}] \quad \forall\, t \in [0, t_\mathrm{f}],
\end{array}
</math>


where <math>p(t) = p_0 + \Delta p\,\sin\!\bigl(2\pi t / t_\mathrm{f}\bigr)</math> is the electricity price at time <math>t</math>.
The cyclic boundary condition <math>x(t_\mathrm{f}) = x_0</math> ensures that the battery returns to its initial state of charge at the end of the horizon.
Charging (<math>u > 0</math>) incurs a positive cost <math>p(t)\,u(t) > 0</math>, while discharging (<math>u < 0</math>) generates revenue.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>E_{\max}</math> || align=right | 1 || Battery capacity (normalised)
|-
| align=center | <math>P_{\max}</math> || align=right | 1 || Maximum charge/discharge power
|-
| align=center | <math>p_0</math> || align=right | 1 || Mean electricity price
|-
| align=center | <math>\Delta p</math> || align=right | 0.5 || Price amplitude
|-
| align=center | <math>x_0</math> || align=right | 0.5 || Initial (and final) state of charge
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 1 || Length of the scheduling horizon
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Battery_Storage.png | State of charge and optimal power schedule for a reference solution.
</gallery>

== Miscellaneous and Further Reading ==
The battery storage optimal control problem arises in the operation of grid-scale energy storage, demand-response programs, and behind-the-meter storage systems.
When efficiency losses are incorporated (separate charging efficiency <math>\eta_c \in (0,1]</math> and discharging efficiency <math>\eta_d \in (0,1]</math>), the state equation becomes
<math>\dot{x} = \eta_c\,u^+ - u^-/\eta_d</math>,
where <math>u^+ = \max(u,0)</math> and <math>u^- = \max(-u,0)</math>, and the problem acquires a mixed-integer character that can be handled by complementarity constraints.
A detailed formulation and further discussion can be found in [[#OCPjl|[1]]].

== References ==
[1] Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl 

[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Energy systems]]

User:RobertLampel

2026-09-03T11:48:11Z

RobertLampel:

Tumor Anti Anginogenesis

2026-09-03T11:47:54Z

RobertLampel: RobertLampel moved page Tumor Anti Anginogenesis to Tumor Anti Angiogenesis: Misspelled title

#REDIRECT [[Tumor Anti Angiogenesis]]

Tumor Anti Angiogenesis

2026-09-03T11:47:54Z

RobertLampel: RobertLampel moved page Tumor Anti Anginogenesis to Tumor Anti Angiogenesis: Misspelled title

{{Dimensions
|nd = 1
|nx = 3
|nw = 1
}}

The '''tumor anti-angiogenesis problem''' is a classical benchmark in biomedical optimal control. This description is taken from [[#OCPjl | [1]]].

It is based on the mathematical model of Hahnfeldt et al. [[#Hahnfeldt|[2]]] for tumor growth under angiogenic signalling, as studied from an optimal control perspective by Ledzewicz and Schättler [[#Ledzewicz|[3]]].
Angiogenesis is the process by which tumors recruit a network of blood vessels to sustain their growth; anti-angiogenic therapy targets this process by administering inhibitor agents.
The two biological states are the tumor volume and the carrying capacity of the vascular endothelium (which sets the maximum supportable tumor size).
A third state tracks the cumulative inhibitor dose, enforcing a total-dose constraint.
The control is the instantaneous inhibitor dose rate.
The objective is to minimise the tumor volume at the final time.
The problem is known to exhibit a rich optimal control structure, including bang–bang arcs, singular arcs, and boundary arcs corresponding to the dose constraint.

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{u} && p(t_\mathrm{f}) \\[6pt]
\text{subject to} \\
\quad \dot{p}(t) &=& -\xi\, p(t)\,\ln\!\left(\dfrac{p(t)}{q(t)}\right), \\[6pt]
\quad \dot{q}(t) &=& b\, p(t) - \mu\, q(t) - \left(d\, p(t)^{2/3} + e\, u(t)\right) q(t), \\[6pt]
\quad \dot{y}(t) &=& u(t), \\
\quad (p(0),\, q(0),\, y(0)) &=& (p_0,\, q_0,\, 0), \\
\quad y(t_\mathrm{f}) &\leq& A, \\
\quad u(t) &\in& [0,\, a] \quad \forall\, t \in [0, t_\mathrm{f}],
\end{array}
</math>


where <math>p(t)</math> is the tumor volume (mm3), <math>q(t)</math> is the endothelial carrying capacity (mm3), and <math>y(t)</math> is the cumulative inhibitor dose (mg/kg) administered up to time <math>t</math>.
The tumor dynamics follow the Gompertz law <math>\dot{p} = -\xi\,p\ln(p/q)</math>, which models logistic-like growth bounded by the carrying capacity <math>q</math>.
The carrying capacity decreases due to natural endothelial cell loss (<math>\mu\,q</math>), inhibitor-induced endothelial cell loss (<math>e\,u\,q</math>), and stimulated shedding proportional to tumor surface area (<math>d\,p^{2/3}\,q</math>), while increasing due to endothelial recruitment by the tumor (<math>b\,p</math>).
The state <math>y(t)</math> enforces the total-dose constraint <math>y(t_\mathrm{f}) \leq A</math> as an isoperimetric inequality.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>\xi</math> || align=right | 0.084 || Tumor growth rate (day−1)
|-
| align=center | <math>b</math> || align=right | 5.85 || Endothelial birth rate coefficient (day−1)
|-
| align=center | <math>\mu</math> || align=right | 0.02 || Endothelial natural loss rate (day−1)
|-
| align=center | <math>d</math> || align=right | 0.00873 || Inhibition coefficient for endothelial shedding (mm−2 day−1)
|-
| align=center | <math>e</math> || align=right | 0.66 || Inhibitor effectiveness on endothelium (kg mg−1 day−1)
|-
| align=center | <math>a</math> || align=right | 75 || Maximum inhibitor dose rate (mg kg−1 day−1)
|-
| align=center | <math>A</math> || align=right | 15 || Maximum cumulative inhibitor dose (mg kg−1)
|-
| align=center | <math>p_0</math> || align=right | 12000 || Initial tumor volume (mm3)
|-
| align=center | <math>q_0</math> || align=right | 15000 || Initial endothelial carrying capacity (mm3)
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 1.2 || Horizon of the control problem (days)
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image: Tumor_Anti_Angiogenesis.png | Tumor volume, carrying capacity, cumulative dose, and optimal inhibitor schedule for a reference solution.
</gallery>

== Miscellaneous and Further Reading ==
The equilibrium <math>p^* = q^*</math> corresponds to a tumor at maximum sustainable size.
Without treatment, <math>p(t)</math> converges to this steady state.
Ledzewicz and Schättler [[#Ledzewicz|[3]]] characterise the optimal control using Pontryagin's minimum principle and show that the optimal policy consists of a sequence of full-dose (<math>u = a</math>), no-dose (<math>u = 0</math>), and singular arcs, depending on the initial conditions and remaining dose budget.
This formulation and detailed analysis can be found in [[#OCPjl|[1]]], [[#Hahnfeldt|[2]]], and [[#Ledzewicz|[3]]].

== References ==
[1] Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl 
[2] Hahnfeldt, P., Panigrahy, D., Folkman, J., & Hlatky, L. (1999). Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Research, 59(19), 4770–4775. 
[3] Ledzewicz, U., & Schättler, H. (2007). Antiangiogenic therapy in cancer treatment as an optimal control problem. SIAM Journal on Control and Optimization, 46(3), 1052–1079. 

[[Category:ODE model]]
[[Category:Biomedical model]]
[[Category:Singular arc]]

File:Tumor Anti Angiogenesis.png

2026-09-03T11:45:34Z

RobertLampel:

Tumor Anti Angiogenesis

2026-09-03T11:45:23Z

RobertLampel: Created page with "{{Dimensions |nd = 1 |nx = 3 |nw = 1 }} The '''tumor anti-angiogenesis problem''' is a classical benchmark in biomedical optimal control. This description is taken from [1]. It is based on the mathematical model of Hahnfeldt et al. [2] for tumor growth under angiogenic signalling, as studied from an optimal control perspective by Ledzewicz and Schättler [3]. Angiogenesis is the process by which tumors rec..."

{{Dimensions
|nd = 1
|nx = 3
|nw = 1
}}

The '''tumor anti-angiogenesis problem''' is a classical benchmark in biomedical optimal control. This description is taken from [[#OCPjl | [1]]].

It is based on the mathematical model of Hahnfeldt et al. [[#Hahnfeldt|[2]]] for tumor growth under angiogenic signalling, as studied from an optimal control perspective by Ledzewicz and Schättler [[#Ledzewicz|[3]]].
Angiogenesis is the process by which tumors recruit a network of blood vessels to sustain their growth; anti-angiogenic therapy targets this process by administering inhibitor agents.
The two biological states are the tumor volume and the carrying capacity of the vascular endothelium (which sets the maximum supportable tumor size).
A third state tracks the cumulative inhibitor dose, enforcing a total-dose constraint.
The control is the instantaneous inhibitor dose rate.
The objective is to minimise the tumor volume at the final time.
The problem is known to exhibit a rich optimal control structure, including bang–bang arcs, singular arcs, and boundary arcs corresponding to the dose constraint.

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{u} && p(t_\mathrm{f}) \\[6pt]
\text{subject to} \\
\quad \dot{p}(t) &=& -\xi\, p(t)\,\ln\!\left(\dfrac{p(t)}{q(t)}\right), \\[6pt]
\quad \dot{q}(t) &=& b\, p(t) - \mu\, q(t) - \left(d\, p(t)^{2/3} + e\, u(t)\right) q(t), \\[6pt]
\quad \dot{y}(t) &=& u(t), \\
\quad (p(0),\, q(0),\, y(0)) &=& (p_0,\, q_0,\, 0), \\
\quad y(t_\mathrm{f}) &\leq& A, \\
\quad u(t) &\in& [0,\, a] \quad \forall\, t \in [0, t_\mathrm{f}],
\end{array}
</math>


where <math>p(t)</math> is the tumor volume (mm3), <math>q(t)</math> is the endothelial carrying capacity (mm3), and <math>y(t)</math> is the cumulative inhibitor dose (mg/kg) administered up to time <math>t</math>.
The tumor dynamics follow the Gompertz law <math>\dot{p} = -\xi\,p\ln(p/q)</math>, which models logistic-like growth bounded by the carrying capacity <math>q</math>.
The carrying capacity decreases due to natural endothelial cell loss (<math>\mu\,q</math>), inhibitor-induced endothelial cell loss (<math>e\,u\,q</math>), and stimulated shedding proportional to tumor surface area (<math>d\,p^{2/3}\,q</math>), while increasing due to endothelial recruitment by the tumor (<math>b\,p</math>).
The state <math>y(t)</math> enforces the total-dose constraint <math>y(t_\mathrm{f}) \leq A</math> as an isoperimetric inequality.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>\xi</math> || align=right | 0.084 || Tumor growth rate (day−1)
|-
| align=center | <math>b</math> || align=right | 5.85 || Endothelial birth rate coefficient (day−1)
|-
| align=center | <math>\mu</math> || align=right | 0.02 || Endothelial natural loss rate (day−1)
|-
| align=center | <math>d</math> || align=right | 0.00873 || Inhibition coefficient for endothelial shedding (mm−2 day−1)
|-
| align=center | <math>e</math> || align=right | 0.66 || Inhibitor effectiveness on endothelium (kg mg−1 day−1)
|-
| align=center | <math>a</math> || align=right | 75 || Maximum inhibitor dose rate (mg kg−1 day−1)
|-
| align=center | <math>A</math> || align=right | 15 || Maximum cumulative inhibitor dose (mg kg−1)
|-
| align=center | <math>p_0</math> || align=right | 12000 || Initial tumor volume (mm3)
|-
| align=center | <math>q_0</math> || align=right | 15000 || Initial endothelial carrying capacity (mm3)
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 1.2 || Horizon of the control problem (days)
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image: Tumor_Anti_Angiogenesis.png | Tumor volume, carrying capacity, cumulative dose, and optimal inhibitor schedule for a reference solution.
</gallery>

== Miscellaneous and Further Reading ==
The equilibrium <math>p^* = q^*</math> corresponds to a tumor at maximum sustainable size.
Without treatment, <math>p(t)</math> converges to this steady state.
Ledzewicz and Schättler [[#Ledzewicz|[3]]] characterise the optimal control using Pontryagin's minimum principle and show that the optimal policy consists of a sequence of full-dose (<math>u = a</math>), no-dose (<math>u = 0</math>), and singular arcs, depending on the initial conditions and remaining dose budget.
This formulation and detailed analysis can be found in [[#OCPjl|[1]]], [[#Hahnfeldt|[2]]], and [[#Ledzewicz|[3]]].

== References ==
[1] Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl 
[2] Hahnfeldt, P., Panigrahy, D., Folkman, J., & Hlatky, L. (1999). Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Research, 59(19), 4770–4775. 
[3] Ledzewicz, U., & Schättler, H. (2007). Antiangiogenic therapy in cancer treatment as an optimal control problem. SIAM Journal on Control and Optimization, 46(3), 1052–1079. 

[[Category:ODE model]]
[[Category:Biomedical model]]
[[Category:Singular arc]]

File:Ramsey Growth.png

2026-09-03T11:44:20Z

RobertLampel:

Ramsey Growth

2026-09-03T11:44:06Z

RobertLampel: Created page with "{{Dimensions |nd = 1 |nx = 1 |nw = 1 }} The '''Ramsey growth problem''' is a foundational benchmark in optimal control and macroeconomic theory. This description is taken from [1]. First formulated by Frank Ramsey in 1928 [2] and later extended by Cass [3] and Koopmans [4], the problem determines the socially optimal path of capital accumulation and consumption for a representative household economy...."

File:SEIR Epidemic.png

2026-09-03T11:43:20Z

RobertLampel:

SEIR Epidemic

2026-09-03T11:43:09Z

RobertLampel: Created page with "{{Dimensions |nd = 1 |nx = 4 |nw = 1 }} The '''SEIR epidemic optimal control problem''' is a classical benchmark in mathematical epidemiology and optimal control of biological systems. This description is taken from [1]. It models the spread of an infectious disease through a homogeneous population divided into four compartments: susceptible (<math>S</math>), exposed (<math>E</math>), infectious (<math>I</math>), and recovered (<math>R..."

{{Dimensions
|nd = 1
|nx = 4
|nw = 1
}}

The '''SEIR epidemic optimal control problem''' is a classical benchmark in mathematical epidemiology and optimal control of biological systems. This description is taken from [[#OCPjl | [1]]].

It models the spread of an infectious disease through a homogeneous population divided into four compartments: susceptible (<math>S</math>), exposed (<math>E</math>), infectious (<math>I</math>), and recovered (<math>R</math>).
The control represents a time-varying vaccination (or isolation) rate applied to the susceptible compartment.
The objective is to minimise a weighted combination of the cumulative infectious burden and the cost of the intervention over a finite horizon, yielding a trade-off between disease suppression and resource expenditure.
The problem is nonlinear and non-convex, and typically admits a singular arc structure in the optimal control.

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{u} && \int_0^{t_\mathrm{f}} \Bigl[ A\, I(t) + \tfrac{1-A}{2}\, u(t)^2 \Bigr]\, \mathrm{d}t \\[6pt]
\text{subject to} \\
\quad \dot{S}(t) &=& -\beta\, S(t)\, I(t) - u(t)\, S(t), \\
\quad \dot{E}(t) &=& \beta\, S(t)\, I(t) - \sigma\, E(t), \\
\quad \dot{I}(t) &=& \sigma\, E(t) - \gamma\, I(t), \\
\quad \dot{R}(t) &=& \gamma\, I(t) + u(t)\, S(t), \\
\quad x(0) &=& (S_0,\; E_0,\; I_0,\; R_0)^T, \\
\quad u(t) &\in& [0,\, u_{\max}] \quad \forall\, t \in [0, t_\mathrm{f}],
\end{array}
</math>


where <math>x = (S, E, I, R)^T</math>.
All compartments are fractions of the total population, so <math>S(t) + E(t) + I(t) + R(t) = 1</math> for all <math>t</math>.
The term <math>-u(t)\,S(t)</math> removes susceptible individuals through vaccination (or effective isolation), moving them directly to the recovered compartment at rate <math>u(t)\,S(t)</math>.
The quadratic penalty <math>\tfrac{1-A}{2}\,u^2</math> models the increasing marginal cost of intervention.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>\beta</math> || align=right | 0.5 || Transmission rate (day−1)
|-
| align=center | <math>\sigma</math> || align=right | 0.2 || Progression rate from exposed to infectious (day−1)
|-
| align=center | <math>\gamma</math> || align=right | 0.1 || Recovery rate (day−1)
|-
| align=center | <math>A</math> || align=right | 0.5 || Weight on infection burden vs. control cost
|-
| align=center | <math>u_{\max}</math> || align=right | 0.5 || Maximum vaccination/isolation rate (day−1)
|-
| align=center | <math>S_0</math> || align=right | 0.99 || Initial susceptible fraction
|-
| align=center | <math>E_0</math> || align=right | 0.01 || Initial exposed fraction
|-
| align=center | <math>I_0</math> || align=right | 0 || Initial infectious fraction
|-
| align=center | <math>R_0</math> || align=right | 0 || Initial recovered fraction
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 100 || Horizon of the control problem (days)
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image: SEIR_Epidemic.png | Compartment trajectories and optimal vaccination rate for a reference solution.
</gallery>

== Miscellaneous and Further Reading ==
The basic reproduction number for the uncontrolled SEIR model is <math>\mathcal{R}_0 = \beta/\gamma</math>.
With the parameters above, <math>\mathcal{R}_0 = 5</math>, corresponding to a highly transmissible pathogen.
The optimal control problem was studied in detail by Lenhart and Workman [[#Lenhart|[2]]], where necessary conditions via Pontryagin's minimum principle are derived and adjoint equations are solved numerically.
This formulation and further discussion can be found in [[#OCPjl|[1]]] and [[#Lenhart|[2]]].

== References ==
[1] Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl 
[2] Lenhart, S., & Workman, J. T. (2007). Optimal Control Applied to Biological Models. CRC Press. 

[[Category:ODE model]]
[[Category:Epidemiological model]]
[[Category:Biological systems]]

Brachistochrone

2026-09-03T11:42:35Z

RobertLampel: /* Reference Solutions */

{{Dimensions
|nd = 1
|nx = 3
|nw = 1
}}

The '''brachistochrone problem''' (from Greek ''brachístos chrónos'', "shortest time") is one of the founding problems of the calculus of variations, and serves as a classical benchmark in optimal control. This description is taken from [[#OCPjl | [1]]].

First posed by Johann Bernoulli in 1696 as a public challenge, the problem asks for the shape of a frictionless curve connecting two fixed points in a vertical plane such that a particle released from rest at the upper point reaches the lower point in minimum time under gravity.
As an optimal control problem, the system state consists of the horizontal position, vertical position, and speed of the particle, while the control is the instantaneous angle of the velocity vector with respect to the vertical.
The terminal time is free and is the quantity to be minimised.
The analytical solution is a cycloid, obtainable through Pontryagin's minimum principle or the classical Euler–Lagrange equations.

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{u,\, t_\mathrm{f}} && t_\mathrm{f} \\[6pt]
\text{subject to} \\
\quad \dot{x}_1(t) &=& x_3(t)\,\sin(u(t)), \\
\quad \dot{x}_2(t) &=& x_3(t)\,\cos(u(t)), \\
\quad \dot{x}_3(t) &=& g\,\cos(u(t)), \\
\quad x(0) &=& (0,\; 0,\; 0)^T, \\
\quad x_1(t_\mathrm{f}) &=& x_{1,\mathrm{f}}, \\
\quad x_2(t_\mathrm{f}) &=& x_{2,\mathrm{f}}, \\
\quad u(t) &\in& \left[-\tfrac{\pi}{2},\, \tfrac{\pi}{2}\right] \quad \forall\, t \in [0, t_\mathrm{f}],
\end{array}
</math>


where <math>x_1(t)</math> is the horizontal position, <math>x_2(t)</math> is the downward vertical position, <math>x_3(t)</math> is the speed of the particle, and <math>u(t)</math> is the angle of the velocity vector measured from the downward vertical.
The speed dynamics follow from conservation of energy under gravity: <math>\dot{x}_3 = g\cos(u)</math>.
Note that <math>x_3(0) = 0</math>, so the particle starts from rest.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>g</math> || align=right | 9.81 || Gravitational acceleration (m s−2)
|-
| align=center | <math>x_{1,\mathrm{f}}</math> || align=right | 1 || Target horizontal position (m)
|-
| align=center | <math>x_{2,\mathrm{f}}</math> || align=right | 1 || Target downward vertical position (m)
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image: Brachistochrone.png | Optimal trajectory and control for the brachistochrone problem (cycloid solution).
</gallery>

== Miscellaneous and Further Reading ==
The analytical solution is a cycloid: the locus traced by a point on the rim of a circle rolling along a flat surface.
It can be parametrised as
<math>x_1(\theta) = r(\theta - \sin\theta)</math>, <math>x_2(\theta) = r(1 - \cos\theta)</math>,
where the radius <math>r</math> is determined by the boundary conditions.
The brachistochrone problem is a standard example in textbooks on optimal control and the calculus of variations, and is used as a test case for direct transcription methods due to its free terminal time and the mild singularity at <math>t = 0</math> where the speed vanishes.
This formulation and further discussion can be found in [[#OCPjl|[1]]] and [[#Bryson|[2]]].

== References ==
[1] Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl 
[2] Bryson, A. E., & Ho, Y.-C. (1975). Applied Optimal Control: Optimization, Estimation, and Control. Taylor & Francis. 

[[Category:ODE model]]
[[Category:Free terminal time]]
[[Category:Classical mechanics]]

File:Brachistochrone.png

2026-09-03T11:42:06Z

RobertLampel:

Brachistochrone

2026-09-03T11:41:56Z

RobertLampel: Created page with "{{Dimensions |nd = 1 |nx = 3 |nw = 1 }} The '''brachistochrone problem''' (from Greek ''brachístos chrónos'', "shortest time") is one of the founding problems of the calculus of variations, and serves as a classical benchmark in optimal control. This description is taken from [1]. First posed by Johann Bernoulli in 1696 as a public challenge, the problem asks for the shape of a frictionless curve connecting two fixed points in a vert..."

{{Dimensions
|nd = 1
|nx = 3
|nw = 1
}}

The '''brachistochrone problem''' (from Greek ''brachístos chrónos'', "shortest time") is one of the founding problems of the calculus of variations, and serves as a classical benchmark in optimal control. This description is taken from [[#OCPjl | [1]]].

First posed by Johann Bernoulli in 1696 as a public challenge, the problem asks for the shape of a frictionless curve connecting two fixed points in a vertical plane such that a particle released from rest at the upper point reaches the lower point in minimum time under gravity.
As an optimal control problem, the system state consists of the horizontal position, vertical position, and speed of the particle, while the control is the instantaneous angle of the velocity vector with respect to the vertical.
The terminal time is free and is the quantity to be minimised.
The analytical solution is a cycloid, obtainable through Pontryagin's minimum principle or the classical Euler–Lagrange equations.

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{u,\, t_\mathrm{f}} && t_\mathrm{f} \\[6pt]
\text{subject to} \\
\quad \dot{x}_1(t) &=& x_3(t)\,\sin(u(t)), \\
\quad \dot{x}_2(t) &=& x_3(t)\,\cos(u(t)), \\
\quad \dot{x}_3(t) &=& g\,\cos(u(t)), \\
\quad x(0) &=& (0,\; 0,\; 0)^T, \\
\quad x_1(t_\mathrm{f}) &=& x_{1,\mathrm{f}}, \\
\quad x_2(t_\mathrm{f}) &=& x_{2,\mathrm{f}}, \\
\quad u(t) &\in& \left[-\tfrac{\pi}{2},\, \tfrac{\pi}{2}\right] \quad \forall\, t \in [0, t_\mathrm{f}],
\end{array}
</math>


where <math>x_1(t)</math> is the horizontal position, <math>x_2(t)</math> is the downward vertical position, <math>x_3(t)</math> is the speed of the particle, and <math>u(t)</math> is the angle of the velocity vector measured from the downward vertical.
The speed dynamics follow from conservation of energy under gravity: <math>\dot{x}_3 = g\cos(u)</math>.
Note that <math>x_3(0) = 0</math>, so the particle starts from rest.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>g</math> || align=right | 9.81 || Gravitational acceleration (m s−2)
|-
| align=center | <math>x_{1,\mathrm{f}}</math> || align=right | 1 || Target horizontal position (m)
|-
| align=center | <math>x_{2,\mathrm{f}}</math> || align=right | 1 || Target downward vertical position (m)
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Brachistochrone.png | Optimal trajectory and control for the brachistochrone problem (cycloid solution).
</gallery>

== Miscellaneous and Further Reading ==
The analytical solution is a cycloid: the locus traced by a point on the rim of a circle rolling along a flat surface.
It can be parametrised as
<math>x_1(\theta) = r(\theta - \sin\theta)</math>, <math>x_2(\theta) = r(1 - \cos\theta)</math>,
where the radius <math>r</math> is determined by the boundary conditions.
The brachistochrone problem is a standard example in textbooks on optimal control and the calculus of variations, and is used as a test case for direct transcription methods due to its free terminal time and the mild singularity at <math>t = 0</math> where the speed vanishes.
This formulation and further discussion can be found in [[#OCPjl|[1]]] and [[#Bryson|[2]]].

== References ==
[1] Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl 
[2] Bryson, A. E., & Ho, Y.-C. (1975). Applied Optimal Control: Optimization, Estimation, and Control. Taylor & Francis. 

[[Category:ODE model]]
[[Category:Free terminal time]]
[[Category:Classical mechanics]]

Battery Storage

2026-09-03T11:41:13Z

RobertLampel: /* Reference Solutions */

File:Battery Storage.png

2026-09-03T11:40:38Z

RobertLampel:

Battery Storage

2026-09-03T11:40:17Z

RobertLampel: /* Reference Solutions */

{{Dimensions
|nd = 1
|nx = 1
|nw = 1
}}

The '''battery storage problem''' is a classical benchmark in optimal control for energy systems. This description is taken from [[#OCPjl | [1]]].

It models the optimal operation of a grid-connected battery over a finite time horizon.
The single state is the energy stored in the battery (state of charge), and the single control is the net power flow.
Positive control corresponds to charging (energy flows into the battery from the grid), while negative control corresponds to discharging (energy flows from the battery to the grid).
The objective is to minimise the total electricity cost, given by the integral of the product of a time-varying price signal and the net power drawn from the grid, subject to capacity constraints on the stored energy and power limits on the control.
The optimal strategy is to charge when prices are low and discharge when prices are high, yielding an energy-arbitrage policy that exhibits bang–bang structure when there is no regularisation term.

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{u} && \int_0^{t_\mathrm{f}} p(t)\, u(t)\, \mathrm{d}t \\[6pt]
\text{subject to} \\
\quad \dot{x}(t) &=& u(t), \\
\quad x(0) &=& x_0, \\
\quad x(t_\mathrm{f}) &=& x_0, \\
\quad x(t) &\in& [0,\, E_{\max}] \quad \forall\, t \in [0, t_\mathrm{f}], \\
\quad u(t) &\in& [-P_{\max},\, P_{\max}] \quad \forall\, t \in [0, t_\mathrm{f}],
\end{array}
</math>


where <math>p(t) = p_0 + \Delta p\,\sin\!\bigl(2\pi t / t_\mathrm{f}\bigr)</math> is the electricity price at time <math>t</math>.
The cyclic boundary condition <math>x(t_\mathrm{f}) = x_0</math> ensures that the battery returns to its initial state of charge at the end of the horizon.
Charging (<math>u > 0</math>) incurs a positive cost <math>p(t)\,u(t) > 0</math>, while discharging (<math>u < 0</math>) generates revenue.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>E_{\max}</math> || align=right | 1 || Battery capacity (normalised)
|-
| align=center | <math>P_{\max}</math> || align=right | 1 || Maximum charge/discharge power
|-
| align=center | <math>p_0</math> || align=right | 1 || Mean electricity price
|-
| align=center | <math>\Delta p</math> || align=right | 0.5 || Price amplitude
|-
| align=center | <math>x_0</math> || align=right | 0.5 || Initial (and final) state of charge
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 1 || Length of the scheduling horizon
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Battery_Storage.png| State of charge and optimal power schedule for a reference solution.
</gallery>

== Miscellaneous and Further Reading ==
The battery storage optimal control problem arises in the operation of grid-scale energy storage, demand-response programs, and behind-the-meter storage systems.
When efficiency losses are incorporated (separate charging efficiency <math>\eta_c \in (0,1]</math> and discharging efficiency <math>\eta_d \in (0,1]</math>), the state equation becomes
<math>\dot{x} = \eta_c\,u^+ - u^-/\eta_d</math>,
where <math>u^+ = \max(u,0)</math> and <math>u^- = \max(-u,0)</math>, and the problem acquires a mixed-integer character that can be handled by complementarity constraints.
A detailed formulation and further discussion can be found in [[#OCPjl|[1]]].

== References ==
[1] Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl 

[[Category:ODE model]]
[[Category:Energy systems]]

Battery Storage

2026-09-03T11:40:02Z

RobertLampel: Created page with "{{Dimensions |nd = 1 |nx = 1 |nw = 1 }} The '''battery storage problem''' is a classical benchmark in optimal control for energy systems. This description is taken from [1]. It models the optimal operation of a grid-connected battery over a finite time horizon. The single state is the energy stored in the battery (state of charge), and the single control is the net power flow. Positive control corresponds to charging (energy flows into..."

{{Dimensions
|nd = 1
|nx = 1
|nw = 1
}}

The '''battery storage problem''' is a classical benchmark in optimal control for energy systems. This description is taken from [[#OCPjl | [1]]].

It models the optimal operation of a grid-connected battery over a finite time horizon.
The single state is the energy stored in the battery (state of charge), and the single control is the net power flow.
Positive control corresponds to charging (energy flows into the battery from the grid), while negative control corresponds to discharging (energy flows from the battery to the grid).
The objective is to minimise the total electricity cost, given by the integral of the product of a time-varying price signal and the net power drawn from the grid, subject to capacity constraints on the stored energy and power limits on the control.
The optimal strategy is to charge when prices are low and discharge when prices are high, yielding an energy-arbitrage policy that exhibits bang–bang structure when there is no regularisation term.

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{u} && \int_0^{t_\mathrm{f}} p(t)\, u(t)\, \mathrm{d}t \\[6pt]
\text{subject to} \\
\quad \dot{x}(t) &=& u(t), \\
\quad x(0) &=& x_0, \\
\quad x(t_\mathrm{f}) &=& x_0, \\
\quad x(t) &\in& [0,\, E_{\max}] \quad \forall\, t \in [0, t_\mathrm{f}], \\
\quad u(t) &\in& [-P_{\max},\, P_{\max}] \quad \forall\, t \in [0, t_\mathrm{f}],
\end{array}
</math>


where <math>p(t) = p_0 + \Delta p\,\sin\!\bigl(2\pi t / t_\mathrm{f}\bigr)</math> is the electricity price at time <math>t</math>.
The cyclic boundary condition <math>x(t_\mathrm{f}) = x_0</math> ensures that the battery returns to its initial state of charge at the end of the horizon.
Charging (<math>u > 0</math>) incurs a positive cost <math>p(t)\,u(t) > 0</math>, while discharging (<math>u < 0</math>) generates revenue.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>E_{\max}</math> || align=right | 1 || Battery capacity (normalised)
|-
| align=center | <math>P_{\max}</math> || align=right | 1 || Maximum charge/discharge power
|-
| align=center | <math>p_0</math> || align=right | 1 || Mean electricity price
|-
| align=center | <math>\Delta p</math> || align=right | 0.5 || Price amplitude
|-
| align=center | <math>x_0</math> || align=right | 0.5 || Initial (and final) state of charge
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 1 || Length of the scheduling horizon
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:BatteryStorage.png| State of charge and optimal power schedule for a reference solution.
</gallery>

== Miscellaneous and Further Reading ==
The battery storage optimal control problem arises in the operation of grid-scale energy storage, demand-response programs, and behind-the-meter storage systems.
When efficiency losses are incorporated (separate charging efficiency <math>\eta_c \in (0,1]</math> and discharging efficiency <math>\eta_d \in (0,1]</math>), the state equation becomes
<math>\dot{x} = \eta_c\,u^+ - u^-/\eta_d</math>,
where <math>u^+ = \max(u,0)</math> and <math>u^- = \max(-u,0)</math>, and the problem acquires a mixed-integer character that can be handled by complementarity constraints.
A detailed formulation and further discussion can be found in [[#OCPjl|[1]]].

== References ==
[1] Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl 

[[Category:ODE model]]
[[Category:Energy systems]]

User:RobertLampel

2026-09-03T11:39:50Z

RobertLampel:

Compartmental OED

2026-03-26T10:18:55Z

RobertLampel: /* Mathematical formulation */

{{Dimensions
|nd = 1
|nx = 12
|nw = 1
}}

The '''Compartmental OED problem''' looks for an optimal measurement strategy to determine three parameters in a one-dimensional [[:Category:ODE model|ODE model]], where we can directly measure the single state. The following description is taken from [[#compartmental| [1]]].

Here we consider the open one-compartment model with first-order absorption input fitted by Button (unpublished Ph.D. thesis, Texas A&M University, 1979) to the results of an experiment in which six horses each received 15 mg/kg of theophylline as aminophylline by intragastric administration.

The optimal integer control functions shows [[:Category:Chattering|bang bang]] behavior.

== Mathematical formulation ==
For the three-dimensional parameter <math>p = (\theta_1, \theta_2, \theta_3)</math> the original initial value problem is given by
<math>
\dot{y}(t) = f(t, p) = \theta_3 \cdot (-\theta_1 \cdot \exp(-\theta_1 \cdot t) + \theta_2 \cdot \exp(-\theta_2 \cdot t)), \quad y(0) = 0.
</math>

We assume both <math>x_0</math> and <math>t_f</math> to be fixed and are only interested in when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the state directly, i.e. <math>h(x(t)) = x(t)</math>.

Now we formulate the OED problem:

<math>
\begin{array}{lll}
\displaystyle \min_{y,G,F,z,w} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{y}(t) & = & f(t, p) \\
\quad \dot{G}(t) & = & f_p(y(t),p) \\
\quad \dot{F}(t) & = & w(t)(h_y(y(t))G(t))^T(h_y(y(t))G(t)) \\
\quad \dot{z}(t) & = & w(t), \\
\quad y(0) & = & y_0 \\
\quad G(0) & = & 0 \\
\quad F(0) & = & 0, \\
\quad z(0) & = & 0 \\
\quad w(t) & \in & \mathcal{W} \\
\quad z(t_f) & \leq & M
\end{array}
</math>


== Parameters ==
These fixed values are used within the model:

<math>
y_0 = 0; \quad t_f = 40; \quad \mathcal{W} = [0,1]; \quad M = 5; \quad p = (0.05884,4.298,21.80)
</math>


== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Compartmental OED.png| Sensitivities and measurement control for <math>\theta=(0.05884, 4.298, 21.80)</math>.
</gallery>

== References ==
[1] Optimum Experimental Designs for Properties of a Compartmental Model, A. Atkinson, K. Chaloner, A. Herzberg, J. Juritz, https://www.jstor.org/stable/pdf/2532547.pdf 


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]

Jackson OED

2026-03-26T10:17:14Z

RobertLampel: /* Mathematical formulation */

{{Dimensions
|nd = 1
|nx = 13
|nw = 3
}}

The '''Jackson OED problem''' is a variation of the [[:Jackson]] problem. It looks for optimal time intervals to measure the three states in order to minimize the uncertainty of a follow-up parameter estimation problem for the two unknown parameters.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. It also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

== Mathematical formulation ==

We are interested in estimating the parameters <math>k_1</math> and <math>k_2</math> of the initial value problem

<math>
\begin{array}{rcl}
\dot{x_1}(t) &=& -u(t) (k_1 x_1(t) - k_2 x_2(t)), && t \in [0,t_f], \quad x_1(0) = 1, \\
\dot{x_2}(t) &=& u(t) (k_1 x_1(t) - k_2 x_2(t)) - (1-u(t)) k_3 x_2(t), && t \in [0,t_f], \quad x_2(0) = 0, \\
\dot{x_3}(t) &=& (1-u(t)) k_3 x_2(t), && t \in [0,t_f], \quad x_3(0) = 0.
\end{array}
</math>


The initial values and <math>t_f</math> are fixed. We are interested in how to choose the control <math>u</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states <math>x_1</math> and <math>x_2</math> directly, <math>h^1(x(t)) = x_1(t)</math> and <math>h^2(x(t)) = x_2(t)</math>. We use two different sampling functions, <math>w_1(\cdot)</math> and <math>w_2(\cdot)</math> in the same experimental setting. This can be seen either as a two-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (k_1, k_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=1}^{n_o} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t), \\
\quad x(0) & = & x_0 \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{2 \times 2}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 1,2,</math> for each observed function of states.

== Parameters ==

These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>k_1</math> || align=right | 1 || Interaction between <math>x_1</math> and <math>x_2</math>
|-
| align=center | <math>k_2</math> || align=right | 10 || Interaction between <math>x_1</math> and <math>x_2</math>
|-
| align=center | <math>k_3</math> || align=right | 1 || Growth of <math>x_3</math> under complementary control
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 1 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_\mathrm{reg}</math> || align=right | 0.01 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{U}</math> || align=right | [0,1] || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | [0,1] || Bounds of measurement function
|-
| align=center | <math>M_1, M_2</math> || align=right | 0.2 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Jackson_OED.png| States, control, and sampling functions for a local optimum. Both measurement functions overlap.
</gallery>

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]

Jackson OED

2026-03-26T10:16:59Z

RobertLampel: /* Mathematical formulation */

{{Dimensions
|nd = 1
|nx = 13
|nw = 3
}}

The '''Jackson OED problem''' is a variation of the [[:Jackson]] problem. It looks for optimal time intervals to measure the three states in order to minimize the uncertainty of a follow-up parameter estimation problem for the two unknown parameters.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. It also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

== Mathematical formulation ==

We are interested in estimating the parameters <math>k_1</math> and <math>k_2</math> of the initial value problem

<math>
\begin{array}{rcl}
\dot{x_1}(t) &=& -u(t) (k_1 x_1(t) - k_2 x_2(t)), && t \in [0,t_f], \quad x_1(0) = 1, \\
\dot{x_2}(t) &=& u(t) (k_1 x_1(t) - k_2 x_2(t)) - (1-u(t)) k_3 x_2(t), && t \in [0,t_f], \quad x_2(0) = 0, \\
\dot{x_3}(t) &=& (1-u(t)) k_3 x_2(t), && t \in [0,t_f], \quad x_3(0) = 0.
\end{array}
</math>


The initial values and <math>t_f</math> are fixed. We are interested in how to choose the control <math>u</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states <math>x_1</math> and <math>x_2</math> directly, <math>h^1(x(t)) = x_1(t)</math> and <math>h^2(x(t)) = x_2(t)</math>. We use two different sampling functions, <math>w_1(\cdot)</math> and <math>w_2(\cdot)</math> in the same experimental setting. This can be seen either as a two-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (k_1, k_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=1}^{n_o} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t), \\
\quad x(0) & = & x_0 \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{2 \times 2}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 1,2</math> for each observed function of states.

== Parameters ==

These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>k_1</math> || align=right | 1 || Interaction between <math>x_1</math> and <math>x_2</math>
|-
| align=center | <math>k_2</math> || align=right | 10 || Interaction between <math>x_1</math> and <math>x_2</math>
|-
| align=center | <math>k_3</math> || align=right | 1 || Growth of <math>x_3</math> under complementary control
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 1 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_\mathrm{reg}</math> || align=right | 0.01 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{U}</math> || align=right | [0,1] || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | [0,1] || Bounds of measurement function
|-
| align=center | <math>M_1, M_2</math> || align=right | 0.2 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Jackson_OED.png| States, control, and sampling functions for a local optimum. Both measurement functions overlap.
</gallery>

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]

Lotka Shared OED

2026-03-26T10:16:25Z

RobertLampel: /* Mathematical formulation */

{{Dimensions
|nd = 1
|nx = 21
|nw = 4}}

The '''Lotka Shared Experimental Design problem''' looks for an optimal strategy to be performed on a fixed time horizon to control the biomasses multiple species as described in [[:LV Shared Resource]]. The goal here, however, is to minimize the uncertainty of a follow-up parameter estimation problem. When measurements of the three state variables are performed becomes a degree of freedom.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. The ODE from [[LV Shared Resource]] is extended such that it also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

The optimal integer control functions shows [[:Category:Chattering|bang bang]] behavior.

== Mathematical formulation ==

We are interested in estimating the parameters <math>\alpha_0, \alpha_1,</math> and <math>\alpha_2</math> of the Lotka-Volterra type predator-prey initial value problem

<math>
\begin{array}{rcl}
\dot{x}_0(t) & = & x_0(t) - \alpha_0 x_0(t) x_1(t) - x_0(t) x_2(t), && t \in [0,t_f], \quad x_0(0) = 1.5, \\
\dot{x}_1(t) & = & - x_1(t) + \alpha_1 x_0(t) x_1(t) - c_1 x_1(t) u(t), && t \in [0,t_f], \quad x_1(0) = 0.5, \\
\dot{x}_2(t) & = & -x_2(t) + \alpha_2 x_0(t) x_2(t) - c_2 x_2(t) u(t), && t \in [0,t_f], \quad x_2(0) = 1,
\end{array}
</math>

where <math>u(\cdot)</math> is a control that may or may not be fixed. The other parameters, the initial values and <math>t_f = 20</math> are fixed. We are interested in how to choose <math>u</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=0,1,2</math>. We use three different sampling functions, <math>w_i(\cdot), i=0,1,2,</math> in the same experimental setting. This can be seen either as a three-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (\alpha_0, \alpha_1, \alpha_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=0}^{2} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t) \\
\quad x(0) & = & (1.5, 0.5, 1)^T \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{3 \times 3}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 0,1,2,</math> for each observed function of states.

== Parameters ==

These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>\alpha_0</math> || align=right | 1.0 ||
|-
| align=center | <math>\alpha_1</math> || align=right | 1.0 ||
|-
| align=center | <math>\alpha_2</math> || align=right | 1.2 ||
|-
| align=center | <math>c_1</math> || align=right | 0.1 ||
|-
| align=center | <math>c_2</math> || align=right | 0.4 ||
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 20 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_\mathrm{reg}</math> || align=right | 0.1 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{U}</math> || align=right | <math>[0,1]^3</math> || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | <math>[0,1]^3</math> || Bounds of measurement function
|-
| align=center | <math>M_0, M_1, M_2</math> || align=right | 4 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Lotka_Shared_OED.png| States, control, and sampling functions for a local optimum.
</gallery>

== Source Code ==

Model descriptions are available in

* [[:Category:Casadi | CasADi]] at https://github.com/rlampel/PySHeRLOC

== Miscellaneous and Further Reading ==
The Lotka Volterra fishing problem was introduced by Sebastian Sager in a proceedings paper <bib id="Sager2006" /> and revisited in his PhD thesis <bib id="Sager2005" />. These are also the references to look for more details. The experimental design problem has been described in the habilitation thesis of Sager, <bib id="Sager2011d" />.

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]
[[Category:Population dynamics]]

Three Tank OED

2026-03-26T10:15:58Z

RobertLampel: /* Mathematical formulation */

{{Dimensions
|nd = 1
|nx = 21
|nw = 6
}}

The '''Three Tank OED problem''' is a variation of the [[:Three Tank multimode problem]]. It looks for optimal time intervals to measure the two states in order to minimize the uncertainty of a follow-up parameter estimation problem for the two unknown parameters.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. It also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

== Mathematical formulation ==

We are interested in estimating the parameters <math>c_1, c_2,</math> and <math>c_3</math> of the initial value problem

<math>
\begin{array}{rcl}
\dot{x}_1(t) & =& -\sqrt{x_1(t)}+c_1 u_1(t) + c_2 u_2(t) - u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_1(0) = 2, \\
\dot{x}_2(t) & =& \sqrt{x_1(t)}-\sqrt{x_2(t)}, && t \in [0,t_f], \quad x_2(0) = 2, \\
\dot{x}_3(t) & =& \sqrt{x_2(t)}-\sqrt{x_3(t)} + u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_3(0) = 2.
\end{array}
</math>

Additionally, the controls <math>u_1,u_2,</math> and <math>u_3</math> are constrained by:

<math>
\sum_{i=1}^3 u_i(t) = 1 \quad \forall t\in [0,t_f]
</math>

The initial values and <math>t_f = 12</math> are fixed. We are interested in how to choose the controls <math>u_i, \ i=1,2,3,</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=1,2,3</math>. We use three different sampling functions, <math>w_i(\cdot), \ i=1,2,3,</math> in the same experimental setting. This can be seen either as a three-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (p_1, p_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=1}^{3} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t) \\
\quad x(0) & = & (2,2,2)^T \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad \sum_{i=1}^3 u_i(t) & = & 1 \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{3 \times 3}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 1,2,3,</math> for each observed function of states.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>c_1</math> || align=right | 1 || Unknown parameter
|-
| align=center | <math>c_2</math> || align=right | 2 || Unknown parameter
|-
| align=center | <math>c_3</math> || align=right | 0.8 || Unknown parameter
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 12 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_{\mathrm{reg}}</math> || align=right | 0.1 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{U}</math> || align=right | <math>[0,1]^3</math> || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | <math>[0,1]^3</math> || Bounds of measurement function
|-
| align=center | <math>M_1, M_2, M_3</math> || align=right | 2 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Three_Tank_OED.png| States, control, and sampling functions for a local optimum.
</gallery>

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]

File:Three Tank OED.png

2026-03-26T10:14:48Z

RobertLampel: RobertLampel uploaded a new version of File:Three Tank OED.png

File:Three Tank OED.png

2026-03-26T10:13:10Z

RobertLampel:

Three Tank OED

2026-03-26T10:12:54Z

RobertLampel: /* Reference Solutions */

{{Dimensions
|nd = 1
|nx = 21
|nw = 6
}}

The '''Three Tank OED problem''' is a variation of the [[:Three Tank multimode problem]]. It looks for optimal time intervals to measure the two states in order to minimize the uncertainty of a follow-up parameter estimation problem for the two unknown parameters.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. It also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

== Mathematical formulation ==

We are interested in estimating the parameters <math>c_1, c_2,</math> and <math>c_3</math> of the initial value problem

<math>
\begin{array}{rcl}
\dot{x}_1(t) & =& -\sqrt{x_1(t)}+c_1 u_1(t) + c_2 u_2(t) - u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_1(0) = 2, \\
\dot{x}_2(t) & =& \sqrt{x_1(t)}-\sqrt{x_2(t)}, && t \in [0,t_f], \quad x_2(0) = 2, \\
\dot{x}_3(t) & =& \sqrt{x_2(t)}-\sqrt{x_3(t)} + u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_3(0) = 2.
\end{array}
</math>

Additionally, the controls <math>u_1,u_2,</math> and <math>u_3</math> are constrained by:

<math>
\sum_{i=1}^3 u_i(t) = 1 \quad \forall t\in [0,t_f]
</math>

The initial values and <math>t_f = 12</math> are fixed. We are interested in how to choose the controls <math>u_i, \ i=1,2,3,</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=1,2,3</math>. We use three different sampling functions, <math>w^i(\cdot), \ i=1,2,3,</math> in the same experimental setting. This can be seen either as a three-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (p_1, p_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=1}^{3} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t) \\
\quad x(0) & = & (2,2,2)^T \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad \sum_{i=1}^3 u_i(t) & = & 1 \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{3 \times 3}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 1,2,3,</math> for each observed function of states.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>c_1</math> || align=right | 1 || Unknown parameter
|-
| align=center | <math>c_2</math> || align=right | 2 || Unknown parameter
|-
| align=center | <math>c_3</math> || align=right | 0.8 || Unknown parameter
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 12 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_{\mathrm{reg}}</math> || align=right | 0.1 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{U}</math> || align=right | <math>[0,1]^3</math> || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | <math>[0,1]^3</math> || Bounds of measurement function
|-
| align=center | <math>M_1, M_2, M_3</math> || align=right | 2 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Three_Tank_OED.png| States, control, and sampling functions for a local optimum.
</gallery>

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]

Three Tank OED

2026-03-26T10:03:49Z

RobertLampel: /* Mathematical formulation */

{{Dimensions
|nd = 1
|nx = 21
|nw = 6
}}

The '''Three Tank OED problem''' is a variation of the [[:Three Tank multimode problem]]. It looks for optimal time intervals to measure the two states in order to minimize the uncertainty of a follow-up parameter estimation problem for the two unknown parameters.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. It also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

== Mathematical formulation ==

We are interested in estimating the parameters <math>c_1, c_2,</math> and <math>c_3</math> of the initial value problem

<math>
\begin{array}{rcl}
\dot{x}_1(t) & =& -\sqrt{x_1(t)}+c_1 u_1(t) + c_2 u_2(t) - u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_1(0) = 2, \\
\dot{x}_2(t) & =& \sqrt{x_1(t)}-\sqrt{x_2(t)}, && t \in [0,t_f], \quad x_2(0) = 2, \\
\dot{x}_3(t) & =& \sqrt{x_2(t)}-\sqrt{x_3(t)} + u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_3(0) = 2.
\end{array}
</math>

Additionally, the controls <math>u_1,u_2,</math> and <math>u_3</math> are constrained by:

<math>
\sum_{i=1}^3 u_i(t) = 1 \quad \forall t\in [0,t_f]
</math>

The initial values and <math>t_f = 12</math> are fixed. We are interested in how to choose the controls <math>u_i, \ i=1,2,3,</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=1,2,3</math>. We use three different sampling functions, <math>w^i(\cdot), \ i=1,2,3,</math> in the same experimental setting. This can be seen either as a three-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (p_1, p_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=1}^{3} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t) \\
\quad x(0) & = & (2,2,2)^T \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad \sum_{i=1}^3 u_i(t) & = & 1 \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{3 \times 3}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 1,2,3,</math> for each observed function of states.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>c_1</math> || align=right | 1 || Unknown parameter
|-
| align=center | <math>c_2</math> || align=right | 2 || Unknown parameter
|-
| align=center | <math>c_3</math> || align=right | 0.8 || Unknown parameter
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 12 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_{\mathrm{reg}}</math> || align=right | 0.1 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{U}</math> || align=right | <math>[0,1]^3</math> || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | <math>[0,1]^3</math> || Bounds of measurement function
|-
| align=center | <math>M_1, M_2, M_3</math> || align=right | 2 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Van_der_Pol_OED.png| States, control, and sampling functions for a local optimum.
</gallery>

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]

Three Tank OED

2026-03-26T10:02:39Z

RobertLampel: /* Mathematical formulation */

{{Dimensions
|nd = 1
|nx = 21
|nw = 6
}}

The '''Three Tank OED problem''' is a variation of the [[:Three Tank multimode problem]]. It looks for optimal time intervals to measure the two states in order to minimize the uncertainty of a follow-up parameter estimation problem for the two unknown parameters.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. It also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

== Mathematical formulation ==

We are interested in estimating the parameters <math>c_1, c_2,</math> and <math>c_3</math> of the initial value problem

<math>
\begin{array}{rcl}
\dot{x}_1(t) & =& -\sqrt{x_1(t)}+c_1 u_1(t) + c_2 u_2(t) - u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_1(0) = 2, \\
\dot{x}_2(t) & =& \sqrt{x_1(t)}-\sqrt{x_2(t)}, && t \in [0,t_f], \quad x_2(0) = 2, \\
\dot{x}_3(t) & =& \sqrt{x_2(t)}-\sqrt{x_3(t)} + u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_3(0) = 2.
\end{array}
</math>

Additionally, the controls <math>u_1,u_2,</math> and <math>u_3</math> are constrained by:

<math>
\sum_{i=1}^3 u_i(t) = 1 \quad \forall t\in [0,t_f]
</math>

The initial values and <math>t_f = 12</math> are fixed. We are interested in how to choose the controls <math>u_i, \ i=1,2,3,</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=1,2,3</math>. We use three different sampling functions, <math>w^i(\cdot), \ i=1,2,3,</math> in the same experimental setting. This can be seen either as a three-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (p_1, p_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=1}^{3} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t) \\
\quad x(0) & = & (2,2,2)^T \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad \sum_{i=1}^3 u_i(t) & = & 1 \\
\quad x_1(t) & \in & \mathcal{X} \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{3 \times 3}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 1,2,3,</math> for each observed function of states.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>c_1</math> || align=right | 1 || Unknown parameter
|-
| align=center | <math>c_2</math> || align=right | 2 || Unknown parameter
|-
| align=center | <math>c_3</math> || align=right | 0.8 || Unknown parameter
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 12 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_{\mathrm{reg}}</math> || align=right | 0.1 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{U}</math> || align=right | <math>[0,1]^3</math> || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | <math>[0,1]^3</math> || Bounds of measurement function
|-
| align=center | <math>M_1, M_2, M_3</math> || align=right | 2 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Van_der_Pol_OED.png| States, control, and sampling functions for a local optimum.
</gallery>

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]

Three Tank OED

2026-03-26T10:02:18Z

RobertLampel: /* Mathematical formulation */

{{Dimensions
|nd = 1
|nx = 21
|nw = 6
}}

The '''Three Tank OED problem''' is a variation of the [[:Three Tank multimode problem]]. It looks for optimal time intervals to measure the two states in order to minimize the uncertainty of a follow-up parameter estimation problem for the two unknown parameters.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. It also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

== Mathematical formulation ==

We are interested in estimating the parameters <math>c_1, c_2,</math> and <math>c_3</math> of the initial value problem

<math>
\begin{array}{rcl}
\dot{x}_1(t) & =& -\sqrt{x_1(t)}+c_1 u_1(t) + c_2 u_2(t) - u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_1(0) = 2, \\
\dot{x}_2(t) & =& \sqrt{x_1(t)}-\sqrt{x_2(t)}, && t \in [0,t_f], \quad x_2(0) = 2, \\
\dot{x}_3(t) & =& \sqrt{x_2(t)}-\sqrt{x_3(t)} + u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_3(0) = 2.
\end{array}
</math>

Additionally, the controls <math>u_1,u_2,</math> and <math>u_3</math> are constrained by:

<math>
\sum_{i=1}^3 u_i(t) = 1 \quad \forall t\in [0,t_f]
</math>

The initial values and <math>t_f = 12</math> are fixed. We are interested in how to choose the controls <math>u_i, \ i=1,2,3,</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=1,2,3</math>. We use three different sampling functions, <math>w^i(\cdot), \ i=1,2,3,</math> in the same experimental setting. This can be seen either as a three-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (p_1, p_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=1}^{3} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t) \\
\quad x(0) & = & (2,2,2)^T \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad \sum_{i=1}^3 u_i(t) & = & 1
\quad x_1(t) & \in & \mathcal{X} \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{3 \times 3}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 1,2,3,</math> for each observed function of states.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>c_1</math> || align=right | 1 || Unknown parameter
|-
| align=center | <math>c_2</math> || align=right | 2 || Unknown parameter
|-
| align=center | <math>c_3</math> || align=right | 0.8 || Unknown parameter
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 12 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_{\mathrm{reg}}</math> || align=right | 0.1 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{U}</math> || align=right | <math>[0,1]^3</math> || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | <math>[0,1]^3</math> || Bounds of measurement function
|-
| align=center | <math>M_1, M_2, M_3</math> || align=right | 2 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Van_der_Pol_OED.png| States, control, and sampling functions for a local optimum.
</gallery>

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]

Three Tank OED

2026-03-26T09:58:29Z

RobertLampel: /* Mathematical formulation */

{{Dimensions
|nd = 1
|nx = 21
|nw = 6
}}

The '''Three Tank OED problem''' is a variation of the [[:Three Tank multimode problem]]. It looks for optimal time intervals to measure the two states in order to minimize the uncertainty of a follow-up parameter estimation problem for the two unknown parameters.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. It also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

== Mathematical formulation ==

We are interested in estimating the parameters <math>c_1, c_2,</math> and <math>c_3</math> of the initial value problem

<math>
\begin{array}{rcl}
\dot{x}_1(t) & =& -\sqrt{x_1(t)}+c_1 u_1(t) + c_2 u_2(t) - u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_1(0) = 2, \\
\dot{x}_2(t) & =& \sqrt{x_1(t)}-\sqrt{x_2(t)}, && t \in [0,t_f], \quad x_2(0) = 2, \\
\dot{x}_3(t) & =& \sqrt{x_2(t)}-\sqrt{x_3(t)} + u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_3(0) = 2.
\end{array}
</math>

Additionally, the controls <math>u_1,u_2,</math> and <math>u_3</math> are constrained by:

<math>
\sum_{i=1}^3 u_i(t) = 1 \quad \forall t\in [0,t_f]
</math>

The initial values and <math>t_f = 12</math> are fixed. We are interested in how to choose the controls <math>u_i, \ i=1,2,3,</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=1,2,3</math>. We use three different sampling functions, <math>w^i(\cdot), \ i=1,2,3,</math> in the same experimental setting. This can be seen either as a three-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (p_1, p_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=1}^{3} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t) \\
\quad x(0) & = & (2,2,2)^T \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad x_1(t) & \in & \mathcal{X} \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{3 \times 3}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 1,2,3,</math> for each observed function of states.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>c_1</math> || align=right | 1 || Unknown parameter
|-
| align=center | <math>c_2</math> || align=right | 2 || Unknown parameter
|-
| align=center | <math>c_3</math> || align=right | 0.8 || Unknown parameter
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 12 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_{\mathrm{reg}}</math> || align=right | 0.1 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{U}</math> || align=right | <math>[0,1]^3</math> || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | <math>[0,1]^3</math> || Bounds of measurement function
|-
| align=center | <math>M_1, M_2, M_3</math> || align=right | 2 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Van_der_Pol_OED.png| States, control, and sampling functions for a local optimum.
</gallery>

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]

Lotka Shared OED

2026-03-26T09:58:11Z

RobertLampel: /* Mathematical formulation */

{{Dimensions
|nd = 1
|nx = 21
|nw = 4}}

The '''Lotka Shared Experimental Design problem''' looks for an optimal strategy to be performed on a fixed time horizon to control the biomasses multiple species as described in [[:LV Shared Resource]]. The goal here, however, is to minimize the uncertainty of a follow-up parameter estimation problem. When measurements of the three state variables are performed becomes a degree of freedom.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. The ODE from [[LV Shared Resource]] is extended such that it also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

The optimal integer control functions shows [[:Category:Chattering|bang bang]] behavior.

== Mathematical formulation ==

We are interested in estimating the parameters <math>\alpha_0, \alpha_1,</math> and <math>\alpha_2</math> of the Lotka-Volterra type predator-prey initial value problem

<math>
\begin{array}{rcl}
\dot{x}_0(t) & = & x_0(t) - \alpha_0 x_0(t) x_1(t) - x_0(t) x_2(t), && t \in [0,t_f], \quad x_0(0) = 1.5, \\
\dot{x}_1(t) & = & - x_1(t) + \alpha_1 x_0(t) x_1(t) - c_1 x_1(t) u(t), && t \in [0,t_f], \quad x_1(0) = 0.5, \\
\dot{x}_2(t) & = & -x_2(t) + \alpha_2 x_0(t) x_2(t) - c_2 x_2(t) u(t), && t \in [0,t_f], \quad x_2(0) = 1,
\end{array}
</math>

where <math>u(\cdot)</math> is a control that may or may not be fixed. The other parameters, the initial values and <math>t_f = 20</math> are fixed. We are interested in how to choose <math>u</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=0,1,2</math>. We use three different sampling functions, <math>w^i(\cdot), i=0,1,2,</math> in the same experimental setting. This can be seen either as a three-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (\alpha_0, \alpha_1, \alpha_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=0}^{2} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t) \\
\quad x(0) & = & (1.5, 0.5, 1)^T \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{3 \times 3}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 0,1,2,</math> for each observed function of states.

== Parameters ==

These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>\alpha_0</math> || align=right | 1.0 ||
|-
| align=center | <math>\alpha_1</math> || align=right | 1.0 ||
|-
| align=center | <math>\alpha_2</math> || align=right | 1.2 ||
|-
| align=center | <math>c_1</math> || align=right | 0.1 ||
|-
| align=center | <math>c_2</math> || align=right | 0.4 ||
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 20 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_\mathrm{reg}</math> || align=right | 0.1 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{U}</math> || align=right | <math>[0,1]^3</math> || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | <math>[0,1]^3</math> || Bounds of measurement function
|-
| align=center | <math>M_0, M_1, M_2</math> || align=right | 4 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Lotka_Shared_OED.png| States, control, and sampling functions for a local optimum.
</gallery>

== Source Code ==

Model descriptions are available in

* [[:Category:Casadi | CasADi]] at https://github.com/rlampel/PySHeRLOC

== Miscellaneous and Further Reading ==
The Lotka Volterra fishing problem was introduced by Sebastian Sager in a proceedings paper <bib id="Sager2006" /> and revisited in his PhD thesis <bib id="Sager2005" />. These are also the references to look for more details. The experimental design problem has been described in the habilitation thesis of Sager, <bib id="Sager2011d" />.

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]
[[Category:Population dynamics]]

Three Tank OED

2026-03-26T09:58:00Z

RobertLampel: /* Mathematical formulation */

{{Dimensions
|nd = 1
|nx = 21
|nw = 6
}}

The '''Three Tank OED problem''' is a variation of the [[:Three Tank multimode problem]]. It looks for optimal time intervals to measure the two states in order to minimize the uncertainty of a follow-up parameter estimation problem for the two unknown parameters.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. It also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

== Mathematical formulation ==

We are interested in estimating the parameters <math>c_1, c_2,</math> and <math>c_3</math> of the initial value problem

<math>
\begin{array}{rcl}
\dot{x}_1(t) & =& -\sqrt{x_1(t)}+c_1 u_1(t) + c_2 u_2(t) - u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_1(0) = 2, \\
\dot{x}_2(t) & =& \sqrt{x_1(t)}-\sqrt{x_2(t)}, && t \in [0,t_f], \quad x_2(0) = 2, \\
\dot{x}_3(t) & =& \sqrt{x_2(t)}-\sqrt{x_3(t)} + u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_3(0) = 2.
\end{array}
</math>

Additionally, the controls <math>u_1,u_2,</math> and <math>u_3</math> are constrained by:

<math>
\sum_{i=1}^3 u_i(t) = 1 \quad \forall t\in [0,t_f]
</math>

The initial values and <math>t_f = 12</math> are fixed. We are interested in how to choose the controls <math>u_i, \ i=1,2,3,</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=1,2,3</math>. We use three different sampling functions, <math>w^i(\cdot), \ i=1,2,3,</math> in the same experimental setting. This can be seen either as a three-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (p_1, p_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=1}^{3} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t) \\
\quad x(0) & = & (2,2,2) \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad x_1(t) & \in & \mathcal{X} \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{3 \times 3}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 1,2,3,</math> for each observed function of states.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>c_1</math> || align=right | 1 || Unknown parameter
|-
| align=center | <math>c_2</math> || align=right | 2 || Unknown parameter
|-
| align=center | <math>c_3</math> || align=right | 0.8 || Unknown parameter
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 12 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_{\mathrm{reg}}</math> || align=right | 0.1 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{U}</math> || align=right | <math>[0,1]^3</math> || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | <math>[0,1]^3</math> || Bounds of measurement function
|-
| align=center | <math>M_1, M_2, M_3</math> || align=right | 2 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Van_der_Pol_OED.png| States, control, and sampling functions for a local optimum.
</gallery>

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]

Lotka Shared OED

2026-03-26T09:57:09Z

RobertLampel: /* Mathematical formulation */

{{Dimensions
|nd = 1
|nx = 21
|nw = 4}}

The '''Lotka Shared Experimental Design problem''' looks for an optimal strategy to be performed on a fixed time horizon to control the biomasses multiple species as described in [[:LV Shared Resource]]. The goal here, however, is to minimize the uncertainty of a follow-up parameter estimation problem. When measurements of the three state variables are performed becomes a degree of freedom.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. The ODE from [[LV Shared Resource]] is extended such that it also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

The optimal integer control functions shows [[:Category:Chattering|bang bang]] behavior.

== Mathematical formulation ==

We are interested in estimating the parameters <math>\alpha_0, \alpha_1,</math> and <math>\alpha_2</math> of the Lotka-Volterra type predator-prey initial value problem

<math>
\begin{array}{rcl}
\dot{x}_0(t) & = & x_0(t) - \alpha_0 x_0(t) x_1(t) - x_0(t) x_2(t), && t \in [0,t_f], \quad x_0(0) = 1.5, \\
\dot{x}_1(t) & = & - x_1(t) + \alpha_1 x_0(t) x_1(t) - c_1 x_1(t) u(t), && t \in [0,t_f], \quad x_1(0) = 0.5, \\
\dot{x}_2(t) & = & -x_2(t) + \alpha_2 x_0(t) x_2(t) - c_2 x_2(t) u(t), && t \in [0,t_f], \quad x_2(0) = 1,
\end{array}
</math>

where <math>u(\cdot)</math> is a control that may or may not be fixed. The other parameters, the initial values and <math>t_f = 20</math> are fixed. We are interested in how to choose <math>u</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=0,1,2</math>. We use three different sampling functions, <math>w^i(\cdot), i=0,1,2,</math> in the same experimental setting. This can be seen either as a three-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (\alpha_0, \alpha_1, \alpha_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=0}^{2} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t), \\
\quad x(0) & = & (1.5, 0.5, 1)^T \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{3 \times 3}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 0,1,2,</math> for each observed function of states.

== Parameters ==

These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>\alpha_0</math> || align=right | 1.0 ||
|-
| align=center | <math>\alpha_1</math> || align=right | 1.0 ||
|-
| align=center | <math>\alpha_2</math> || align=right | 1.2 ||
|-
| align=center | <math>c_1</math> || align=right | 0.1 ||
|-
| align=center | <math>c_2</math> || align=right | 0.4 ||
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 20 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_\mathrm{reg}</math> || align=right | 0.1 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{U}</math> || align=right | <math>[0,1]^3</math> || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | <math>[0,1]^3</math> || Bounds of measurement function
|-
| align=center | <math>M_0, M_1, M_2</math> || align=right | 4 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Lotka_Shared_OED.png| States, control, and sampling functions for a local optimum.
</gallery>

== Source Code ==

Model descriptions are available in

* [[:Category:Casadi | CasADi]] at https://github.com/rlampel/PySHeRLOC

== Miscellaneous and Further Reading ==
The Lotka Volterra fishing problem was introduced by Sebastian Sager in a proceedings paper <bib id="Sager2006" /> and revisited in his PhD thesis <bib id="Sager2005" />. These are also the references to look for more details. The experimental design problem has been described in the habilitation thesis of Sager, <bib id="Sager2011d" />.

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]
[[Category:Population dynamics]]

Lotka Shared OED

2026-03-26T09:56:11Z

RobertLampel: /* Parameters */

{{Dimensions
|nd = 1
|nx = 21
|nw = 4}}

The '''Lotka Shared Experimental Design problem''' looks for an optimal strategy to be performed on a fixed time horizon to control the biomasses multiple species as described in [[:LV Shared Resource]]. The goal here, however, is to minimize the uncertainty of a follow-up parameter estimation problem. When measurements of the three state variables are performed becomes a degree of freedom.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. The ODE from [[LV Shared Resource]] is extended such that it also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

The optimal integer control functions shows [[:Category:Chattering|bang bang]] behavior.

== Mathematical formulation ==

We are interested in estimating the parameters <math>\alpha_0, \alpha_1,</math> and <math>\alpha_2</math> of the Lotka-Volterra type predator-prey initial value problem

<math>
\begin{array}{rcl}
\dot{x}_0(t) & = & x_0(t) - \alpha_0 x_0(t) x_1(t) - x_0(t) x_2(t), \\
\dot{x}_1(t) & = & - x_1(t) + \alpha_1 x_0(t) x_1(t) - c_1 x_1(t) u(t), \\
\dot{x}_2(t) & = & -x_2(t) + \alpha_2 x_0(t) x_2(t) - c_2 x_2(t) u(t),
\end{array}
</math>

where <math>u(\cdot)</math> is a control that may or may not be fixed. The other parameters, the initial values and <math>t_f = 20</math> are fixed. We are interested in how to choose <math>u</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=0,1,2</math>. We use three different sampling functions, <math>w^i(\cdot), i=0,1,2,</math> in the same experimental setting. This can be seen either as a three-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (\alpha_0, \alpha_1, \alpha_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=0}^{2} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t), \\
\quad x(0) & = & (1.5, 0.5, 1)^T \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{3 \times 3}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 0,1,2,</math> for each observed function of states.

== Parameters ==

These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>\alpha_0</math> || align=right | 1.0 ||
|-
| align=center | <math>\alpha_1</math> || align=right | 1.0 ||
|-
| align=center | <math>\alpha_2</math> || align=right | 1.2 ||
|-
| align=center | <math>c_1</math> || align=right | 0.1 ||
|-
| align=center | <math>c_2</math> || align=right | 0.4 ||
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 20 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_\mathrm{reg}</math> || align=right | 0.1 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{U}</math> || align=right | <math>[0,1]^3</math> || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | <math>[0,1]^3</math> || Bounds of measurement function
|-
| align=center | <math>M_0, M_1, M_2</math> || align=right | 4 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Lotka_Shared_OED.png| States, control, and sampling functions for a local optimum.
</gallery>

== Source Code ==

Model descriptions are available in

* [[:Category:Casadi | CasADi]] at https://github.com/rlampel/PySHeRLOC

== Miscellaneous and Further Reading ==
The Lotka Volterra fishing problem was introduced by Sebastian Sager in a proceedings paper <bib id="Sager2006" /> and revisited in his PhD thesis <bib id="Sager2005" />. These are also the references to look for more details. The experimental design problem has been described in the habilitation thesis of Sager, <bib id="Sager2011d" />.

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]
[[Category:Population dynamics]]

Three Tank OED

2026-03-26T09:52:20Z

RobertLampel: /* Parameters */

{{Dimensions
|nd = 1
|nx = 21
|nw = 6
}}

The '''Three Tank OED problem''' is a variation of the [[:Three Tank multimode problem]]. It looks for optimal time intervals to measure the two states in order to minimize the uncertainty of a follow-up parameter estimation problem for the two unknown parameters.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. It also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

== Mathematical formulation ==

We are interested in estimating the parameters <math>c_1, c_2,</math> and <math>c_3</math> of the initial value problem

<math>
\begin{array}{rcl}
\dot{x}_1(t) & =& -\sqrt{x_1(t)}+c_1 u_1(t) + c_2 u_2(t) - u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_1(0) = 2, \\
\dot{x}_2(t) & =& \sqrt{x_1(t)}-\sqrt{x_2(t)}, && t \in [0,t_f], \quad x_2(0) = 2, \\
\dot{x}_3(t) & =& \sqrt{x_2(t)}-\sqrt{x_3(t)} + u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_3(0) = 2.
\end{array}
</math>

Additionally, the controls <math>u_1,u_2,</math> and <math>u_3</math> are constrained by:

<math>
\sum_{i=1}^3 u_i(t) = 1 \quad \forall t\in [0,t_f]
</math>

The initial values and <math>t_f = 12</math> are fixed. We are interested in how to choose the controls <math>u_i, \ i=1,2,3,</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=1,2,3</math>. We use three different sampling functions, <math>w^i(\cdot), \ i=1,2,3,</math> in the same experimental setting. This can be seen either as a three-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (p_1, p_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=1}^{3} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t), \\
\quad x(0) & = & (2,2,2) \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad x_1(t) & \in & \mathcal{X} \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{3 \times 3}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 1,2,3,</math> for each observed function of states.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>c_1</math> || align=right | 1 || Unknown parameter
|-
| align=center | <math>c_2</math> || align=right | 2 || Unknown parameter
|-
| align=center | <math>c_3</math> || align=right | 0.8 || Unknown parameter
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 12 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_{\mathrm{reg}}</math> || align=right | 0.1 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{U}</math> || align=right | <math>[0,1]^3</math> || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | <math>[0,1]^3</math> || Bounds of measurement function
|-
| align=center | <math>M_1, M_2, M_3</math> || align=right | 2 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Van_der_Pol_OED.png| States, control, and sampling functions for a local optimum.
</gallery>

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]

Three Tank OED

2026-03-26T09:51:59Z

RobertLampel: /* Parameters */

{{Dimensions
|nd = 1
|nx = 21
|nw = 6
}}

The '''Three Tank OED problem''' is a variation of the [[:Three Tank multimode problem]]. It looks for optimal time intervals to measure the two states in order to minimize the uncertainty of a follow-up parameter estimation problem for the two unknown parameters.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. It also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

== Mathematical formulation ==

We are interested in estimating the parameters <math>c_1, c_2,</math> and <math>c_3</math> of the initial value problem

<math>
\begin{array}{rcl}
\dot{x}_1(t) & =& -\sqrt{x_1(t)}+c_1 u_1(t) + c_2 u_2(t) - u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_1(0) = 2, \\
\dot{x}_2(t) & =& \sqrt{x_1(t)}-\sqrt{x_2(t)}, && t \in [0,t_f], \quad x_2(0) = 2, \\
\dot{x}_3(t) & =& \sqrt{x_2(t)}-\sqrt{x_3(t)} + u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_3(0) = 2.
\end{array}
</math>

Additionally, the controls <math>u_1,u_2,</math> and <math>u_3</math> are constrained by:

<math>
\sum_{i=1}^3 u_i(t) = 1 \quad \forall t\in [0,t_f]
</math>

The initial values and <math>t_f = 12</math> are fixed. We are interested in how to choose the controls <math>u_i, \ i=1,2,3,</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=1,2,3</math>. We use three different sampling functions, <math>w^i(\cdot), \ i=1,2,3,</math> in the same experimental setting. This can be seen either as a three-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (p_1, p_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=1}^{3} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t), \\
\quad x(0) & = & (2,2,2) \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad x_1(t) & \in & \mathcal{X} \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{3 \times 3}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 1,2,3,</math> for each observed function of states.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>c_1</math> || align=right | 1 || Unknown parameter
|-
| align=center | <math>c_2</math> || align=right | 2 || Unknown parameter
|-
| align=center | <math>c_3</math> || align=right | 0.8 || Unknown parameter
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 12 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_{\mathrm{reg}}</math> || align=right | 0.1 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{U}</math> || align=right | <math>[0,1]^3</math> || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | [0,1] || Bounds of measurement function
|-
| align=center | <math>M_1, M_2, M_3</math> || align=right | 2 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Van_der_Pol_OED.png| States, control, and sampling functions for a local optimum.
</gallery>

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]

Three Tank OED

2026-03-26T09:49:25Z

RobertLampel: /* Parameters */

{{Dimensions
|nd = 1
|nx = 21
|nw = 6
}}

The '''Three Tank OED problem''' is a variation of the [[:Three Tank multimode problem]]. It looks for optimal time intervals to measure the two states in order to minimize the uncertainty of a follow-up parameter estimation problem for the two unknown parameters.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. It also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

== Mathematical formulation ==

We are interested in estimating the parameters <math>c_1, c_2,</math> and <math>c_3</math> of the initial value problem

<math>
\begin{array}{rcl}
\dot{x}_1(t) & =& -\sqrt{x_1(t)}+c_1 u_1(t) + c_2 u_2(t) - u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_1(0) = 2, \\
\dot{x}_2(t) & =& \sqrt{x_1(t)}-\sqrt{x_2(t)}, && t \in [0,t_f], \quad x_2(0) = 2, \\
\dot{x}_3(t) & =& \sqrt{x_2(t)}-\sqrt{x_3(t)} + u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_3(0) = 2.
\end{array}
</math>

Additionally, the controls <math>u_1,u_2,</math> and <math>u_3</math> are constrained by:

<math>
\sum_{i=1}^3 u_i(t) = 1 \quad \forall t\in [0,t_f]
</math>

The initial values and <math>t_f = 12</math> are fixed. We are interested in how to choose the controls <math>u_i, \ i=1,2,3,</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=1,2,3</math>. We use three different sampling functions, <math>w^i(\cdot), \ i=1,2,3,</math> in the same experimental setting. This can be seen either as a three-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (p_1, p_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=1}^{3} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t), \\
\quad x(0) & = & (2,2,2) \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad x_1(t) & \in & \mathcal{X} \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{3 \times 3}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 1,2,3,</math> for each observed function of states.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>c_1</math> || align=right | 1 || Unknown parameter
|-
| align=center | <math>c_2</math> || align=right | 2 || Unknown parameter
|-
| align=center | <math>c_3</math> || align=right | 0.8 || Unknown parameter
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 12 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_{\mathrm{reg}}</math> || align=right | 0.1 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{U}</math> || align=right | [0,1] || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | [0,1] || Bounds of measurement function
|-
| align=center | <math>M_1, M_2, M_3</math> || align=right | 2 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Van_der_Pol_OED.png| States, control, and sampling functions for a local optimum.
</gallery>

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]

Three Tank OED

2026-03-26T09:47:11Z

RobertLampel: /* Mathematical formulation */

{{Dimensions
|nd = 1
|nx = 21
|nw = 6
}}

The '''Three Tank OED problem''' is a variation of the [[:Three Tank multimode problem]]. It looks for optimal time intervals to measure the two states in order to minimize the uncertainty of a follow-up parameter estimation problem for the two unknown parameters.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. It also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

== Mathematical formulation ==

We are interested in estimating the parameters <math>c_1, c_2,</math> and <math>c_3</math> of the initial value problem

<math>
\begin{array}{rcl}
\dot{x}_1(t) & =& -\sqrt{x_1(t)}+c_1 u_1(t) + c_2 u_2(t) - u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_1(0) = 2, \\
\dot{x}_2(t) & =& \sqrt{x_1(t)}-\sqrt{x_2(t)}, && t \in [0,t_f], \quad x_2(0) = 2, \\
\dot{x}_3(t) & =& \sqrt{x_2(t)}-\sqrt{x_3(t)} + u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_3(0) = 2.
\end{array}
</math>

Additionally, the controls <math>u_1,u_2,</math> and <math>u_3</math> are constrained by:

<math>
\sum_{i=1}^3 u_i(t) = 1 \quad \forall t\in [0,t_f]
</math>

The initial values and <math>t_f = 12</math> are fixed. We are interested in how to choose the controls <math>u_i, \ i=1,2,3,</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=1,2,3</math>. We use three different sampling functions, <math>w^i(\cdot), \ i=1,2,3,</math> in the same experimental setting. This can be seen either as a three-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (p_1, p_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=1}^{3} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t), \\
\quad x(0) & = & (2,2,2) \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad x_1(t) & \in & \mathcal{X} \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{3 \times 3}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 1,2,3,</math> for each observed function of states.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>p_1</math> || align=right | 1 || Unknown parameter
|-
| align=center | <math>p_2</math> || align=right | 1 || Unknown parameter
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 10 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_{\mathrm{reg}}</math> || align=right | 0.1 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{X}</math> || align=right | [-0.5,<math>\infty</math>] || Bounds of <math>x_1</math>
|-
| align=center | <math>\mathcal{U}</math> || align=right | [-1,1] || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | [0,1] || Bounds of measurement function
|-
| align=center | <math>M_1, M_2</math> || align=right | 2 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Van_der_Pol_OED.png| States, control, and sampling functions for a local optimum.
</gallery>

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]