Category:ODE model

This category includes all problems constrained by the solution of ordinary differential equations (ODE). In particular, no algebraic variables and derivatives with respect to one independent variable only (typically time) are present in the model equations for ${\displaystyle F(\cdot )}$.

The mixed-integer optimal control problem is of the form

${\displaystyle \begin{array}{ll}{\displaystyle \underset{x(\cdot ),u(\cdot ),v(\cdot ),q,\rho }{\min }}& \varphi (x(t_f),q,\rho )\\ \text{s.t.}& \dot{x}(t)& =& f(x(t),u(t),v(t),q,\rho ),\\ & 0& \le & c(x(t),u(t),v(t),q,\rho ),\\ & 0& =& r^{\text{eq}}(x(t_0),x(t_1),\dots ,x(t_m),q,\rho ),\\ & 0& \le & r^{\text{ieq}}(x(t_0),x(t_1),\dots ,x(t_m),q,\rho ),\\ & v(t)& \in & \mathrm{\Omega }:=\{v^1,v^2,\dots ,v^{n_{\omega }}\},\\ & \rho & \in & \mathrm{P}:=\{{\rho }^1,{\rho }^2,\dots ,{\rho }^{n_{\mathrm{P}}}\},\end{array}}$

for ${\displaystyle t\in [t_0,t_f]}$ almost everywhere.

${\displaystyle x(\cdot )}$ denotes the differential states, ${\displaystyle u(\cdot )}$ denotes the continuous control functions, ${\displaystyle v(\cdot )}$ denotes the integer control functions, ${\displaystyle q}$ denotes the continuous (constant-in-time) control values, ${\displaystyle \rho }$ denotes the integer (constant-in-time) control values.

The multipoint constraints ${\displaystyle r^{\cdot }(\cdot )}$ are defined on a time grid ${\displaystyle t_0\le t_1\le \dots \le t_m=t_f}$. The Mayer term functional ${\displaystyle \varphi :{\mathbb{ℝ}}^{n_x+n_q}\to \mathbb{ℝ}}$, the path- and control constraints ${\displaystyle c:{\mathbb{ℝ}}^{n_x\times n_u+n_v+n_q}\to {\mathbb{ℝ}}^{n_c}}$ and the constraint functions ${\displaystyle r^{\cdot }:{\mathbb{ℝ}}^{(m+1)n_x+n_q}\to {\mathbb{ℝ}}^{n_{r\cdot }}}$ are assumed to be sufficiently often differentiable.

The equality constraints ${\displaystyle r^{\text{eq}}(\cdot )}$ will often fix the initial values, i.e., ${\displaystyle x(0)=x_0}$, or impose a periodicity constraint.

Extensions

• For some problems the functions may as well depend explicitely on the time ${\displaystyle t}$.
• The differential equations might depend on state-dependent switches.
• The variables may include boolean variables.
• The underlying process might be a multistage process.
• The dynamics might be unstable.
• There might be an underlying network topology.
• The integer control functions might have been (re)formulated by means of an outer convexification.

Note that a Lagrange term ${\displaystyle {\displaystyle \underset{t_0}{\overset{t_f}{\int }}}L(x(t),u(t),v(t),q,\rho )\mathrm{d}t}$ can be transformed into a Mayer-type objective functional.