Difference between revisions of "Category:ODE model"

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The differential equations might depend on [[:Category:State-dependent switches | state-dependent switches]].
 
The differential equations might depend on [[:Category:State-dependent switches | state-dependent switches]].
  
Note that a Lagrange term <math>\int_{t_0}^{t_f} L( x(t), u(t), v(t), q, \rho)</math> can be transformed into a Mayer-type objective functional.
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Note that a Lagrange term <math>\int_{t_0}^{t_f} L( x(t), u(t), v(t), q, \rho) \; \mathrm{d} t</math> can be transformed into a Mayer-type objective functional.
  
 
[[Category:Model characterization]]
 
[[Category:Model characterization]]

Revision as of 06:05, 29 November 2008

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 mathematical model.

The mixed-integer optimal control problem is of the form


\begin{array}{llcl}
 \displaystyle \min_{x(\cdot), u(\cdot), v(\cdot), q, \rho} & \phi(x(t_f), q, \rho) \\[1.5ex]
 \mbox{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), \\[1.5ex]
 & 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), \\[1.5ex]
 & v(t) &\in& \Omega := \{v^1, v^2, \dots, v^{n_\omega} \},\\
 & \rho &\in& \Rho := \{\rho^1, \rho^2, \dots, \rho^{n_\Rho} \},
\end{array}

for t \in [t_0, t_f] almost everywhere.

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

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

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

Extensions

For some problems the functions may as well depend explicitely on the time t.

The differential equations might depend on state-dependent switches.

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

Subcategories

This category has only the following subcategory.

Pages in category "ODE model"

The following 50 pages are in this category, out of 50 total.