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 are present in the mathematical model.

The mixed-integer optimal control problem is of the form

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

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 ${\displaystyle \varphi :{\mathbb{ℝ}}^{n_x}\to \mathbb{ℝ}}$, the path- and control constraints ${\displaystyle c:{\mathbb{ℝ}}^{n_x\times n_u\times n_v}\to {\mathbb{ℝ}}^{n_c}}$ and the constraint functions ${\displaystyle r^{\cdot }:{\mathbb{ℝ}}^{(m+1)n_x}\to {\mathbb{ℝ}}^{n_{r\cdot }}}$ 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 of periodicity constraint.