Category:ODE model

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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 F(\cdot).

The mixed-integer optimal control problem is of the form

 \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} \},

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.


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.


This category has only the following subcategory.