Van der Pol Oscillator (binary variant)

Van der Pol Oscillator (binary variant)
State dimension:	1
Differential states:	2
Discrete control functions:	3
Interior point equalities:	2

This site describes a Van der Pol Oscillator variant with three binary controls instead of only one continuous control.

Mathematical formulation

The mixed-integer optimal control problem is given by

$\begin{array}{lll} \min_{x, y, w} & \int_{t_{0}}^{t_{f}} & (x (t)^{2} + y (t)^{2} d t \\ s . t . & \dot{x} & = y, \\ \dot{y} & = \sum_{i = 1}^{3} c_{i} w_{i} (1 - x^{2}) y - x, \\ x (0) & = 1, \\ y (0) & = 0, \\ 1 & = \sum_{i = 1}^{3} w_{i} (t), \\ w_{i} (t) & \in {0, 1}, i = 1 \dots 3 . \end{array}$

Parameters

These fixed values are used within the model:

$[t_{0}, t_{f}] = [0, 20], c_{1} = - 1, c_{2} = 0.75, c_{3} = - 2 .$

Reference Solutions

If the problem is relaxed, i.e., we demand that $w (t)$ be in the continuous interval $[0, 1]$ instead of the binary choice ${0, 1}$ , the optimal solution can be determined by means of direct optimal control.

The optimal objective value of the relaxed problem with $n_{t} = 6000, n_{u} = 60$ is $1.30167235$ . The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is $1.30273681$ .

Reference solution plots
Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and $n_{t} = 6000, n_{u} = 60$ .
Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and $n_{t} = 6000, n_{u} = 60$ . The relaxed controls were approximated by Combinatorial Integral Approximation.