Van der Pol Oscillator (binary variant)

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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

\min\limits_{x,y,w}  & \int\limits_{t_0}^{t_f} & (x(t)^2+y(t)^2 dt\\
s.t. & 	 \dot x & = y,\\
&	\dot y & =  \sum\limits_{i=1}^{3} c_{i}\;  w_i \;(1-x^2) y-x,\\
& x(0) & =1,\\
& y(0) & =0,\\
& 1 &= \sum\limits_{i=1}^{3}w_i(t), \\
 & w_i(t) &\in  \{0, 1\}, \quad i=1\ldots 3.


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.

Source Code

Model description is available in