Fuller's initial value problem

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Fuller's initial value problem
State dimension: 1
Differential states: 2
Discrete control functions: 1
Interior point equalities: 2

This site describes a Fuller's problem variant with no terminal constraints and additional Mayer term for penalizing deviation from given reference values.

Mathematical formulation

For t \in [t_0, t_f] almost everywhere the mixed-integer optimal control problem is given by

 \displaystyle \min_{x, w} &  \int_{t_0}^{t_f} x_0^2  \; &\mathrm{d} t& + (x(t_f)-x_T)^2 \\[1.5ex]
 \mbox{s.t.} & \dot{x}_0 & = & x_1, \\
 & \dot{x}_1 & = & 1 - 2 \; w, \\[1.5ex]
 & x(0) &=& x_S, \\
 & w(t) &\in&  \{0, 1\}.


We use x_S = x_T = (0.01, 0)^T and (t_0,t_f) = (0,1).

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=150  is 1.45412214e-05. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is 2.40273813e-05.