Fuller's initial value multimode problem

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Fuller's initial value multimode problem
State dimension: 1
Differential states: 2
Discrete control functions: 4
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. Furthermore, this variant comprises four binary controls instead of only one control.

Mathematical formulation

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

$\begin{array}{llcl} \displaystyle \min_{x, w} & \int_{t_0}^{t_f} x_0^2 & \; \mathrm{d} t & + (x(t_f)-x_T)^2 \\[1.5ex] \mbox{s.t.} & \dot{x}_0 & = & x_1+ \sum\limits_{i=1}^{4} c_{0,i} \omega_i, \\ & \dot{x}_1 & = & 1 + \sum\limits_{i=1}^{4} c_{1,i} \omega_i, \\[1.5ex] & 1 &=& \sum\limits_{i=1}^{4}w_i(t), \\ & x(0) &=& x_S, \\ & w(t) &\in& \{0, 1\}. \end{array}$

Parameters

We use $x_S = x_T = (0.01, 0)^T$ together with:

$\begin{array}{rcl} [t_0, t_f] &=& [0, 1],\\ (c_{0,1}, c_{1,1}) &=& (0, -2),\\ (c_{0,2}, c_{1,2}) &=& (0, -0.5),\\ (c_{0,3}, c_{1,3}) &=& (0, -3),\\ (c_{0,4}, c_{1,4}) &=& (0, 0). \end{array}$

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 $1.08947605e-05$. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is $0.000422127329$.