# 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 $0.000422127329$.