# 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

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

## Parameters

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 $2.40273813e-05$.