Fuller's initial value problem

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 $2.40273813e-05$.

• Reference solution plots
• 

  Optimal relaxed states determined by an direct approach with ampl_mintoc (Radau collocation) and $n_t=6000, \, n_u=150$.

• 

  Optimal relaxed controls.

• 

  Optimal differential states trajectories of binary controls determined by an direct approach (Radau collocation) with ampl_mintoc and $n_t=6000, \, n_u=150$. The relaxed controls were approximated by Combinatorial Integral Approximation.

• 

  Optimal binary controls.