Third order state constraints

Robbins [1] considered the following family of problems:

$\begin{array}{ll} Min \displaystyle \frac{1}{2} \int_0^T \alpha y(t) + \frac{1}{2} u(t)^2 \quad dt; \quad {y}^{(3)}(t) = u(t); \;\; y(t) \geq 0. \end{array}$

It has been proved by Robbins [1] that, for appropriate initial conditions, the exact solution has infinitely many isolated contact points, such that the length of unconstrained arcs decreases geometrically. Detailed computations can be found in [2]. Therefore the isolated contact points have an accumulation point; the latter is followed by the trivial singular arc $u=0$ , $y=0$ . It is not easy to reproduce numerically this behaviour, since the unconstrained arcs rapidly become too small to be captured by a given time discretization. We display in Figure1 the value of the first state component and of the control.

Numerical simulations: problem state_constraint_3
Discretization: Runge-Kutta 4 with 100 steps.
We take here $\alpha=3$ , $T=10$ , $y(0)=(1,-2,0)$ .

Figure 1

References:

[1] H. M. Robbins. Junction phenomena for optimal control with state-variable inequality constraints of third order. J. of Optimization Theory and Applications, 31:85–99, 1980.

[2] Audrey Hermant. Sur l’algorithme de tir pour les problèmes de commande optimale avec contraintes sur l’état. PhD thesis, Ecole Polytechnique X, 2008.

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