Third order state constraints

Robbins [1] considered the following family of problems:

Latex formula

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 Latex formula, Latex formula. 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 Latex formula, Latex formula, Latex formula.


Figure 1


[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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