# 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

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.