Robbins  considered the following family of problems:
It has been proved by Robbins  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 . Therefore the isolated contact points have an accumulation point; the latter is followed by the trivial singular arc , . 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 , , .
 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.