Robbins [1] considered the following family of problems:

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 , . 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 , , .*

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.