We consider a second order singular regulator problem, see Aly , or :
The difference with Fuller’s problem is that the cost function includes a penalization of the “speed” . We observe in figure 1 the occurrence of a singular arc, the optimal control being of the form bang (-1) – singular.
Discretization: Runge-Kutta 4 with 1000 steps.
We take here , .
 G.M. Aly. The computation of optimal singular control. International Journal of Control, 28 (5):681-688, 1978.