Second order singular regulator

We consider a second order singular regulator problem, see Aly [1], or [2]:

$Latex formula$

The difference with Fuller’s problem is that the cost function includes a penalization of the “speed” $Latex formula$. We observe in figure 1 the occurrence of a singular arc, the optimal control being of the form bang (-1) – singular.

Numerical simulations:
Discretization: Runge-Kutta 4 with 1000 steps.
We take here $Latex formula$, $Latex formula$.

Figure 1

References:

[1] G.M. Aly. The computation of optimal singular control. International Journal of Control, 28 (5):681-688, 1978.

[2] S. Aronna, F.J. Bonnans, and P. Martinon. A well-posed shooting algorithm for optimal control problems with singular arcs. J. Optim. Theory Appl., 2013. Online First, Jan. 2013.