Second order singular regulator

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

$Min \int_0^T x^2(t) + \dot x^2(t) \quad dt; \quad \ddot x(t) = u(t) \in [-1,1].$

The difference with Fuller’s problem is that the cost function includes a penalization of the “speed” $\dot x(t)$ . 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 $T=5$ , $x(0)=0, \dot x(0)=1$ .

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.

Second order singular regulator

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