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.