Lohéac. J, Zuazua E. Norm saturating property of time optimal controls for wave-type equations IFAC-PapersOnLine, Vol. 49, No.8 (2016), pp. 37-42DOI: 10.1016/j.ifacol.2016.07.415
Abstract: We consider a time optimal control problem with point target for a class of infinite dimensional systems governed by abstract wave operators. In order to ensure the existence of a time optimal control, we consider controls of energy bounded by a prescribed constant E > 0. Even when this control constraint is absent, in many situations, due to the hyperbolicity of the system under consideration, a target point cannot be reached in arbitrarily small time and there exists a minimal universal controllability time $T_* > 0$, so that for every points $y_0$ and $y_1$ and every time $T > T_*$, there exists a control steering $y_0$ to $y_1$ in time T. Simultaneously this may be impossible if $T < T_*$ for some particular choices of $y_0$ and $y_1$. In this note we point out the impact of the strict positivity of the minimal time $T_*$ on the structure of the norm of time optimal controls. In other words, the question we address is the following: If T is the minimal time, what is the L2-norm of the associated time optimal control? For different values of $y_0$, $y_1$ and E, we can have $τ ≤ T_*$ or $τ > T_*$. If $τ > T_*$, the time optimal control is unique, given by an adjoint problem and its L2-norm is E, in the classical sense. In this case, the time optimal control is also a norm optimal control. But when $τ < T_*$, we show, analyzing the string equation with Dirichlet boundary control, that, surprisingly, there exist time optimal controls which are not of maximal norm E.