Norm saturating property of time optimal controls for wave-type equations

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 y0 and y1 and every time T > T*, there exists a control steering y0 to y1 in time T. Simultaneously this may be impossible if T < T* for some particular choices of y0 and y1.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 y0, y1 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.