Norm saturating property of time optimal controls for wave-type equations
Résumé
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 τ is the minimal time, what is the L 2-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 L 2-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.
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