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

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 L-2-norm of the associated time optimal control? For different values of y(0), y(1) and E, we can have tau <= T-* or tau > T-*. If tau > 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 tau < 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. (C) 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.