**Lohéac. J, Zuazua E.** Norm saturating property of time optimal controls for wave-type equations

2nd Workshop on Control of Systems Governed by Partial Differential Equations 2016 Bertinoro, Italy, 13—15 June 2016, IFAC-PapersOnLine, 49 (8) 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 , so that for every points and and every time , there exists a control steering to in time T. Simultaneously this may be impossible if for some particular choices of and .

In this note we point out the impact of the strict positivity of the minimal time 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 , and E, we can have or . If , 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 , we show, analyzing the string equation with Dirichlet boundary control, that, surprisingly, there exist time optimal controls which are not of maximal norm E.