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

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 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 \le 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.

   Read Full Paper…