Norm saturating property of time optimal controls for wave-type equations. This work was partially supported by Grants FA9550-14-1-0214 of the EOARD-AFOSR, FA9550-15-1-0027 of AFOSR and the MTM2014-52347 Grants of the MINECO (Spain)

Loheac J, Zuazua E (2016)

Publication Type: Conference contribution

Publication year: 2016

Journal

IFAC-PapersOnLine International Federation of Automatic Control

Publisher: Elsevier B.V.

Book Volume: 49

Pages Range: 37-42

Conference Proceedings Title: IFAC-PapersOnLine

DOI: 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 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.

Involved external institutions

Université Nantes Angers Le Mans FR France (FR) Universidad Autónoma de Madrid (UAM) ES Spain (ES)

How to cite

APA:

Loheac, J., & Zuazua, E. (2016). Norm saturating property of time optimal controls for wave-type equations. This work was partially supported by Grants FA9550-14-1-0214 of the EOARD-AFOSR, FA9550-15-1-0027 of AFOSR and the MTM2014-52347 Grants of the MINECO (Spain). In IFAC-PapersOnLine (pp. 37-42). Elsevier B.V..

MLA:

Loheac, Jerome, and Enrique Zuazua. "Norm saturating property of time optimal controls for wave-type equations. This work was partially supported by Grants FA9550-14-1-0214 of the EOARD-AFOSR, FA9550-15-1-0027 of AFOSR and the MTM2014-52347 Grants of the MINECO (Spain)." Proceedings of the IFAC-PapersOnLine Elsevier B.V., 2016. 37-42.

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