Work

Optimal time for the null-controllability of linear hyperbolic systems in one dimensional space

Year: 2018

Type: article

Abstract: We are concerned about the controllability of a general linear hyperbolic system of the form $\partial_t w (t, x) = \Sigma(x) \partial_x w (t, x) + \gamma C(x) w(t, x) $ ($\gamma \in \mathbb{R}$) in one space dimension using boundary controls on one side. More precisely, we establish the optimal time for the null and exact controllability of the hyperbolic system for generic $\gamma$. We also present examples which yield that the generic requirement is necessary. In the case of constant $\Sigma$ and of two positive directions, we prove that the null-controllability is attained for any time greater than the optimal time for all $\gamma \in \mathbb{R}$ and for all $C$ which is analytic if the slowest negative direction can be alerted by both positive directions. We also show that the null-controllability is attained at the optimal time by a feedback law when $C \equiv 0$. Our approach is based on the backstepping method paying a special attention on the construction of the kernel and the selection of controls.

Source: Infoscience (Ecole Polytechnique Fédérale de Lausanne)

DOI: 10.1137/18m1185600

Authors: Coron, Jean-Michel, Nguyên, Hoài-Minh

Institutions: Sorbonne Université, Laboratoire Jacques-Louis Lions, CAGE: Contrôle et géométrie, École Polytechnique Fédérale de Lausanne

Language: English

Topic: Stability and Controllability of Differential Equations

Subfield: Control and Systems Engineering

Field: Engineering

Domain: Physical Sciences

Open Access status: green

Funders: Agence Nationale de la Recherche, Eidgenössische Technische Hochschule Zürich

Award: Finite Time Control and Estimation for System of Systems