Exact boundary controllability and exact boundary synchronization for a coupled system of wave equations with coupled Robin boundary controls

¹ School of Mathematical Sciences, Fudan University, 200433 Shanghai, China ; Shanghai Key Laboratory for Contemporary Applied Mathematic; Nonlinear Mathematical Modeling and Methods Laboratory.
² School of Mathematics, Southeast University, 211189 Nanjing, China.
³ Institut de Recherche Mathématique Avancée, Université de Strasbourg, 67084 Strasbourg, France.
⁴ School of Mathematical Sciences, Qufu Normal University, 273165 Qufu, China.

^*** Corresponding author: xinglu@seu.edu.cn

Received: 18 August 2019
Accepted: 10 July 2020

Abstract

In this paper, we consider the exact boundary controllability and the exact boundary synchronization (by groups) for a coupled system of wave equations with coupled Robin boundary controls. Owing to the difficulty coming from the lack of regularity of the solution, we confront a bigger challenge than that in the case with Dirichlet or Neumann boundary controls. In order to overcome this difficulty, we use the regularity results of solutions to the mixed problem with Neumann boundary conditions by Lasiecka and Triggiani [J. Differ. Equ. 94 (1991) 112–164] to get the regularity of solutions to the mixed problem with coupled Robin boundary conditions. Thus we show the exact boundary controllability of the system, and by a method of compact perturbation, we obtain the non-exact boundary controllability of the system with fewer boundary controls on some special domains. Based on this, we further study the exact boundary synchronization (by groups) for the same system, the determination of the exactly synchronizable state (by groups), as well as the necessity of the compatibility conditions of the coupling matrices.