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

TL;DR

This paper establishes the exact boundary controllability and synchronization for coupled wave equations with Robin boundary controls, overcoming regularity challenges and analyzing controllability limits on specific domains.

Contribution

It introduces new regularity results for Robin boundary conditions and demonstrates controllability and synchronization properties for coupled wave systems.

Findings

Proves exact boundary controllability for coupled wave equations with Robin controls.

Shows non-exact controllability with fewer controls on certain domains.

Analyzes synchronization and coupling matrix conditions.

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 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.