On the observability inequality of coupled wave equations: the case without boundary

¹ School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, 519082, PR China.
² CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France and Sorbonne Université, Université Paris-Diderot SPC, CNRS, Laboratoire Jacques-Louis Lions, 75005 Paris.
³ School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, PR China.

^* Corresponding author: wzq@fudan.edu.cn

Received: 21 May 2019
Accepted: 6 January 2020

Abstract

In this paper, we study the observability and controllability of wave equations coupled by first or zero order terms on a compact manifold. We adopt the approach in Dehman-Lebeau’s paper [B. Dehman and G. Lebeau, SIAM J. Control Optim. 48 (2009) 521–550.] to prove that: the weak observability inequality holds for wave equations coupled by first order terms on compact manifold without boundary if and only if a class of ordinary differential equations related to the symbol of the first order terms along the Hamiltonian flow are exactly controllable. We also compute the higher order part of the observability constant and the observation time. By duality, we obtain the controllability of the dual control system in a finite co-dimensional space. This gives the full controllability under the assumption of unique continuation of eigenfunctions. Moreover, these results can be applied to the systems of wave equations coupled by zero order terms of cascade structure after an appropriate change of unknowns and spaces. Finally, we provide some concrete examples as applications where the unique continuation property indeed holds.