On the Observability Inequality of Coupled Wave Equations: the Case without Boundary

TL;DR

This paper investigates the observability and controllability of coupled wave equations on compact manifolds without boundary, establishing conditions based on the controllability of associated ODEs and providing explicit constants and examples.

Contribution

It extends the understanding of observability inequalities for coupled wave equations without boundary, linking them to the controllability of related ODEs and deriving explicit constants.

Findings

Weak observability holds iff related ODEs are controllable

Higher order observability constants are computed

Controllability is achieved in a finite co-dimensional space

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 \cite{DehmanLebeau09} 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.