Controllability of Two Coupled Wave Equations on a Compact Manifold

Abstract

We consider the exact controllability problem on a compact manifold Ω for two coupled wave equations, with a control function acting on one of them only. Action on the second wave equation is obtained through a coupling term. First, when the two waves propagate with the same speed, we introduce the time \({T_{\omega \rightarrow \mathcal{O} \rightarrow \omega}}\) for which all geodesics traveling in Ω go through the control region ω, then through the coupling region \({\mathcal{O}}\), and finally come back in ω. We prove that the system is controllable if and only if both ω and \({\mathcal{O}}\) satisfy the Geometric Control Condition and the control time is larger than \({T_{\omega \rightarrow \mathcal{O} \rightarrow \omega}}\). Second, we prove that the associated HUM control operator is a pseudodifferential operator and we exhibit its principal symbol. Finally, if the two waves propagate with different speeds, we give sharp sufficient controllability conditions on the functional spaces, the geometry of the sets ω and \({\mathcal{O}}\), and the minimal time.