Internal Controllability of Systems of Semilinear Coupled One-Dimensional Wave Equations with One Control

Abstract

We study systems of two coupled wave equations in one space dimension, with one control, spatially supported on an arbitrarily small interval. We obtain the controllability of such systems under certain conditions on the coupling. To do this we apply the “fictitious control method” in two cases: general systems with a controllable linearized system, and a particular case where the linearized system is not controllable, namely, a cubic coupling. In the latter case, our proof requires finding nontrivial trajectories of the control system that go from 0 to 0 and having a controllable linearized system. We build these trajectories by adapting (in one space dimension) a construction developed by Jean-Michel Coron, Sergio Guerrero, and Lionel Rosier for the study of coupled parabolic systems.