Internal controllability of systems of semilinear coupled one-dimensional wave equations with one control

Christophe Zhang (1, 2, 3)

1 D-MATH - Department of Mathematics [ETH Zurich]
2 CaGE - Control And GEometry
3 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions

We prove the internal controllability of some systems of two coupled wave equations in one space dimension, with one control, under certain conditions on the coupling. To do this we apply the " fictitious control method " in two cases: general systems with a " non-degenerate " coupling, and a particular case where the coupling is " degenerate " , namely a cubic coupling. In the latter case, our proof requires to find nontrivial trajectories of the control system that go from 0 to 0. We build these trajectories by adapting (in 1 space dimension) a construction developed by Jean-Michel Coron, Sergio Guerrero and Lionel Rosier for the study of coupled parabolic systems.

Mots clés: Fictitious control method; Coupled systems; Algebraic solvability; Exact internal controllability; Wave equations; Return method

Domaines: Mathématiques [math]; Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Christophe Zhang. Internal controllability of systems of semilinear coupled one-dimensional wave equations with one control. SIAM Journal on Control and Optimization, 2018, 56 (4), pp.3092 - 3127. ⟨10.1137/17m1128885⟩. ⟨hal-01517461v3⟩