Stabilization of the wave equation by on-off and positive-negative feedbacks

Motivated by several works on the stabilization of the oscillator by on-off feedbacks, we study the related problem for the one-dimensional wave equation, damped by an on-off feedback a(t)u(t). We obtain results that are radically different from those known in the case of the oscillator. We consider periodic functions a: typically a is equal to 1 on (0; T), equal to 0 on (T;qT) and is qT-periodic. We study the boundary case and next the locally distributed case, and we give optimal results of stability. In both cases, we prove that there are explicit exceptional values of T for which the energy of some solutions remains constant with time. If T is different from those exceptional values, the energy of all solutions decays exponentially to zero. This number of exceptional values is countable in the boundary case and finite in the distributed case. When the feedback is acting on the boundary, we also study the case of postive-negative feedbacks: a(t) =a(0) > 0 on (0; T), and a(t) = b(0) < 0 on(T;qT), and we give the necessary and sufficient condition under which the energy (that is no more nonincreasing with time) goes to zero or goes to infinity. The proofs of these results are based on congruence properties and on a theorem of Weyl in the boundary case, and on new observability inequalities for the undamped wave equation, weakening the usual "optimal time condition" in the locally distributed case. These new inequalities provide also new exact controllability results.