Abstract :

We consider a linear wave equation, on a bounded interval, with bilinear control and Neumann boundary conditions. We study the controllability of this nonlinear control system, locally around a constant reference trajectory. We prove that the following results hold generically. For every T >2, this system is locally controllable in H^3 * H^2, in time T, with controls in L^2. For T=2, this system is locally controllable up to codimension one in H^3 * H^2, in time T, with controls in L^2: the reachable set is (locally) a non flat submanifold of H^3 * H^2 with codimension one. For every T<2, this system is not locally controllable, more precisely, the reachable set, with controls in L^2, is contained in a non flat submanifold of H^3 * H^2, with infinite codimension. The proof of these results relies on the inverse mapping theorem and second order expansions.