Local controllability and non-controllability for a 1D wave equation with bilinear control

We consider a linear wave equation, on the interval (0,1), 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 H3×H2, in time T, with controls in L2((0,T),R).
• For T=2, this system is locally controllable up to codimension one in H3×H2, in time T, with controls in L2((0,T),R): the reachable set is (locally) a non-flat submanifold of H3×H2 with codimension one.
• For every T<2, this system is not locally controllable, more precisely, the reachable set, with controls in L2((0,T),R), is contained in a non-flat submanifold of H3×H2, with infinite codimension. The proof of these results relies on the inverse mapping theorem and second order expansions.