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.

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For every T>2, this system is locally controllable in H³×H², in time T, with controls in L²((0,T),R).

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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.

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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.