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

Abstract

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 H 3 × H 2 , in time T, with controls in L 2 ( ( 0 , T ) , R ) . • 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 ( ( 0 , T ) , R ) : 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 ( ( 0 , T ) , R ) , 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.