Bilinear control of a degenerate hyperbolic equation

We consider the linear degenerate wave equation, on the interval (0, 1) wtt - (xα wx)x = p(t) μ (x) w, with bilinear control p and Neumann boundary conditions. We study the controllability of this nonlinear control system, locally around a constant reference trajectory, the ground state. We prove that, generically with respect to μ, any target close to the ground state in the H3× H2 topology (suitably adapted to the underlying degenerate operator) is reachable in time T > frac42-α, with controls in L2((0, T ),mathbb R). Under some classical and generic assumption on μ, we prove that there exists a threshold value for time, T0= frac42-α, such that the reachable set is: - a neighborhood of the ground state if T>T0, - contained in a C1-submanifold of infinite codimension if T<T0 - a C1-submanifold of codimension 1 if α ∈ [0,1), and a neighborhood of the ground state if α ∈ (1,2) if T=T0, the case α =1 remaining open. This extends to the degenerate case the work [K. Beauchard, Local controllability and non-controllability for a 1D wave equation with bilinear control. J. Differential Equations, 250(4), 2064-2098, 2011] concerning the bilinear control of the classical wave equation (α =0), and adapts to bilinear controls the work [F. Alabau-Boussouira, P. Cannarsa, and G. Leugering. Control and stabilization of degenerate wave equations. SIAM J. Control Optim., 55(3), 2052-2087, 2017] on the degenerate wave equation where additive control are considered. Our proofs are based on a careful analysis of the spectral problem, and on Ingham type results, which are extensions of the Kadec's frac14 theorem.