Bilinear control of a degenerate hyperbolic equation

Abstract

We consider the linear degenerate wave equation, on the interval (0, 1)wtt − (x α 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".Under some classical and generic assumption on µ, we prove that there exists a threshold value for time, T 0 = 4 2−α , such that the reachable set is1), and a neighborhood of the ground state if α ∈ (1, 2) if T = T 0 , the case α = 1 remaining open. This extends to the degenerate case the work of Beauchard [6] concerning the bilinear control of the classical wave equation (α = 0), and adapts to bilinear controls the work of Alabau-Boussouira, Cannarsa and Leugering [1] 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 1 4 theorem.