Bilinear Control of a Degenerate Hyperbolic Equation

.We consider the linear degenerate wave equation, on the interval \((0, 1)\) \(w_{tt} - (x^\alpha w_x)_x = p(t) \mu (x) w\), 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 assumptions on \(\mu\), we prove that there exists a threshold value for time, \(T_{\alpha }= \frac{4}{2-\alpha }\), such that the reachable set is a neighborhood of the ground state. If \(T\lt T_{\alpha}\) it is contained in a \(C^1\)-submanifold of infinite codimension. Finally, it \(T=T_\alpha\) and \(\alpha\in[0,1)\) it is a \(C^1\)-submanifold of codimension 1 and if \(\alpha\in(1,2)\) the reachable set is a neighborhood of the ground state. The case \(\alpha=1\) remains open. This extends to the degenerate case the work [K. Beauchard, J. Differential Equations, 250 (2011), pp. 2064–2098] and adapts to bilinear controls the work [F. Alabau-Boussouira, P. Cannarsa, and G. Leugering, SIAM J. Control Optim., 55 (2017), pp. 2052–2087]. Our proofs are based on a careful analysis of the spectral problem and on Ingham type results, which are extensions of Kadec's \(\frac{1}{4}\) theorem.Keywordsdegenerate hyperbolic equationsbilinear controlBessel functionsground stateRiesz basisMSC codes35L8093B0393B6033C1042C40