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. We prove that, generically with respect to $\mu$, any target close to the ground state in the $H^3\times H^2$ topology (suitably adapted to the underlying degenerate operator) is reachable in time $T > \frac{4}{2-\alpha}$, with controls in $L^2((0, T ),\mathbb R)$. Under some classical and generic assumption on $\mu$, we prove that there exists a threshold value for time, $T_0= \frac{4}{2-\alpha}$, such that the reachable set is: - a neighborhood of the ground state if $T>T_0$, - contained in a $C^1$-submanifold of infinite codimension if $T<T_0$ - a $C^1$-submanifold of codimension $1$ if $\alpha \in [0,1)$, and a neighborhood of the ground state if $\alpha \in (1,2)$ if $T=T_0$, the case $\alpha =1$ 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 ($\alpha =0$), 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 $\frac{1}{4}$ theorem.