Bilinear control of a degenerate hyperbolic equation

TL;DR

This paper investigates the controllability of a degenerate wave equation with bilinear control, establishing thresholds and conditions for local controllability near the ground state depending on the degeneracy parameter and control time.

Contribution

It extends previous work on wave control to the degenerate case, providing new controllability thresholds and geometric descriptions of the reachable sets based on the degeneracy parameter.

Findings

Controllability is achievable for target states near the ground state if time exceeds a critical threshold.

The structure of the reachable set varies with control time and degeneracy parameter, including neighborhoods and submanifolds.

The results generalize classical wave control results to degenerate equations using spectral analysis and Ingham type theorems.

Abstract

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.