One-dimensional wave equation with set-valued boundary damping: well-posedness, asymptotic stability, and decay rates

Abstract

This paper is concerned with the analysis of a 1D wave equation $z_{tt}-z_{xx}=0$ on $[0,1]$ with a Dirichlet condition at $x=0$ and a damping acting at $x=1$ of the form $(z_t(t,1),-z_x(t,1))\in\Sigma$ for $t\geq 0$, where $\Sigma$ is a given subset of $\mathbb R^2$. The study is performed within an $L^p$ functional framework, $p\in [1, +\infty]$. We determine conditions on $\Sigma$ ensuring existence and uniqueness of solutions of that wave equation, its strong stability and uniform global asymptotic stability of the solutions. In the latter case, we study the corresponding decay rates and their optimality. We first establish a correspondence between the solutions of that wave equation and the iterated sequences of a discrete-time dynamical system in terms of which we investigate the above mentioned issues. This enables us to provide a necessary and sufficient condition on $\Sigma$ ensuring existence and uniqueness of solutions of the wave equation and an efficient strategy for determining optimal decay rates when $\Sigma$ verifies a generalized sector condition. In case the boundary damping is subject to perturbations, we derive sharp results regarding asymptotic perturbation rejection and input-to-state issues.

Stabilization of the wave equation through nonlinear Dirichlet actuation

“…Here, using the terminology of [21], the feedback is allowed to be weak, i.e., g(s)/s can go to 0 as |s| goes to infinity, as it is the case when g represents a saturation mapping; then, loss of uniformity is to be expected. More precisely, coming back to the Neumann problem, the one-dimensional version of (1.6) with g given by (1.7) is known to possess weak solutions that decay to zero (in the natural energy space H 1 (Ω) × L 2 (Ω)) slower than any exponential or polynomial, whereas strong solutions decay exponentially to zero but in a non-uniform way -see [21,Theorem 4.1] or also [1,Theorem 4.33]. Proving a similar result in our Dirichlet case would be interesting.…”

“…In the one-dimensional settings, arguments based on Riemann invariants are available, and the decay of the energy can be analyzed via appropriate iterated sequences. See for instance [1], where g is allowed to be a multivalued monotone mapping, or [21], where it is proved, in particular, that exponential or polynomial uniform decay cannot be achieved when g represents a pointwise saturation mapping -see also [22] or [16] for a stability analysis in the saturated case.…”

Boundary sliding mode control of a system of linear hyperbolic equations: A Lyapunov approach

“…Indeed, most of the existing works give their stability results in the Hilbertian space L 2 . Let us mention however some recent works in that direction: [12], [3] or [11]. We also emphasize on the fact that our approach is adjustable to more general SMC methods such as adaptive approaches.…”