Abstract
This paper is concerned with the analysis of a one dimensional wave equation ztt − zxx = 0 on [0, 1] with a Dirichlet condition at x = 0 and a damping acting at x = 1 which takes the form (zt(t, 1), −zx(t, 1)) ∈ Σ for every t ≥ 0, where Σ is a given subset of ℝ2. The study is performed within an Lp functional framework, p ∈ [1, +∞]. We aim at determining conditions on Σ ensuring existence and uniqueness of solutions of that wave equation as well as strong stability and uniform global asymptotic stability of its solutions. In the latter case, we also study the decay rates of the solutions and their optimality. We first establish a one-to-one 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 simple necessary and sufficient condition on Σ ensuring existence and uniqueness of solutions of the wave equation as well as an efficient strategy for determining optimal decay rates when Σ verifies a generalized sector condition. As an application, we solve two conjectures stated in the literature, the first one seeking a specific optimal decay rate and the second one associated with a saturation type of damping. In case the boundary damping is subject to perturbations, we derive sharp results regarding asymptotic perturbation rejection and input-to-state issues.
Highlights
The norm · Xp, previously used, for instance, in [14], is equivalent to the standard norm in Xp, but it has the advantage of being well-adapted to the analysis of wave equations, since it is expressed in terms of Riemann invariants and it is nonincreasing as soon as Σ satisfies a damping condition
After the present introduction describing the contents of our paper and gathering the main notations, Section 2 is devoted to the description of the precise correspondence between the wave equation described by (1.1) and the discrete-time dynamical system given in (1.4), as well as the list of meaningful hypotheses one can assume on Σ and auxiliary results on the set-valued map S
We focus on the case n = 2 due to Proposition 2.18(e) as well as to the fact that, from Theorem 4.14, S[2] being a strict damping is a necessary condition for the strong stability of (1.1), which is not the case for S due to Example 4.4
Summary
Keywords and phrases: Wave equation, set-valued boundary condition, saturation, well-posedness, stability, asymptotic behavior. Defining Σ as the set of pairs (x, y) ∈ R2 such that y ∈ σi(x) for some i ∈ I, one obtains that any solution of the wave equation with the boundary condition (1.3) is a solution of (1.1) This construction is the analogue of the classical transformation of finite-dimensional switched systems into differential inclusions (see, e.g., [4, 11, 18])
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