Saturated Boundary Feedback Stabilization of a Linear Wave Equation

Xu, Cheng‐Zhong; Xu, Guan

doi:10.1137/15m1034350

Cited by 9 publications

(11 citation statements)

References 17 publications

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“…Finally, we revisit the case where Σ is equal to the graph of the sign set-valued map sgn and extend all the results obtained in [28] regarding this question. In particular, we provide optimal results for existence and uniqueness of solutions of (1) in any X p , p ∈ [1, +∞] without relying on semigroup theory and we characterize the ω-limit set of every solution of (1) in an explicit manner in terms of the initial condition.…”

Section: Resultsmentioning

confidence: 61%

“…Besides the linear cone condition, several results establishing UGAS have been obtained, cf. [18,20,21,28] where σ verifies a linear cone condition for large s and is either of polynomial type or weaker than any polynomial in a neighborhood of the origin, see also [27] for a extensive list of references. It has to be noticed that many of these studies deal with wave equations in dimension not necessarily equal to one and, for all of them, the estimates are obtained by refined arguments based on the multiplier method or highly nontrivial Lyapunov functionals.…”

Section: Existing Resultsmentioning

confidence: 99%

“…Thanks to this fact, it is possible to obtain existence and uniqueness of solutions of (1) for a large variety of sets Σ. For instance, this allows us to consider Σ as graphs of discontinuous functions completed by vertical segments at jump discontinuities without facing the usual issues addressed in [12,28]. An example of the above discussion is the sign function, illustrated in Figure 4 and treated in more details in Section 6.…”

Section: Existence and Uniqueness Of Solutions In X Pmentioning

confidence: 99%

“…Indeed, using an inductive argument, the above inequality holds trivially for n = 0, while the induction step follows by considering separately the cases where |k Remark 65. To the best of our knowledge, few results on (74) have been obtained and the most precise ones can be found in [28], which considers the case where Σ is the graph of a saturation function and essentially deals with disturbance rejection. Theorem 63(a) improves the results of that reference since the disturbance there is assumed to be matching and regular (both d(•) and d (•) belong to D 1 ) and Remark 64 shows that we can achieve ISS-type of results in X ∞ .…”

Section: Boundary Perturbationsmentioning

confidence: 99%

“…Note that considering Σ more general that the mere graph of a function is a possible alternative to model the fact that the function σ is subject to uncertainties or discontinuities, as in the case for instance, where Σ is equal to the graph of the sign set-valued map sgn : R ⇒ R defined by sgn(s) = {s/|s|} for nonzero s and sgn(0) = [−1, 1], cf. [28]. The use of a set Σ in (1) can also model switching boundary conditions, such as those considered in [3,8,13,17].…”

Section: Introductionmentioning

confidence: 99%

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

2021

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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.

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Section: Resultsmentioning

confidence: 61%