L-asymptotic stability analysis of a 1D wave equation with a nonlinear damping

Chitour, Yacine; Marx, Swann; Prieur, Christophe

doi:10.1016/j.jde.2020.06.007

Cited by 17 publications

(30 citation statements)

References 25 publications

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“…Other functional frameworks have been considered recently [2,7,15], where the functional spaces are of L p -type, p ∈ [1, +∞], but these works consider 1D wave equations with localized distributed damping, i.e., z tt − z xx = −a(x)σ(z t ). Recall that the semigroup generated by the D'Alembertian z := z tt − ∆z with Dirichlet boundary conditions on an open bounded subset in R n , n ≥ 2, is not defined for any suitable extension of the Hilbertian framework to L p -type spaces for p = 2, as explained in [25].…”

Section: Existing Resultsmentioning

confidence: 99%

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: Existing Resultsmentioning

confidence: 99%

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

2021

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“…In other words, the items (i) and (ii) of Theorem 1 will be equivalently proved in the coordinates (A.1). Using the Duhamel formula (i.e., variation of constants formula), one can write the solution to (A.1) with an integral formula which exists thanks to a Banach fixed point theorem, as it has been done in [37]. Thus, one can prove that solutions of the system (A.1) exist and are unique for some small interval of time, uniformly in the initial time t 0 and in the initial conditions (ζ 0 , e 0 , φ 0 ).…”

Section: Discussionmentioning

confidence: 99%

Repetitive control design based on forwarding for nonlinear minimum-phase systems

2021

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We propose a new design for a repetitive control scheme for nonlinear minimum-phase systems with arbitrary relative degree and globally Lipschitz nonlinearities. We represent the delay of the repetitive control scheme as a transport equation and we propose a new forwarding-based (partial) state-feedback design that uses not only the boundary information of the delay, but the entire state of the transport equation representing the delay. Through a rigorous mathematical analysis, we show that, from a theoretical point of view, asymptotic convergence of the desired regulated output can be achieved with the proposed control design.

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“…A typical example of such a nonlinearity is the saturation, but we consider a larger class of nonlinearity, namely cone-bounded nonlinearities, as in [21]. Note that very few is known about methods for the stabilization of nonlinear PDEs, but let us mention [23], which deals with the stabilization of a nonlinear KdV subject to a saturation, [12,31], which addresses the stabilization problem of some wave equations with saturated controller; or [24,34], which builds Lyapunov functionals for abstract systems. However, these papers do not deal with coupled systems.…”

Section: Introductionmentioning

confidence: 99%

Forwarding techniques for the global stabilization of dissipative infinite-dimensional systems coupled with an ODE

Marx

Brivadis

Astolfi

2021

Math. Control Signals Syst.

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This paper deals with the stabilization of a coupled system composed by an infinite-dimensional system and an ODE. Moreover, the control, which appears in the dynamics of the ODE, is subject to a general class of nonlinearities. Such a situation may arise, for instance, when the actuator admits a dynamics. The openloop ODE is exponentially stable and the open-loop infinite-dimensional system is dissipative, i.e., the energy is nonincreasing, but its equilibrium point is not necessarily attractive. The feedback design is based on an extension of a finite-dimensional method, namely the forwarding method. We propose some sufficient conditions that imply the well-posedness and the global asymptotic stability of the closed-loop system. As illustration, we apply these results to a transport equation coupled with an ODE.

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L-asymptotic stability analysis of a 1D wave equation with a nonlinear damping

Cited by 17 publications

References 25 publications

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

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

Repetitive control design based on forwarding for nonlinear minimum-phase systems

Forwarding techniques for the global stabilization of dissipative infinite-dimensional systems coupled with an ODE

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