Nonlinear Neumann boundary stabilization of the wave equation using rotated multipliers

Cornilleau, Pierre; Lohéac, Jean-Pierre; Osses, Axel

doi:10.1007/s10883-010-9088-6

Cited by 16 publications

(10 citation statements)

References 7 publications

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“…In such a situation the type of approximating problems we consider, appear naturally in control theory/stabilization of waves, see [15,44,59,47,48], or in homogenization of vibration problems with inclusions near the boundary, see [51,28,29,17] and references therein. On the other hand, the limit problem appears in the boundary control theory, see [40,41,39,67,50,19] and references therein. Second, in Section 7 we explore the situation in which the damping and linear potential concentrate around a smooth, compact orientable hypersurface without boundary, M, contained in Ω, that is not touching the boundary Γ.…”

Section: Pde Problems With Concentrating Terms Near the Boundary 2149mentioning

confidence: 99%

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PDE problems with concentrating terms near the boundary

Jiménez-Casas¹,

Rodriguez-Bernal²

2020

Communications on Pure &Amp; Applied Analysis

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In this paper we study several PDE problems where certain linear or nonlinear termsin the equation concentrate in the domain, typically (but not exclusively) near the boundary. We analyze some linear and nonlinear elliptic models, linear and nonlinear parabolic ones as well as some damped wave equations. We show that in all these singularly perturbed problems, the concentrating terms give rise in the limit to a modification in the original boundary condition of the problem. Hence we describe in each case which is the singular limit problem and analyze the convergence of solutions. 1. Introduction. In this paper we consider several types of PDE problems in which certain terms in the equations concentrate, as a parameter ε → 0. Typically, near the boundary of the domain. This implies that the problem under consideration are subjected to singular perturbations that drastically change the nature of the problem, when passing to the limit. In all the problems considered the goal is then to identify the form of the limit problem and to describe the process of convergence of solutions, as ε → 0. To make the notations apparent, we will consider Ω, an open bounded smooth set in IR N with a C 2 boundary ∂Ω. Let Γ = ∂Ω and consider the subset of Ω ω ε = {x ∈ Ω : dist(x, Γ) < ε} for sufficiently small ε, say 0 < ε < ε 0. Notice that this set has measure |ω ε | ∼ ε|Γ| for small values of ε. For sufficiently small σ ≥ 0 we can define the "parallel" interior boundary Γ σ = {y − σ n(y), y ∈ Γ} where n(x) denotes the outward normal vector. Note that Γ 0 = Γ and that for small ε, the set ω ε is a neighborhood of Γ in Ω, that collapses to the boundary when the parameter ε goes to zero.

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Section: Pde Problems With Concentrating Terms Near the Boundary 2149mentioning

confidence: 99%

“…Then, we can assume, by taking subsequences if necessary, the following convergences as ε → 0: 19) see Lemma 5.2 ii). With these assumptions consider the mild solutions, u ε (t) and u 0 (t) of (6.1) and (6.2), respectively, constructed in Sections 6.1 and 6.2.…”

Section: Passing To the Limitmentioning

confidence: 99%

PDE problems with concentrating terms near the boundary

Jiménez-Casas¹,

Rodriguez-Bernal²

2020

Communications on Pure &Amp; Applied Analysis

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“…Moreover, Lebeau and Robbiano (see [23]) have shown that, in the case where the Neumann boundary condition is applied on the entire boundary, a weak condition on the feedback (which does not satisfy Geometric Control Condition) provides logarithmic decay of regular solutions. On the other hand, multiplier techniques (see [15,8]) give some results of exponential stabilization (even if the part of the boundary driven by the homogeneous Dirichlet condition touches the region where the feedback is applied) but under very strong assumptions on the form of the boundary conditions. Our goal here is to obtain some stabilization of logarithmic type under weak assumptions for the boundary conditions.…”

Section: General Backgroundmentioning

confidence: 99%

Carleman estimates for the Zaremba Boundary Condition and Stabilization of Waves

Cornilleau¹,

Robbiano²

2014

ajm

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In this paper, we shall prove a Carleman estimate for the so-called Zaremba problem. Using some techniques of interpolation and spectral estimates, we deduce a result of stabilization for the wave equation by means of a linear Neumann feedback on the boundary. This extends previous results from the literature: indeed, our logarithmic decay result is obtained while the part where the feedback is applied contacts the boundary zone driven by an homogeneous Dirichlet condition. We also derive a controllability result for the heat equation with the Zaremba boundary condition.

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“…If (H1) holds, the geometric optics condition insures the exponential stability of (5.7) with γ = 1, see [6]. If (H2) holds, the exponential stability of (5.7) with γ = 1 follows from Cornilleau et al [9,Remark 2].…”

Section: The Wave Equationmentioning

confidence: 99%

Recovering the initial state of an infinite-dimensional system using observers

2010

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. Recovering the initial state of an infinitedimensional system using observers. Automatica, Elsevier, 2010, 46 (10) Abstract: Let A be the generator of a strongly continuous semigroup T on the Hilbert space X, and let C be a linear operator from D(A) to another Hilbert space Y (possibly unbounded with respect to X, not necessarily admissible). We consider the problem of estimating the initial state z 0 ∈ D(A) (with respect to the norm of X) from the output function y(t) = CT t z 0 , given for all t in a bounded interval [0, τ ]. We introduce the concepts of estimatability and backward estimatability for (A, C) (in a more general way than currently available in the literature), we introduce forward and backward observers, and we provide an iterative algorithm for estimating z 0 from y. This algorithm generalizes various algorithms proposed recently for specific classes of systems and it is an attractive alternative to methods based on inverting the Gramian. Our results lead also to a very general formulation of Russell's principle, i.e., estimatability and backward estimatability imply exact observability. This general formulation of the principle does not require T to be invertible. We illustrate our estimation algorithms on systems described by wave and Schrödinger equations, and we provide results from numerical simulations.

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Nonlinear Neumann boundary stabilization of the wave equation using rotated multipliers

Cited by 16 publications

References 7 publications

PDE problems with concentrating terms near the boundary

PDE problems with concentrating terms near the boundary

Carleman estimates for the Zaremba Boundary Condition and Stabilization of Waves

Recovering the initial state of an infinite-dimensional system using observers

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