Explicit observability estimate for the wave equation with potential and its application

Zhang, X u

doi:10.1098/rspa.2000.0553

Cited by 79 publications

(79 citation statements)

References 12 publications

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“…Our approach, stimulated by [10] (see also [6,8,17,18]), is different from that in [1], which instead employed the classical local Carleman estimate and therefore needs C ∞ -regularity for the data. It would be quite interesting to establish better decay rate (than logarithmic decay) for system (1.5) under further conditions (without geometric optics condition).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Logarithmic Decay of Hyperbolic Equations with Arbitrary Small Boundary Damping

2010

Series in Contemporary Applied Mathematics

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In this paper, we study the logarithmic stability for the hyperbolic equations by arbitrary boundary observation. Based on Carleman estimate, we first prove an estimate of the resolvent operator of such equation. Then we prove the logarithmic stability estimate for the hyperbolic equations without any assumption on an observation subboundary.2000 Mathematics Subject Classification. Primary 93B05; Secondary 93B07, 35B37.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Logarithmic Decay of Hyperbolic Equations with Arbitrary Small Boundary Damping

2010

Series in Contemporary Applied Mathematics

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“…The proof of proposition 5 can be found in [11]. There, the way the constant C 2 depends on c ∞ is explicitly indicated.…”

Section: Fernández-cara González-burgos and De Teresamentioning

confidence: 99%

“…From the expressions of the function ζ and the constants C 1 and C 2 that can be found in [6], [5] and [11], it is not difficult to deduce an estimate of the constant C in (30) in terms of a ∞ and b ∞ . More precisely, C can be taken of the form…”

Section: Fernández-cara González-burgos and De Teresamentioning

confidence: 99%

Null-exact controllability of a semilinear cascade system of parabolic-hyperbolic equations

Fernández-Cara¹,

González-Burgos²,

Teresa³

2006

Communications on Pure &Amp; Applied Analysis

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Abstract. This paper is concerned with the null-exact controllability of a cascade system formed by a semilinear heat and a semilinear wave equation in a cylinder Ω×(0, T ). More precisely, we intend to drive the solution of the heat equation (resp. the wave equation) exactly to zero (resp. exactly to a prescribed but arbitrary final state). The control acts only on the heat equation and is supported by a set of the form ω × (0, T ), where ω ⊂ Ω. In the wave equation, the restriction of the solution to the heat equation to another set O × (0, T ) appears. The nonlinear terms are assumed to be globally Lipschitz-continuous. In the main result in this paper, we show that, under appropriate assumptions on T , ω and O, the equations are simultaneously controllable.

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“…There is an extensive literature providing observability results for wave, plate and Schrödinger equations, among other models, and by various methods including microlocal analysis [3,4], multipliers techniques [29,40], Carleman estimates [25,48], Ingham type inequalities [27,20], etc. Our goal in this paper is to develop a theory allowing to get observability results for space semidiscrete systems as a direct consequence of those corresponding to the continuous ones, thus avoiding technical developments in the discrete setting.…”

Section: Introductionmentioning

confidence: 99%

Resolvent estimates in controllability theory and applications to the discrete wave equation

Ervedoza¹

2011

Journées Équations Aux Dérivées Partielles

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Resolvent estimates in controllability theory and applications to the discrete wave equationResolvent estimates in controllability theory and applications to the discrete wave equation Sylvain Ervedoza AbstractWe briefly present the difficulties arising when dealing with the controllability of the discrete wave equation, which are, roughly speaking, created by highfrequency spurious waves which do not travel. It is by now well-understood that such spurious waves can be dealt with by applying some convenient filtering technique. However, the scale of frequency in which we can guarantee that none of these non-traveling waves appears is still unknown in general. Though, using Hautus tests, which read the controllability of a given system in terms of resolvent estimates, we are able to prove that these spurious waves do not appear before some frequency scale. This document is based on the articles [12,13,14].

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Explicit observability estimate for the wave equation with potential and its application

Abstract: By means of the Carleman-type estimate, we obtain an explicit observability estimate for the wave equation with a potential. As its application, we get the exact internal controllability of the semilinear wave equations in any space dimensions.

Cited by 79 publications

References 12 publications

Logarithmic Decay of Hyperbolic Equations with Arbitrary Small Boundary Damping

Logarithmic Decay of Hyperbolic Equations with Arbitrary Small Boundary Damping

Null-exact controllability of a semilinear cascade system of parabolic-hyperbolic equations

Resolvent estimates in controllability theory and applications to the discrete wave equation

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