A Note on the Semiglobal Controllability of the Semilinear Wave Equation

Joly, Romain; Laurent, Camille

doi:10.1137/120891174

Cited by 11 publications

(13 citation statements)

References 30 publications

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“…This type of non-linear stabilisation results is also closely related to the problem of global control of the non-linear wave equation, see [9], [16] and [17]. For example, one gets the following result in dimension d = 1.…”

Section: Global Attractor and Stabilisation For The Non-linear Equationmentioning

confidence: 83%

Exponential decay for the damped wave equation in unbounded domains

Burq

Joly

2016

Commun. Contemp. Math.

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We study the decay of the semigroup generated by the damped wave equation in an unbounded domain. We first prove under the natural geometric control condition the exponential decay of the semigroup. Then we prove under a weaker condition the logarithmic decay of the solutions (assuming that the initial data are smoother). As corollaries, we obtain several extensions of previous results of stabilisation and control.Onétudie la décroissance du semi-groupe des ondes amorties dans un domaine non borné. Notre premier résultat est que, sous une hypothèse naturelle de contrôle géométrique, le semigroupe décroît exponentiellement vite. On démontre ensuite sous une hypothèse plus faible la décroissance logarithmique des solutions associéesà des données initiales plus régulières. On obtient en corollaire plusieurs généralisations de résultats de stabilisation et de contrôle.

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Section: Global Attractor and Stabilisation For The Non-linear Equationmentioning

confidence: 83%

Exponential decay for the damped wave equation in unbounded domains

Burq

Joly

2016

Commun. Contemp. Math.

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“…2). There exists some r > 0 and some constant C(T 0 ) > 0 such that (10.9) inf u∈L 2 ((0,T0)×ω) y(0)=y 0 y(T0)=y In the particular case of the semilinear wave equation, the above assumption is satisfied 21 when, in addition to (10.3), f (0) = 0 (this is generally needed for ensuring (10.9) -(10.10)), and under specific geometric assumptions on ω ⊂ Ω, which needs to satisfy a slightly stronger condition than just GCC, namely the so-called multiplier condition (see [194,193,105], and [54, Remark 10], as well as the setting presented in the latter paper). We state this as an explicit hypothesis in order to render transparent the needed elements for generalizing the strategy to other systems 22 .…”

Section: Theorem 810 ([145]mentioning

confidence: 99%

“…We state this as an explicit hypothesis in order to render transparent the needed elements for generalizing the strategy to other systems 22 . 21 A subtile point regarding possible extensions to non-globally Lipschitz nonlinearities is that the controllability time T 0 may depend on the initial datum y 0 (see [105]). But in the big picture of turnpike this is not necessarily an issue, since we are looking at T 1.…”

Section: Theorem 810 ([145]mentioning

confidence: 99%

Turnpike in optimal control of PDEs, ResNets, and beyond

Geshkovski¹,

Zuazua²

2022

Preprint

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The turnpike property in contemporary macroeconomics asserts that if an economic planner seeks to move an economy from one level of capital to another, then the most efficient path, as long as the planner has enough time, is to rapidly move stock to a level close to the optimal stationary or constant path, then allow for capital to develop along that path until the desired term is nearly reached, at which point the stock ought to be moved to the final target. Motivated in part by its nature as a resource allocation strategy, over the past decade, the turnpike property has also been shown to hold for several classes of partial differential equations arising in mechanics. When formalized mathematically, the turnpike theory corroborates the insights from economics: for an optimal control problem set in a finite-time horizon, optimal controls and corresponding states, are close (often exponentially), during most of the time, except near the initial and final time, to the optimal control and corresponding state for the associated stationary optimal control problem. In particular, the former are mostly constant over time. This fact provides a rigorous meaning to the asymptotic simplification that some optimal control problems appear to enjoy over long time intervals, allowing the consideration of the corresponding stationary problem for computing and applications. We review a slice of the theory developed over the past decade -the controllability of the underlying system is an important ingredient, and can even be used to devise simple turnpike-like strategies which are nearly optimal-, and present several novel applications, including, among many others, the characterization of Hamilton-Jacobi-Bellman asymptotics, and stability estimates in deep learning via residual neural networks.

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“…Theorem 1 extends to the multi-dimensional case the result of [29] devoted to the one dimensional case under the condition lim sup |r|→∞ |g(r)| |r| ln 2 |r| = 0, relaxed later on in [2], following [9], and in [21]. The exact controllability for subcritical nonlinearities is obtained in [7] assuming the sign condition rg(r) ≥ 0 for every r ∈ R. This latter assumption has been weakened in [12] to an asymptotic sign condition leading to a semi-global controllability result in the sense that the final data (z 0 , z 1 ) is prescribed in a precise subset of V . In this respect, we also mention in the one dimensional case [6] where a positive boundary controllability result is proved for a steady-state initial and final data specific class of initial and final data and for T large enough by a quasi-static deformation approach.…”

Section: Introductionmentioning

confidence: 96%

Constructive proof of the exact controllability for semi-linear wave equations

Lemoine,

Münch

2021

Preprint

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The exact distributed controllability of the semilinear wave equation ∂tty − ∆y + g(y) = f 1ω posed over multi-dimensional and bounded domains, assuming that g ∈ C 1 (R) satisfies the growth condition lim sup r→∞ g(r)/(|r| ln 1/2 |r|) = 0 has been obtained by Fu, Yong and Zhang in 2007. The proof based on a non constructive Leray-Schauder fixed point theorem makes use of precise estimates of the observability constant for a linearized wave equation. Assuming that g ′ does not grow faster than β ln 1/2 |r| at infinity for β > 0 small enough and that g ′ is uniformly Hölder continuous on R with exponent s ∈ (0, 1], we design a constructive proof yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order 1 + s after a finite number of iterations.

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A Note on the Semiglobal Controllability of the Semilinear Wave Equation

Abstract: International audienc

Cited by 11 publications

References 30 publications

Exponential decay for the damped wave equation in unbounded domains

Exponential decay for the damped wave equation in unbounded domains

Turnpike in optimal control of PDEs, ResNets, and beyond

Constructive proof of the exact controllability for semi-linear wave equations

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