Optimal Shape and Location of Sensors for Parabolic Equations with Random Initial Data

Privat, Yannick; Trélat, Emmanuel; Zuazua, Enrique

doi:10.1007/s00205-014-0823-0

Cited by 58 publications

(91 citation statements)

References 64 publications

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“…We can observe the expected stationarity property of the sequence of optimal domains ω N from N = 4 on (i.e., 16 eigenmodes). These results can be established as well for more general parabolic equations (see [19]), involving in particular anomalous diffusion equations and Stokes equations. Let us mention in particular an interesting feature occuring for the anomalous diffusion equation ∂ t y+(− ) α y = 0 in Ω, where (− ) α is some positive power of the Dirichlet-Laplacian, with arbitrary boundary conditions implying y |∂Ω = 0.…”

Section: Theorem 4 ([19])mentioning

confidence: 69%

“…It is the idea developed in [16], [15], [17], [19] where the problem of the optimal location of an observation subset ω among all possible subsets of a given measure or volume fraction of Ω was addressed and solved for wave and Schrödinger equations and also for general parabolic equations. A relevant spectral criterion, viewed as a measure of eigenfunction concentration and not depending on the initial conditions was considered, in order to design an optimal observation or control set in an uniform way, independent of the data and solutions under consideration, as explained next.…”

Section: A the Contextmentioning

confidence: 99%

“…Theorem 5 ( [19]): For every N ∈ IN * , there exists a unique set ω N such that χ ω N ∈ U L maximizes the functional…”

Section: Theorem 4 ([19])mentioning

confidence: 99%

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Optimal shape and location of sensors or actuators in PDE models

Privat

Trélat

Zuazua

2014

2014 American Control Conference

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Abstract-We investigate the problem of optimizing the shape and location of sensors and actuators for evolution systems driven by distributed parameter systems or partial differential equations (PDE), like for instance the wave equation, the Schrödinger equation, or a parabolic system, on an arbitrary domain Ω, in any space dimension, and with suitable boundary conditions if there is a boundary, which can be of Dirichlet, Neumann, mixed or Robin. This kind of problem is frequently encountered in applications where one aims, for instance, at maximizing the quality of reconstruction of the solution, using only a partial observation. From the mathematical point of view, using probabilistic considerations we model this problem as that of maximizing the so-called randomized observability constant, over all possible subdomains of Ω having a prescribed measure. The spectral analysis of this problem reveals intimate connections with the theory of quantum chaos. More precisely, we provide a solution to this problem when the domain Ω satisfies suitable quantum ergodicity assumptions.

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Section: Theorem 4 ([19])mentioning

confidence: 69%

Section: A the Contextmentioning

confidence: 99%

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Optimal shape and location of sensors or actuators in PDE models

Privat

Trélat

Zuazua

2014

2014 American Control Conference

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“…To see this it is sufficient to multiply by the adjoint state p in the controlled equation (11) and to integrate by parts.…”

Section: Proof Of the Spectral Controllability Resultsmentioning

confidence: 99%

Randomised observation, control and stabilization of waves

2016

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The problems of observing, controlling and stabilizing wave processes arise in many different contexts ranging from structural mechanics to seismic waves. In a suitable functional setting, they are closely interconnected and sometimes completely equivalent. In a series of previous articles we have addressed the problem of the optimal design of sensors for purely conservative wave models. We analyzed a relaxed version of the optimal observation problem, considering the expectation of solutions under a randomisation procedure, rather than that where all possible solutions are considered in a purely deterministic setting. From an analytical point of view, this randomisation procedure had the advantage of leading to a spectral diagonalisation of the observations. In this way, using fine asymptotic spectral properties of the Laplacian, we disclosed the links between the geometric properties of the domain where waves propagate and the existence of optimal locations for the sensors or, by the contrary, the emergence of relaxation phenomena. Here we show that spectral randomised observability is equivalent to the property of spectral controllability by means of a discrete set of lumped controls acting everywhere on the domain, and distributed according to the shape of the eigenfunctions. Our results on optimal observation then find natural equivalents on the problem of optimal spectral control. We also give an interpretation of these results in terms of a feedback stabilization property, ensuring the exponential decay of the energy of solutions as time tends to infinity.

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“…Such issues have been widely discussed in a recent series of articles [19,20]. The authors propose a new average observability concept inspired by [3] and introduce the notion of randomized observability constant.…”

Section: Modeling Of the Problemmentioning

confidence: 99%

Optimal design of boundary observers for the wave equation

2014

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Abstract. In this article, we consider the wave equation on a domain of IR n with Lipschitz boundary. For every observable subset Γ of the boundary ∂Ω (endowed with the usual Hausdorff measure H n−1 on ∂Ω), the observability constant provides an account for the quality of the reconstruction in some inverse problem. Our objective is here to determine what is, in some appropriate sense, the best observation domain. After having defined a randomized observability constant, more relevant tan the usual one in applications, we determine the optimal value of this constant over all possible subsets Γ of prescribed area H n−1 (Γ) = LH n−1 (∂Ω), with L ∈ (0, 1), under appropriate spectral assumptions on Ω. We compute the maximizers of a relaxed version of the problem, and then study the existence of an optimal set of particular domains Ω. We then define and study an approximation of the problem with a finite number of modes, showing existence and uniqueness of an optimal set, and provide some numerical simulations.1. Setting of the shape optimization problem 1.1. A brief state of the art.The literature on the optimal observation or sensor location problems is abundant in engineering applications, where mainly the optimal location of sensors or controllers is investigated. In many contributions, numerical tools are developed to solve a simplified version of the optimal design problem where either the partial differential equation has been replaced with a discrete approximation, or the class of optimal designs is replaced with a compact finite dimensional set (see for example [1,5,22] and [15] where such problems are investigated in a more general setting).From an engineering point of view, the aim is to optimize the number, the position or the shape of sensors in order to improve the estimation of the state of the system. Fields of applications are very numerous, for example structural acoustics, piezoelectric actuators, or vibration control in mechanical structures just to name a few of them, are interested in such a study.In this study, we model this issue by maximizing an observability constant with respect to the shape an location of the sensors. We refer to [21,23] for a general presentation of observability inequalities for wave like systems. We adopt here the "randomized" approach proposed in [18,19] Article published online by EDP Sciences and available at http://www.esaim-proc.org or http://dx

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Optimal Shape and Location of Sensors for Parabolic Equations with Random Initial Data

Cited by 58 publications

References 64 publications

Optimal shape and location of sensors or actuators in PDE models

Optimal shape and location of sensors or actuators in PDE models

Randomised observation, control and stabilization of waves

Optimal design of boundary observers for the wave equation

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