Optimal Observation of the One-dimensional Wave Equation

Privat, Yannick; Trélat, Emmanuel; Zuazua, Enrique

doi:10.1007/s00041-013-9267-4

Cited by 51 publications

(78 citation statements)

References 21 publications

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“…In particular cases it can however be addressed using harmonic analysis. For instance in 1D we have proved in [8] that the supremum is reached if and only if L = 1/2 (and that, in that particular case, there is an infinite number of optimal sets). In multi-D the question is open, and we conjecture that, for generic domains and generic values of L, the supremum is not reached and hence there does not exist any optimal set.…”

Section: Holds True In From the Subset ω If And Only If The Wave Equamentioning

confidence: 99%

“…In this paper we analyse the control and stabilization counterparts of our previous works [8][9][10]12] on the randomised observability for wave processes.…”

Section: Introductionmentioning

confidence: 99%

“…In Sect. 2.1, we have recalled some results of the paper [12], in which it was shown that the spectral observability inequality can also be recast as property of observability of the wave equation in infinite time (see (8)). This allows also to motivate the fact that, at the control level, we come out with a result in which the control is distributed everywhere in the domain as shown in Theorem 1.…”

Section: Additional Comments and Perspectivesmentioning

confidence: 99%

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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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Section: Holds True In From the Subset ω If And Only If The Wave Equamentioning

confidence: 99%

“…In this paper we analyse the control and stabilization counterparts of our previous works [8][9][10]12] on the randomised observability for wave processes.…”

Section: Introductionmentioning

confidence: 99%

Section: Additional Comments and Perspectivesmentioning

confidence: 99%

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Randomised observation, control and stabilization of waves

2016

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“…The second problem for this one-dimensional case was studied in details in [18]. In particular, in this case the non-uniqueness phenomenon can be exactly characterized in terms of Fourier series.…”

Section: Remarkmentioning

confidence: 99%

“…They are based on a thorough investigation through spectral considerations. In [18] we investigated the second problem presented previously in the one-dimensional case. We also quote the article [19] where we study the related problem of finding the optimal location of the support of the control for the one-dimensional wave equation.…”

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confidence: 99%

On the best observation of wave and Schrödinger equations in quantum ergodic billiards

Privat¹,

Trélat²,

Zuazua³

2013

Journées Équations Aux Dérivées Partielles

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International audienceThis paper is a proceedings version of an ongoing work, and has been the object of the talk of the second author at Journées EDP in 2012. In this work we investigate optimal observability properties for wave and Schrödinger equations considered in a bounded open set $\Omega\subset\R^n$, with Dirichlet boundary conditions. The observation is done on a subset $\omega$ of Lebesgue measure $\vert\omega\vert=L\vert\Omega\vert$, where $L\in(0,1)$ is fixed. We denote by $\mathcal{U}_L$ the class of all possible such subsets. Let $T>0$. We consider first the benchmark problem of maximizing the observability energy $\int_0^T\int_\omega\vert y(t,x)^2\, dx\, dt$ over $\mathcal{U}_L$, for fixed initial data. There exists at least one optimal set and we provide some results on its regularity properties. In view of practical issues, it is far more interesting to consider then the problem of maximizing the observability constant. But this problem is difficult and we propose a slightly different approach which is actually more relevant for applications. We define the notion of a randomized observability constant, where this constant is defined as an averaged over all possible randomized initial data. This constant appears as a spectral functional which is an eigenfunction concentration criterion. It can be also interpreted as a time asymptotic observability constant. This maximization problem happens to be intimately related with the ergodicity properties of the domain $\Omega$. We are able to compute the optimal value under strong ergodicity properties on $\Omega$ (namely, Quantum Unique Ergodicity). We then provide comments on ergodicity issues, on the existence of an optimal set, and on spectral approximations

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Remarks on Long Time Versus Steady State Optimal Control

Porretta

Zuazua

2016

Springer INdAM Series

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Optimal Observation of the One-dimensional Wave Equation

Cited by 51 publications

References 21 publications

Randomised observation, control and stabilization of waves

Randomised observation, control and stabilization of waves

On the best observation of wave and Schrödinger equations in quantum ergodic billiards

Remarks on Long Time Versus Steady State Optimal Control

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