A Spillover Phenomenon in the Optimal Location of Actuators

Hébrard, Pascal; Henrot, Antoine

doi:10.1137/s0363012903436247

Cited by 46 publications

(75 citation statements)

References 20 publications

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“…This result is the same as the one in [9,Theorem 3.2]. Note however that the proof in [9] is not completely correct.…”

Section: Corollarysupporting

confidence: 75%

“…Note however that the proof in [9] is not completely correct. In order to prove (25), the authors of [9] use a first-order expansion of…”

Section: Corollarymentioning

confidence: 97%

“…If we fix some initial data (y 0 , y 1 ) ∈ L 2 (0, π) × H −1 (0, π) and some observability time T 2π, we can investigate the problem of maximizing the functional G T (χ ω ) defined by (5) over all possible subsets ω of [0, π] of measure equal to Lπ. According to the expression (9), there exists at least an optimal set ω which can be characterized in terms of a level set of the function…”

Section: Remarkmentioning

confidence: 99%

“…There holds Based on the fact that, for L = 1/2, the optimization problem (17) does not have any solution, and based on the observation that it is more realistic from an engineering point of view to take into consideration only a finite number of modes, it is natural to consider as in [9] a truncated version of (17) involving only the first N modes, for a given N ∈ IN * , and to investigate the optimization problem…”

Section: Corollarymentioning

confidence: 99%

“…Note that the necessity of L N being small enough for (25) to hold has been illustrated on numerical simulations in [9]. The problem of fully understanding the minimization problem (24) and its limit behavior as N tends to infinity for all possible values of L is widely open.…”

Section: Corollarymentioning

confidence: 99%

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Optimal Observation of the One-dimensional Wave Equation

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Zuazua

2013

J Fourier Anal Appl

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In this paper, we consider the homogeneous one-dimensional wave equation on [0, π] with Dirichlet boundary conditions, and observe its solutions on a subset ω of [0, π]. Let L ∈ (0, 1). We investigate the problem of maximizing the observability constant, or its asymptotic average in time, over all possible subsets ω of [0, π] of Lebesgue measure Lπ. We solve this problem by means of Fourier series considerations, give the precise optimal value and prove that there does not exist any optimal set except for L = 1/2. When L = 1/2 we prove the existence of solutions of a convexified minimization problem, proving a no gap result. We then provide and solve a modal approximation of this problem, show the oscillatory character of the optimal sets, the so called spillover phenomenon, which explains the lack of existence of classical solutions for the original problem.

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“…This result is the same as the one in [9,Theorem 3.2]. Note however that the proof in [9] is not completely correct.…”

Section: Corollarysupporting

confidence: 75%

“…Note however that the proof in [9] is not completely correct. In order to prove (25), the authors of [9] use a first-order expansion of…”

Section: Corollarymentioning

confidence: 97%

Section: Remarkmentioning

confidence: 99%

Section: Corollarymentioning

confidence: 99%

Section: Corollarymentioning

confidence: 99%

See 3 more Smart Citations

Optimal Observation of the One-dimensional Wave Equation

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Trélat

Zuazua

2013

J Fourier Anal Appl

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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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