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Optimal observation of the one-dimensional wave equation

Yannick Privat 1, Emmanuel Trélat 2, Enrique Zuazua 3

(2012)

In this paper, we consider the homogeneous one-dimensional wave equation on $[0,\pi]$ with Dirichlet boundary conditions, and observe its solutions on a subset $\omega$ of $[0,\pi]$. Let $L\in(0,1)$. We investigate the problem of maximizing the observability constant, or its asymptotic average in time, over all possible subsets $\omega$ of $[0,\pi]$ of Lebesgue measure $L\pi$. 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 $L \neq 1/2$ 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.

1 :	Institut de Recherche Mathématique de Rennes (IRMAR)
	CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne
2 :	Laboratoire Jacques-Louis Lions (LJLL)
	CNRS : UMR7598 – Université Pierre et Marie Curie [UPMC] - Paris VI
3 :	Basque Center for Applied Mathematics (BCAM)
	Basque Center for Applied Mathematics

Équipe de recherche : Analyse numérique

Domaine

:

Mathématiques/Optimisation et contrôle

Mots Clés : wave equation – observability – optimal design – harmonic analysis

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Contributeur : Emmanuel Trélat <>

Soumis le : Jeudi 15 Mars 2012, 23:52:07

Dernière modification le : Lundi 19 Mars 2012, 09:59:34