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

Yannick Privat[1]; Emmanuel Trélat[2]; Enrique Zuazua[3]

  • [1] IRMAR, ENS Cachan Bretagne Univ. Rennes 1, CNRS, UEB, av. Robert Schuman, 35170 Bruz, France
  • [2] Université Pierre et Marie Curie (Univ. Paris 6) and Institut Universitaire de France, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
  • [3] BCAM - Basque Center for Applied Mathematics, Mazarredo, 14 E-48009 Bilbao-Basque Country-Spain. Ikerbasque, Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, 48011, Bilbao-Basque Country-Spain

Journées Équations aux dérivées partielles (2012)

  • Volume: 18, Issue: 5, page 1-13
  • ISSN: 0752-0360

Abstract

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This paper is a proceedings version of the ongoing work [20], 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 Ω ⊂ ℝ n , with Dirichlet boundary conditions. The observation is done on a subset ω of Lebesgue measure | ω | = L | Ω | , where L ∈ ( 0 , 1 ) is fixed. We denote by 𝒰 L the class of all possible such subsets. Let T > 0 . We consider first the benchmark problem of maximizing the observability energy ∫ 0 T ∫ ω | y ( t , x ) 2 d x d t over 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 Ω . We are able to compute the optimal value under strong ergodicity properties on Ω (namely, Quantum Unique Ergodicity). We then provide comments on ergodicity issues, on the existence of an optimal set, and on spectral approximations.

How to cite

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Privat, Yannick, Trélat, Emmanuel, and Zuazua, Enrique. "On the best observation of wave and Schrödinger equations in quantum ergodic billiards." Journées Équations aux dérivées partielles 18.5 (2012): 1-13. <http://eudml.org/doc/275590>.

@article{Privat2012,
abstract = {This paper is a proceedings version of the ongoing work [20], 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 $\Omega \subset \mathbb\{R\}^n$, 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 $T&gt;0$. 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.},
affiliation = {IRMAR, ENS Cachan Bretagne Univ. Rennes 1, CNRS, UEB, av. Robert Schuman, 35170 Bruz, France; Université Pierre et Marie Curie (Univ. Paris 6) and Institut Universitaire de France, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France; BCAM - Basque Center for Applied Mathematics, Mazarredo, 14 E-48009 Bilbao-Basque Country-Spain. Ikerbasque, Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, 48011, Bilbao-Basque Country-Spain},
author = {Privat, Yannick, Trélat, Emmanuel, Zuazua, Enrique},
journal = {Journées Équations aux dérivées partielles},
keywords = {Wave equation; Schrödinger equation; observability inequality; optimal design; ergodic properties; Quantum Unique Ergodicity; wave equation; spectral decomposition; quantum ergodicity},
language = {eng},
number = {5},
pages = {1-13},
publisher = {Groupement de recherche 2434 du CNRS},
title = {On the best observation of wave and Schrödinger equations in quantum ergodic billiards},
url = {http://eudml.org/doc/275590},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Privat, Yannick
AU - Trélat, Emmanuel
AU - Zuazua, Enrique
TI - On the best observation of wave and Schrödinger equations in quantum ergodic billiards
JO - Journées Équations aux dérivées partielles
PY - 2012
PB - Groupement de recherche 2434 du CNRS
VL - 18
IS - 5
SP - 1
EP - 13
AB - This paper is a proceedings version of the ongoing work [20], 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 $\Omega \subset \mathbb{R}^n$, 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 $T&gt;0$. 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.
LA - eng
KW - Wave equation; Schrödinger equation; observability inequality; optimal design; ergodic properties; Quantum Unique Ergodicity; wave equation; spectral decomposition; quantum ergodicity
UR - http://eudml.org/doc/275590
ER -

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