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

Privat, Yannick; Trélat, Emmanuel; Zuazua, Enrique

Journées équations aux dérivées partielles (2012), article no. 10, 13 p.

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 $\Omega \subset {ℝ}^n$, with Dirichlet boundary conditions. The observation is done on a subset $\omega$ of Lebesgue measure $\left|\omega \right|=L\left|\Omega \right|$, where $L\in (0,1)$ 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 ${\int }_0^T{\int }_{\omega }{|y(t,x)}^{2}dxdt$ 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 $\Omega$. 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.

Classification : 49J30, 49K20, 35L05, 93B05, 93C20
Mots clés : Wave equation, Schrödinger equation, observability inequality, optimal design, ergodic properties, Quantum Unique Ergodicity.

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