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.

[1] N. Anantharaman, Entropy and the localization of eigenfunctions, Ann. of Math. 2 (2008), Vol. 168, 435–475. | MR | Zbl

[2] C. Bardos, G. Lebeau, J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992), no. 5, 1024–1065. | MR | Zbl

[3] F. Bonechi, S. De Bièvre, Controlling strong scarring for quantized ergodic toral automorphisms, Duke Math. J. 117 (2003), no. 3, 571–587. | MR | Zbl

[4] N. Burq, Large-time dynamics for the one-dimensional Schrödinger equation, Proc. Roy. Soc. Edinburgh Sect. A. 141 (2011), no. 2, 227–251. | MR | Zbl

[5] N. Burq, N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math. 173 (2008), no. 3, 449–475. | MR | Zbl

[6] G. Chen, S.A. Fulling, F.J. Narcowich, S. Sun, Exponential decay of energy of evolution equations with locally distributed damping, SIAM J. Appl. Math. 51 (1991), no. 1, 266–301. | MR | Zbl

[7] F. Faure, S. Nonnenmacher, S. De Bièvre, Scarred eigenstates for quantum cat maps of minimal periods, Comm. Math. Phys. 239 (2003), no. 3, 449–492. | MR | Zbl

[8] P. Gérard, E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. J. 71 (1993), 559–607. | MR | Zbl

[9] A. Hassell, Ergodic billiards that are not quantum unique ergodic, Ann. Math. (2) 171 (2010), 605–619. | MR | Zbl

[10] A. Hassell, S. Zelditch, Quantum ergodicity of boundary values of eigenfunctions, Comm. Math. Phys. 248 (2004), no. 1, 119–168. | MR | Zbl

[11] P. Hébrard, A. Henrot, Optimal shape and position of the actuators for the stabilization of a string, Syst. Cont. Letters 48 (2003), 199–209. | MR | Zbl

[12] P. Hébrard, A. Henrot, A spillover phenomenon in the optimal location of actuators, SIAM J. Control Optim. 44 2005, 349–366. | MR | Zbl

[13] S. Kerckhoff, H. Masur, J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. (2) 124 (1986), no. 2, 293–311. | MR | Zbl

[14] S. Kumar, J.H. Seinfeld, Optimal location of measurements for distributed parameter estimation, IEEE Trans. Autom. Contr. 23Ê(1978), 690–698. | Zbl

[15] V.F. Lazutkin, On the asymptotics of the eigenfunctions of the Laplacian, Soviet Math. Dokl. 12 (1971), 1569–1572. | Zbl

[16] G. Lebeau, Contrôle de l’equation de Schrödinger, J. Math. Pures Appl. 71 (1992), 267–291. | MR | Zbl

[17] K. Morris, Linear-quadratic optimal actuator location, IEEE Trans. Automat. Control 56 (2011), no. 1, 113–124. | MR

[18] Y. Privat, E. Trélat, E. Zuazua, Optimal observability of the one-dimensional wave equation, Preprint Hal (2012).

[19] Y. Privat, E. Trélat, E. Zuazua, Optimal location of controllers for the one-dimensional wave equation, Preprint Hal (2012). | MR

[20] Y. Privat, E. Trélat, E. Zuazua, Optimal observability of wave and Schrödinger equations in ergodic domains, ongoing work (2012).

[21] Z. Rudnick, P. Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161 (1994), no. 1, 195–213. | MR | Zbl

[22] M. van de Wal, B. Jager, A review of methods for input/output selection, Automatica 37 (2001), no. 4, 487–510. | MR | Zbl

[23] S. Zelditch, Note on quantum unique ergodicity, Proc. Amer. Math. Soc. 132 (2004), 1869–1872. | MR | Zbl

[24] S. Zelditch, M. Zworski, Ergodicity of eigenfunctions for ergodic billiards, Comm. Math. Phys. 175 (1996), no. 3, 673–682. | MR | Zbl