Optimal sensor location for wave and Schrödinger equations

This paper summarizes the research we have carried out recently on the problem of the optimal location of sensors and actuators for wave equa- tions, which has been the object of the talk of the third author at the Hyp2012 Conference held in Padova (Italy). We also address the same issues for the Schro ̈dinger equations and present some possible perspectives of future re- search. We consider the multi-dimensional wave or Schro ̈dinger equations in a bounded domain Ω, with usual boundary conditions (Dirichlet, Neumann or Robin). We investigate the problem of optimal sensor location, in other words, the problem of designing what is the best possible subdomain of a prescribed measure on which one can observe the solutions. We present two mathematical problems modeling this question. The first one, in which the initial data under consideration are fixed, leads to optimal sets whose complexity depends on the regularity of the initial data. In the second one, the optimal set is searched so as to be uniform with respect to all initial data, and leads to a criterium of spec- tral nature, the answer being intimately related to the concentration properties of the eigenfunctions of the Laplacian. Under quantum ergodicity assumptions on the domain Ω we compute the optimal value of this problem, and show that this optimal value can be interpreted as the best possible observability constant of a corresponding time-asymptotic or randomized observability in- equality. Although optimal sets do exist in some specific situations, we show that the existence of an optimal set cannot be expected in general. Finally, we study a spectral approximation of that problem and construct a maximizing sequence of sets