OPTIMAL SHAPE AND LOCATION OF SENSORS OR ACTUATORS IN PDE MODELS

摘要

We report on a series of works done in collaboration with Y. Privat and E. Zuazua, concerning the problem of optimizing the shape and location of sensors and actuators for systems whose evolution is driven by a linear partial differential equation. This problem is frequently encountered in applications where one wants to optimally design sensors in order to maximize the quality of the reconstruction of solutions by using only partial observations, or to optimally design actuators in order to control a given process with minimal efforts. For example, we model and solve the following informal question: what is the optimal shape and location of a thermometer? Note that we want to optimize not only the placement but also the shape of the observation or control subdomain over the class of all possible measurable subsets of the domain having a prescribed Lebesgue measure. By probabilistic considerations we model this optimal design problem as the one of maximizing a spectral functional interpreted as a randomized observability constant, which models optimal observabnility for random initial data. Solving this problem strongly depends on the operator in the PDE model and requires fine knowledge on the asymptotic properties of eigenfunctions of that operator. For parabolic equations like heat, Stokes or anomalous diffusion equations, we prove the existence and uniqueness of a best domain, proved to be regular enough, and whose algorithmic construction depends in general on a finite number of modes. In contrast, for wave or Schr?dinger equations, relaxation may occur and our analysis reveals intimate relations with quantum chaos, more precisely with quantum ergodicity properties of the Laplacian eigenfunctions.