Optimal shape and location of sensors for parabolic equations with random initial data

摘要

In this article, we consider parabolic equations on a bounded open connectedsubset $\Omega$ of $\R^n$. We model and investigate the problem of optimalshape and location of the observation domain having a prescribed measure. Thisproblem is motivated by the question of knowing how to shape and place sensorsin some domain in order to maximize the quality of the observation: forinstance, what is the optimal location and shape of a thermometer? We show thatit is relevant to consider a spectral optimal design problem corresponding toan average of the classical observability inequality over random initial data,where the unknown ranges over the set of all possible measurable subsets of$\Omega$ of fixed measure. We prove that, under appropriate sufficient spectralassumptions, this optimal design problem has a unique solution, depending onlyon a finite number of modes, and that the optimal domain is semi-analytic andthus has a finite number of connected components. This result is in strongcontrast with hyperbolic conservative equations (wave and Schr\"odinger)studied in [56] for which relaxation does occur. We also provide examples ofapplications to anomalous diffusion or to the Stokes equations. In the casewhere the underlying operator is any positive (possible fractional) power ofthe negative of the Dirichlet-Laplacian, we show that, surprisingly enough, thecomplexity of the optimal domain may strongly depend on both the geometry ofthe domain and on the positive power. The results are illustrated with severalnumerical simulations.