Optimal location of sensors and actuators for wave and heat processes

Yannick Privat (1) , Emmanuel Trélat (1) , Enrique Zuazua (1, 2)

1 LJLL - Laboratoire Jacques-Louis Lions
2 Departamento de Matemáticas [Madrid]

We investigate the problem of optimizing the shape and location of sensors and actuators for evolution systems driven by distributed parameter systems or partial differential equations (PDE). We consider wave, Schrödinger and heat equations on an arbitrary domain Ω, in any space dimension, and with suitable boundary conditions (if there is a boundary) which can be of Dirichlet, Neumann, mixed or Robin type. This kind of problem is frequently encountered in applications where one aims, for instance, at maximizing the quality of reconstruction of the solution, using only a partial observation. From the mathematical point of view, using probabilistic considerations we model this problem as that of maximizing the so-called randomized observability constant, over all possible subdomains of Ω having a prescribed measure. The spectral analysis of this problem reveals intimate connections with the theory of quantum chaos. More precisely, we provide a solution to this problem when the domain Ω satisfies suitable quantum ergodicity assumptions. This proceedings paper is a short exposition and survey of results published in a series of works [17, 18, 19, 20, 21], in which we solved the above issues. The second author gave a talk on that subject in a workshop organized in Monastir in 2013.

Domaines: Equations aux dérivées partielles [math.AP]

Yannick Privat, Emmanuel Trélat, Enrique Zuazua. Optimal location of sensors and actuators for wave and heat processes. Pde’s, Dispersion, Scattering theory and Control theory, Jun 2013, Monastir, Tunisia. pp.123--139. ⟨hal-01582648⟩