Optimal shape and location of actuators or sensors in PDE models

We investigate the problem of optimizing the shape and

location of actuators or sensors for evolution systems

driven by a partial differential equation, like for

instance a wave equation, a Schrödinger equation, or a

parabolic system, on an arbitrary domain Omega, in

arbitrary dimension, with boundary conditions if there

is a boundary, which can be of Dirichlet, Neumann,

mixed or Robin. 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 the problem of

maximizing what we call a randomized observability

constant, over all possible subdomains of Omega having

a prescribed measure. The spectral analysis of this

problem reveals intimate connections with the theory of

quantum chaos. More precisely, if the domain Omega

satisfies some quantum ergodic assumptions then we

provide a solution to this problem.



These works are in collaboration with Emmanuel Trélat

(Univ. Paris 6) and Enrique Zuazua (BCAM Bilbao, Spain).