UBC Mathematics Colloquium: Enrique Zuazua

In this lecture we address the problem of the optimal placement of sensors and actuators for wave propagation problems. Using Fourier series representation the problem can be recast as a spectral optimal design problem, involving all the spectrum of the Laplacian. We show that, depending on the complexity of the data to be observed/controlled, several scenarios have to be distinguished. Those in which the solution is a classical set constituted by a finite number of simply connected subdomains, others in which the optimal set is of Cantor type and those leading to relaxation phenomena.

We also explain how closely this topic is related to the fine properties of the high frequency behavior of the eignefunctions of the Laplacian which is intimately linked to the ergodicity properties of the dynamical system generated by the corresponding billiard. We shall also discuss the same problem for heat processes showing that, in that frame, according to intuition, the problem is governed by a finite number of Fourier modes.

These results will be illustrated by numerical simulations. The lecture is conceived for a general audience and unnecessary technicalities will be avoided. It is based on recent joint work in collaboration with Y. Privat and E. Trélat from UMPC, Paris.