In this lecture we address the problem of the optimal placement of sensors and actuators for wave-like equations.
Using Fourier series representations the problem can be recast as an optimal design one involving all the spectrum of the laplacian. We shall develop a complete theory allowing to distinguish, depending on the complexity of the data to be observed/controlled, cases in which the solution is a classical with a finite number of connected components, from others in which the optimal set is of Cantor type or those when relaxation occurs. These results will be illlustrated by numerical simulations.
The lecture is based on recent joint work in collaboration with Y. Privat and E. Trélat.