Distributed control and observation

Over the recent years, mostly under the impetus of the late JL Lions, important progress have been made for the control of distributed systems. This has contributed to the understanding of the duality which exist between the modal analysis and the need of very localized actuators. This duality leads to the phenomena of overspilling (excitation of higher order modes). On the other hand this type of research is closely related to the analysis of the exterior problem for the acoustic equation which was, under the influence of P. Lax, one of the main stimulus for the development of microlocal analysis. In this contribution only the acoustic equation is studied. However most of the idea carry on to other linear equations and to some non linear equations when considered in a perturbative regime where linearization techniques can be used this includes perturbations of given solution of the Navier Stokes equation following the work of Imanuvilov and Fursikov (cf [12]) for a recent reference. Same ideas are used also in identification problems for instance in the migration method for oil recovery.