Exact Controllability, Stabilization and Perturbations for Distributed Systems

Exact controllability is studied for distributed systems, of hyperbolic type or for Petrowsky systems (like plate equations). The control is a boundary control or a local distributed control. Exact controllability consists in trying to drive the system to rest in a given finite time. The solution of the problems depends on the function spaces where the initial data are taken, and also depends on the function space where the control can be chosen. A systematic method (named HUM, for Hilbert Uniqueness Method) is introduced. As the terminology indicates, it is based on Uniqueness results (classical or new) and on Hilbert spaces constructed (in infinitely many ways) by using Uniqueness. A number of applications are indicated. Having a general method for exact controllability implies having a general method for stabilization. This leads to new (and complicated ...) nonlinear Riccati’s type PDEs, to be compared with direct methods (when available). In the last part of the paper, we consider how all this behaves for perturbed systems: singular perturbations, singular domains, homogenization. A number of open questions are indicated.