Exact controllability, stabilization and perturbations for distributed systems. (English) Zbl 0644.49028

Lions, J. L.

SIAM Rev. 30, No. 1, 1-68 (1988).

The paper deals with exact controllability for distributed systems of hyperbolic type or for Petrowsky systems. The control is a boundary control or a local distributed control. The concept of exact controllability is as follows: given T, for all initial data, there is a corresponding control driving the system to a desired state at time T. The existence of a corresponding control depends on the function spaces where the initial data are taken, and also depends on the functions space where the control can be chosen. The author introduces a systematic method (named Hilbert uniqueness method. It is based on uniqueness results and on Hilbert spaces constructed by using uniqueness. Having a general method for exact controllability implies having a general method for stabilization. Based on the Hilbert uniqueness method, the author considers the following topics: (i) exact controllability; (ii) stabilization of systems; (iii) behavior of exact controllability and of stabilization under perturbations such as singular perturbations, singular domains and homogenization.

Reviewer: T.Kobayashi

MSC:

93B03	Attainable sets, reachability
93D99	Stability of control systems
93C20	Control/observation systems governed by partial differential equations
35B25	Singular perturbations in context of PDEs
35B37	PDE in connection with control problems (MSC2000)
35L20	Initial-boundary value problems for second-order hyperbolic equations
46C99	Inner product spaces and their generalizations, Hilbert spaces
49J20	Existence theories for optimal control problems involving partial differential equations
49K40	Sensitivity, stability, well-posedness
93D15	Stabilization of systems by feedback
93B05	Controllability

Keywords:

exact controllability; distributed systems of hyperbolic type; Petrowsky systems; boundary control; local distributed control; Hilbert uniqueness method