Exact controllability, stabilization and perturbations for distributed systems.

*(English)*Zbl 0644.49028The 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 |