Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary.

*(English)*Zbl 0786.93009The observability, controllability and stabilization of the solutions of second-order hyperbolic partial differential equations are studied. Linear problems with variable coefficients are dealt with. The variable coefficients occur through linearization around a nonequilibrium solution.

According to the authors, J. Rauch and M. Taylor [Indiana Univ. Math. J. 24,79-86 (1974; Zbl 0264.35049)] have proved the controllability and observability in domains without boundary. In this paper, the sufficiency is proved for multidimensional problems of control and observation from the boundary. Starting-point in the analysis is the classical energy method, which consists of multiplying the differential equation by properly chosen differential operators. The ray of geometric optics and the method of microlocal analysis of boundary value problems are used in the derivations. Some examples are considered.

According to the authors, J. Rauch and M. Taylor [Indiana Univ. Math. J. 24,79-86 (1974; Zbl 0264.35049)] have proved the controllability and observability in domains without boundary. In this paper, the sufficiency is proved for multidimensional problems of control and observation from the boundary. Starting-point in the analysis is the classical energy method, which consists of multiplying the differential equation by properly chosen differential operators. The ray of geometric optics and the method of microlocal analysis of boundary value problems are used in the derivations. Some examples are considered.

Reviewer: I.H.H.vande Ven (Eindhoven)

##### MSC:

93B05 | Controllability |

35B40 | Asymptotic behavior of solutions to PDEs |

35L20 | Initial-boundary value problems for second-order hyperbolic equations |

93D20 | Asymptotic stability in control theory |

93B07 | Observability |

93C20 | Control/observation systems governed by partial differential equations |