A survey of controllability and stabilization results for partial differential equations

This paper surveys several issues related to the control of partial differential equa- tions (PDE). The main focus is on the exact controllability property, which corresponds to the question of whether the solution of a PDE can be driven to a given state at a given final time by means of a control acting on a subregion of the domain or of the boundary. It is demonstrated that such a property is equivalent to an observability property for the adjoint system. The study of the exact controllability is detailed on several examples, including the wave equation, the heat equation, and the plate equation in dimension one. The controllability of the Korteweg-de Vries equation is also detailed in order to give an insight of the ideas involved in the control of a nonlinear PDE. The last part of the paper is devoted to the stabilization issue and to its connections with the controllability properties. RESUME. Cet article passe en revue diverses questions liees au controle des equations aux deri- vees partielles (EDP). La question principale sur laquelle se focalise le papier est celle de la controlabilite exacte, qui correspond au fait que la solution d'une EDP peut etre amenee a un etat donne au bout d'un temps donne au moyen d'un controle agissant sur une sous-region du

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