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Contrôlabilité et stabilisation frontière pour l'équation de Korteweg-de Vries.

Abstract : In this thesis, we will consider a control system where the state is given by the solution of a Korteweg-de Vries (KdV) equation posed on a bounded interval. We consider the homogeneous Dirichlet boundary conditions and the control acts on the Neumann boundary conditions at the right-end point. We will study two kind of problems strongly related: controllability and stabilization. Chapters 2 and 3 are concerned with the study of controllability on some domains for which the linear system is not controllable. Our aim is to prove that, in despite of this lack of controllability for the linear system, the nonlinearity allows us to obtain the controllability for the nonlinear system. To do that, we will use the development in power series method, which has been introduced, in the infinite-dimensional framework, by J.-M. Coron and E. Crépeau in [J. Eur. Math. Soc. (JEMS) 6, no. 3, pp. 367-398, 2004]. This method consists in moving the system along the missed directions for the linear system by means of higher order development, and then to apply a fixed-point theorem. In chapter 4, we will study the stabilization issue. Our aim is to build some feedback laws such that the closed-loop system has an exponential decay to zero with a decay rate arbitrary. The method used is due to J. M. Urquiza who introduced it in [SIAM J. Control Optim., V. 43, no. 6, pp 2233-2244, 2005]. In order to apply this method, a spectral analysis for the stationary KdV operator is needed.
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Submitted on : Tuesday, October 20, 2009 - 5:19:15 PM
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  • HAL Id : tel-00425316, version 1



Eduardo Cerpa. Contrôlabilité et stabilisation frontière pour l'équation de Korteweg-de Vries.. Mathématiques [math]. Université Paris Sud - Paris XI, 2008. Français. ⟨tel-00425316⟩



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