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Rapid stabilization of Burgers equations and of Korteweg-de Vries equations

Abstract : This thesis is devoted to the study of stabilization of partial differential equations by nonlinear feedbacks. We are interested in the cases where classical linearization and stationary feedback law do not work for stabilization problems, for example KdV equations and Burgers equations. More precisely, it includes three important cases : stabilization of nonlinear systems whose linearized systems are not asymptotically stabilizable ; small-time local stabilization of linear controllable systems ; small-time global stabilization of nonlinear controllable systems. We find a strategy for the small-time global stabilization of the viscous Burgers equation : small-time global approximate stabilization and small-time local stabilization. Moreover, using a quadratic structure, we prove that the KdV system is exponentially stabilizable even in the case of critical lengths.
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Shengquan Xiang. Rapid stabilization of Burgers equations and of Korteweg-de Vries equations. Analysis of PDEs [math.AP]. Sorbonne Université, 2019. English. ⟨NNT : 2019SORUS422⟩. ⟨tel-03004788⟩

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