Vol. 10, No. 5, 2017

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Local exponential stabilization for a class of Korteweg–de Vries equations by means of time-varying feedback laws

Jean-Michel Coron, Ivonne Rivas and Shengquan Xiang

Vol. 10 (2017), No. 5, 1089–1122

We study the exponential stabilization problem for a nonlinear Korteweg-de Vries equation on a bounded interval in cases where the linearized control system is not controllable. The system has Dirichlet boundary conditions at the end-points of the interval and a Neumann nonhomogeneous boundary condition at the right end-point, which is the control. We build a class of time-varying feedback laws for which the solutions of the closed-loop systems with small initial data decay exponentially to 0. We present also results on the well-posedness of the closed-loop systems for general time-varying feedback laws.

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Korteweg–de Vries, time-varying feedback laws, stabilization, controllability
Mathematical Subject Classification 2010
Primary: 93D15, 93D20, 35Q53
Received: 2 May 2016
Revised: 29 September 2016
Accepted: 7 March 2017
Published: 1 July 2017
Jean-Michel Coron
Université Pierre et Marie Curie - Paris 6
UMR 7598 Laboratoire Jacques-Louis Lions
75005 Paris
ETH Zürich
Institute for Theoretical Studies
8092 Zürich
Ivonne Rivas
Universidad del Valle
Departamento de Matemáticas
AA 25360
Shengquan Xiang
Université Pierre et Marie Curie - Paris 6
UMR 7598 Laboratoire Jacques-Louis Lions
75005 Paris
ETH Zürich
Institute for Theoretical Studies and Forschungsinstitut für Mathematik
8092 Zürich