Vol. 10, No. 5, 2017

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ISSN: 1948-206X (e-only)
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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
Abstract

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.

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