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Global stabilization of a Korteweg-de Vries equation with saturating distributed control

Abstract : This article deals with the design of saturated controls in the context of partialdifferential equations. It focuses on a Korteweg–de Vries equation, which is a nonlinear mathematicalmodel of waves on shallow water surfaces. Two different types of saturated controls are considered.The well-posedness is proven applying a Banach fixed-point theorem, using some estimates of thisequation and some properties of the saturation function. The proof of the asymptotic stability of theclosed-loop system is separated in two cases: (i) when the control acts on all the domain, a Lyapunovfunction together with a sector condition describing the saturating input is used to conclude on thestability; (ii) when the control is localized, we argue by contradiction. Some numerical simulationsillustrate the stability of the closed-loop nonlinear partial differential equation.
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Submitted on : Thursday, June 29, 2017 - 9:05:30 PM
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Swann Marx, Eduardo Cerpa, Christophe Prieur, Vincent Andrieu. Global stabilization of a Korteweg-de Vries equation with saturating distributed control. SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2017, 55 (3), pp.1452-1480. ⟨10.1137/16M1061837⟩. ⟨hal-01367622v3⟩



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