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Local Exponential $H^2$ Stabilization of a $2\times2$ Quasilinear Hyperbolic System Using Backstepping

 

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Opened Access Local Exponential $H^2$ Stabilization of a $2\times2$ Quasilinear Hyperbolic System Using Backstepping
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Author: Coron, Jean Michel
Vázquez Valenzuela, Rafael
Krstic, Miroslav
Bastin, Georges
Department: Universidad de Sevilla. Departamento de Ingeniería Aeroespacial y Mecánica de Fluidos
Date: 2013
Published in: SIAM Journal on Control and Optimization, 51 (3), 2005-2035.
Document type: Article
Abstract: In this work, we consider the problem of boundary stabilization for a quasilinear 2 × 2 system of first-order hyperbolic PDEs. We design a new full-state feedback control law, with actuation on only one end of the domain, which achieves H2 exponential stability of the closedloop system. Our proof uses a backstepping transformation to find new variables for which a strict Lyapunov function can be constructed. The kernels of the transformation are found to verify a Goursat-type 4 × 4 system of first-order hyperbolic PDEs, whose well-posedness is shown using the method of characteristics and successive approximations. Once the kernels are computed, the stabilizing feedback law can be explicitly constructed from them.
Cite: Coron, J.M., Vázquez Valenzuela, R., Krstic, M. y Bastin, G. (2013). Local Exponential ^2$ Stabilization of a \times2$ Quasilinear Hyperbolic System Using Backstepping. SIAM Journal on Control and Optimization, 51 (3), 2005-2035.
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URI: http://hdl.handle.net/11441/57910

DOI: 10.1137/120875739

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