Saturated Boundary Feedback Stabilization of a Linear Wave Equation

Abstract : In this paper, we study boundary feedback stabilization of a linear wave equation by saturated linear or nonlinear Neumann control laws. Firstly we prove asymptotic stabilization of the closed-loop system when the feedback control law has a linear growth rate around zero. In particular, we study the effect of spatial dimension on the decay rate of the closed loop system. More precisely, we prove that in the one-dimensional (1D) case the smooth solutions of the closed-loop system decay exponentially to zero as t \rightar \infty ; in the two-dimensional case the smooth solutions decay asymptotically to zero faster than any polynomial (1/t) \alpha \forall \alpha > 0; and in the three-dimensional case the smooth solutions decay to zero like (1/t) 2 as t \rightar \infty. Secondly we study robustness of the stabilization faced with the boundary disturbances. We show that, in the 1D case, every solution of the closed-loop system decays asymptotically to zero provided that the unknown disturbance is in the Sobolev space W 1,1 (0, \infty). Finally we consider a sliding mode output feedback control law that is regarded as the limit case of the Yosida approximation of the sign function. We prove that the resulting system is not asymptotically stable. However each smooth solution of the resulting system is bounded and converges asymptotically to a periodic solution as t \rightar \infty .
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Submitted on : Wednesday, February 27, 2019 - 6:23:54 PM
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Cheng-Zhong Xu, Gen Xu. Saturated Boundary Feedback Stabilization of a Linear Wave Equation. SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2019, 57 (1), pp.290-309. ⟨10.1137/15M1034350⟩. ⟨hal-02051505⟩

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