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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 → ∞ ; in the two-dimensional case the smooth solutions decay asymptotically to zero faster than any polynomial (1/t)α α > 0; and in the three-dimensional case the smooth solutions decay to zero like (1/t)2 as t → ∞. 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 W1,1(0,∞). 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 → ∞. [ABSTRACT FROM AUTHOR]

The paper is concerned with the control of a uid ow system governed by nonlinear hyperbolic partial differential equations. The control and the output observation are located on the boundary. We study local stability of spatially heterogeneous equilibrium states by using Lyapunov approach. We prove that the linearized system is exponentially stable around each subcritical equilibrium state. A systematic design of proportional and integral controllers is proposed for the system based on the linearized model. Robust stabilization of the closed-loop system is proved by using a spectrum method. [ABSTRACT FROM AUTHOR]

We are concerned with the PI control/regulation design for a cascaded network of multi systems governed by hyperbolic partial differential equations. The PI controllers have both control inputs and measured outputs situated on the junctions. The stability analysis of closed-loop network is carried out in the L 2 topology by using the Lyapunov direct method. Then the output regulation is proven based on the stability of closed-loop systems. We finally work out detailed PI controller design for a practical cascaded network of n hydraulic Saint-Venant models as well as numerical simulations to validate theoretical results. [ABSTRACT FROM AUTHOR]