The Feedback Control And Stability Analysis For Several Simplified Saint-venant Equations

Saint-Venant equations can accurately and comprehensively describe the flow law of water in channel system or the movement law of vehicle traffic flow on highway,which has been paid close attention by scholars and engineering experts for a long time.In this paper,the feedback controller design and exponential stability analysis for several simplified SaintVenant equations are studied.The first chapter describes the research background and present situation of Saint-Venant equation,and emphatically introduces the diffusive wave equation after simplified treatment and the one-dimensional hyperbolic partial differential equations model under transformation of coordinate.In the second chapter,a position and delayed position(PDP)feedback controller is designed for the diffusive wave equation and the well-posedness of the closed system solutions is studied by the theory of operator semigroup.Then,the Lyapunov function is constructed by using the energy norm,and the exponential stability of closed system is obtained.Finally,compared the value range of the control parameters with the literature results,the parameter conditions in this paper are more extensive,which fully reflects the superiorty of the PDP feedback controller.In the third chapter,the heat equation is regarded as the dynamic compensation controller to calm the diffusion wave equation,the well-posedness and dissipation of the system are proved,and the detailed spectral analysis with parameter conditions of the system operator is carried out.In the forth chapter,the well-posedness and stability of a one-dimensional hyperbolic partial differential system under proportional feedback control is addressed.Firstly,the explicit dissipative condition for control parameters is obtained by the semigroup approach and norm equivalence theorem.Secondly,the asymptotic expressions of eigenvalues and eigenfunctions of the system operator are given out by the method of spectral analysis.Finally,we find that there exists a group of generalized eigenfunctions that constitutes a Riesz basis for the state space.Hence,the spectrum determined growth condition holds,and then the exponential stability is established.