Control And Stabilization Of A Class Of Rayleigh Beam Equations

Distributed Parameter Systems are the control system with infinite dimensions,and mainly study the systems which described by partial differential equations,functional differential equations,integral differential equations,integral equations or some abstract differential equations defined in Banach or Hilbert space.In recent years,Distributed Parameter Systems have gradually become a research hotspot of scientists due to the demand of engineering systems,for examples,the control of temperature field,elastic vibration,and satellite attitude and orbit coupling system,and so on.It is an important problem in the study of Distributed Parameter Systems to design the feedback control to stabilize the systems.The thesis considers the control design and stabilization of a class of Rayleigh beam equations.We study three issues:(1)the stabilization of an active constrained layer(ACL)beam system consisting of a stiff layer,a viscoelastic layer and a piezoelectric layer which actuated by a voltage source without magnetic effects;(2)the stabilization of a Rayleigh cantilever beam system with axial force;(3)the stabilization of a Rayleigh cantilever beam system with axial force and a concentrated mass at the free end.Firstly,we propose the boundary feedback control for the Rayleigh beam systems.Secondly,we discuss the asymptotic behavior of the eigenvalues and eigenfunctions of the systems by the method of spectral analysis.Finally,we prove the exponential stability of the closed-loop systems using the Riesz basis approach.The thesis is detailedly organized as follows:In Chapter 1,we introduce the engineering background,basic knowledge of beam equations,development situation and preliminaries,and give the structure and main results of the thesis.In Chapter 2,we study the stabilization of an active constrained layer(ACL)beam system consisting of a stiff layer,a viscoelastic layer and a piezoelectric layer which actuated by a voltage source without magnetic effects.The system is modeled as a Rayleigh beam equation coupled with two wave equations.We use the boundary feedback control to stabilize the ACL beam system.we first prove that the system is well-posed in the state space.By an asymptotic technique for tackling the matrix operator pencils,we present the asymptotic expressions for the eigenvalues and eigenfunctions of the system.Finally,we show that the generalized eigenfunctions of the system form a Riesz basis in the state space and the closed-loop system is exponentially stable.In Chapter 3,we study the exponential stability of a Rayleigh beam system with axial force.The system contains the bending moment term with variable coefficient in the equation.We propose a boundary feedback control moment to stabilize the system.We firstly analyse the well-posedness of the system.Secondly,we give the asymptotic expressions for the eigenpairs of the system by using an asymptotic technique.Finally,we prove the exponential stability of the closed-loop system by the Riesz basis approach.In Chapter 4,we consider the stabilization of a Rayleigh beam system with axial force and a concentrated mass at the free end.The system has two characteristics:(1)the bending moment term contained variable coefficient in the equation;(2)a tip mass contained at the boundary.We study the stabilization of the system by the boundary feedback control.Firstly,we give the asymptotic expressions of the eigenvalues and eigenfuctions of the system by applying the technique of spectral analysis.Secondly,we obtain the exponential stability of the closed-loop system by the Riesz basis approach.We give the summary and references of the thesis and present some unsolved problems at the end.