Several Control Problems Of Wave Equations

The purpose of this paper is to discuss several control problems of wave equations,this paper includes four chapters.In chapter 1,we provide a simple research summary of the control problems for wave equation and main research problems of this paper.In chapter 2,we study the exact controllability for a n-dimensional wave equation in non-cylindrical domains ???:where u is the state variable,u" stands for???²u/???t²,and?u⁰,u¹??L²??₀?×H^-1??₀?is any given initial data,here v is the control variable.The exact boundary controllability of the original system is obtained through decay estimates of the dual system.Also,an explicit dependence of the controllability time is given.In chapter 3,we study the hierarchic control for a one-dimensional wave equation in non-cylindrical domains ???:here u is the state variable,w is the control variable and?u₀?x?,u₁?x???^L2??₀?×H^-1??₀?is any given initial data.By u'=u'?x,t?represents the derivative???u/???t and u_xx=u_xx?x,t?the one dimensional Laplace operator???²u/???x².We establish hierarchic control for the equation of the one dimensional wave in non-cylindrical domains by the Stackelberg-Nash Strategy.And by using the multiplier method in non-cylindrical domains,we establish the control.Existence and uniqueness is also proven.The optimality system is given in this chapter.In chapter 4,we study the boundary control of a one-dimensional unstable wave equation with boundary disturbance:where u is the state,q>1 is a constant,U is the control input,d is an unknown disturbance.We use ut to stand for the partial derivative of u with respect to t,u_x the partial derivative with respect to x.The method of backstepping and Lyapunov function are adopted in the design of the state feedback controller.It is shown that the resulting closed-loop system is associated with a nonlinear semigroup and is asymp-totically stable when there is no external disturbance.The existence and uniqueness of the solution are developed by the Galerkin approximation scheme.In addition,we also show that the energy of closed-loop system is convergent to zero as time goes to infinity in the presence of disturbance.