The Exponential Stabilization Of The Heat-wave Coupled System And Its Approximation

In this note,the exponential stabilization of the heat-wave coupled equation with control and collocated observation is discussed.The system is only polynomially stable when the control is zero.Two feedback strategies,static negative proportional feedbacks and dynamic feedbacks,are designed.By the direct method of Lyapunov function,the exponential stability of the closed-loop systems are verified.Moreover,the H_?robustness of two kinds of controls are further analyzed and the related suffcient conditions are given.Furthermore,because the uniform exponential stabilities with respect to the discretized parameter play key roles in the computing of optimal control and the inverse problem of observability,the uniform exponential stabilities of wave equations have been intensively discussed in the past decades.We also address this problem for the closed-loop systems mentioned above and generalize some relevant results to the coupled systems.We firstly reduce the order of the heat-wave equation and then discretized spacial variable by finite difference method,the uniform exponential stabilities of the discretized systems are finally tested by the methods paralleling to the continuous systems.The main work is as follows:In the second chapter,the reduced order difference method is introduced firstly.Secondly,the theory of Linear Operator Semigroups on Hilbert space is introduced by introducing some operators and the properties of semigroups they generate.In the third and fourth chapter,we first introduce the state space and the inner product of the continuous system,then write the continuous system into the form of its equivalent abstract differential equation,then prove the existence and uniqueness of the solution of the equation by the theory of operator semigroup,and then introduce the appropriate Lyapunov function to verify the uniform exponential stability of the continuous system.In the fourth chapter,the proof of continuous system is extended to discrete system.In the fifth chapter,we research H_? boundary control problem and the related suffcient conditions are given.