Uniform Exponential Stability And The State Reconstruction Of The Semi-linear Wave Equation With Variable Coefficients

Semi-linear wave equation with variable coefficients is an important research object in mathematics,cybernetics and engineering.It has important theoretical significance and practical value to make it exponentially stabilized by boundary.By using Lyapunov Krasovskii function method and comparison principle,researchers have given the sufficient conditions of Linear Matrix Inequalities(LMI)type to ensure the exponential stability of semi-linear wave equation with variable coefficients.In this paper,we will study the three aspects of it as follows: First,a new verification method of exponential stability is given by combining Port-Hamiltonian and Lyapunov method.Second,this new method is used to study the uniform exponential stability of the above mentioned system,that it means that a family of systems is obtained by discretizing the spatial variables,and the discretized system is uniformly exponentially stable with respect to the discretized parameters.Third,it is assumed that when the initial value of the system is unknown,the specific calculation formula of the initial value is given by using the observation information.In particular,it should be pointed out that although the methods of verifying exponential stability in this paper are different,the sufficient conditions obtained are the same,and this verification method can be easily used to study uniform exponential stability of discrete systems,and the sufficient conditions of uniform exponential stability of discrete systems are the same as those of continuous systems.This paper consists of four chapters: The first chapter is the introduction,which introduces the research background and significance of this paper.In the second chapter,the semi-linear wave equation with variable coefficients is reduced and transformed into an equivalent Port-Hamiltonian system.By constructing a suitable Lyapunov function,the sufficient conditions for the exponential stability of the continuous system are given.In the third chapter,a family of discrete systems are obtained by using the central difference of the spatial variables of the reduced order system.By using the parallel method in the second chapter,the sufficient conditions for uniform exponential stability of discrete systems are given.In the last chapter,an application of exponential stability is given,that is to study how to give the initial value calculation formula through the boundary observation information when the initial value of the system is unknown.By constructing a forward-backward observers,the iteration sequence constructed converges strongly to the initial value for any given guess value.