Asymptotic Stability Of Damped Elastic Systems In Hilbert Spaces

In this paper,the asymptotic stability of linear damped elastic system and damped elastic system with random term is studied in Hilbert spaces,and the existence of mild solutions of damped elastic system is discussed in Banach spaces.The main content consists of the following five chapters:In chapter 1,it describes the research background and significance of damped elastic system at home and abroad,listing the main problems studied in the following three chapters as well as the important definitions and theorems needed in the research.In chapter 2,it investigates the asymptotic stability of theC₀-semigroup corresponding to the damped elastic system in Hilbert spaces by using operator semigroup theory and Lyapunov equation,which obtains the sufficient conditions that the corresponding solution semigroup of the system is a contraction semigroup and theC₀-semigroup is exponentially stable.According to the Gearhart-Pruss-Greiner lemma,the stability theory of the system is established,and the existing work in this field is extended and developed.In chapter 3,the sufficient conditions for the existence of mild solutions of structural damping elastic system are discussed.In the first part,the existence of positive mild solutions of a structural damping elastic system with initial conditions is discussed mainly in a partially ordered Banach spaces,and the existence theorem of positive mild solution is established according to the fixed point theorem of cone compression.In the second part,the existence of mild solutions of damped elastic system with nonlocal conditions on finite interval is studied by using the operator transformation technique and the fixed point theorem of convex power condensing operator.In chapter 4,the asymptotic stability of mild solutions of structural damping elastic system with random term is discussed in Hilbert spaces,and by the sufficient condition of exponential stability and the definition of p-stability of the solution,the p-asymptotic stability of mild solutions of structural damping elastic random system is obtained.Finally,an application example is given.In chapter 5,it is the summary of the main research content of this paper,and the prospect of the research direction of such issues.