Stability And Optimality Of Timoshenkobeam Systems With Boundary Damping

With the development of robotics and space technology,flexible structural materials with intelligence are widely used.As one of the accurate elastic beam models,Timoshenko beam model plays a key role in practical application.Therefore,the research on the performance of Timoshenko beam is not only of theoretical value,but also of extensive engineering guiding significance.This thesis mainly studies the stability and optimality of two kinds of Timoshenko beam systems with different types of boundary damping.One class has boundary feedback damping,and the other class has fractional boundary damping.For the analysis of system stability,the operator semigroup theory and frequency domain method are mainly used.It is obtained that the first type of system is exponential stable,and the second type of system is non-exponential stable,but under certain conditions,the system satisfies polynomial stability.For the optimality of the system,we use the rolling time domain method to obtain that the optimal state trajectories of these two types of systems satisfy exponential decay.The full text consists of five chapters.The first chapter introduces the development process of control theory,analyzes the development status of stability and optimality of distributed parameter systems,and finally gives the main content of this paper.In the second chapter,some basic concepts and theories are introduced to make the bedding for probing system stability and optimality.In Chapter 3,Timoshenko beam system with boundary feedback damping is considered:(?)where(?)First,transform the system into an abstract Cauchy problem,and the well-posedness of the system is proved by using the operator semigroup theory.Then,The energy of the system is proved to be exponential stable by the frequency domain equivalent condition of exponential stability of the system.Secondly,using the rolling time domain method and Bellman optimality principle,it is proved that the optimal trajectory in the finite time domain has exponential decay,and the suboptimal trajectory in the infinite time domain is exponential decay.In Chapter 4,the following Timoshenko beam system with fractional boundary damping is considered:(?)First,the system is re-described as an augmented system,using operator semigroup theory to obtain the well-posedness of the system.Then,according to the frequency domain equivalent condition of the system stability,when the space dimension is d> 2α+1,It is proved that the system is not uniformly exponential stable.Then,the system is proved to be polynomial stable by estimating the energy of the auxiliary system.Secondly,the rolling time domain method is used to prove the exponential attenuation of the optimal trajectory.The fifth chapter briefly summarizes the contents of this paper and gives the prospect.