Stability Of A Class Of Viscous Damping Timoshenko Beams With Dynamic Boundaries

In recent decades,the "smart material" technology has developed greatly.Naturally,the stability of the boundary value problem of the deformed structure has become one of the research hotspots.The Timoshenko beam,the Euler-Bernoulli beam and the Rayleigh beam are its Important components,especially the Timoshenko beam,can more accurately describe a type of deformation structure model,so it is easier to meet the needs of actual engineering.This paper studies the internal viscous damped beam system with dynamic boundary and the boundary damped beam system with dynamic boundary.For the analysis of system stability,this paper first rewrites the original system into an abstract Cauchy problem,applying operator semigroup theory to obtain the system's well-posedness.Furthermore,the multiplier technique and the counter-evidence method are used to prove that the system studied is consistent and exponentially stable.The full text consists of the following five chapters:In the first chapter,we first review the development history of control theory,and give a brief introduction to it.Then we briefly explain the research background and the relevant research status in China,and finally explain the methods and definitions involved in system stability.The second chapter prepares for the stability of the system studied in the following papers.This chapter mainly introduces the definitions related to the research system and the basic inequalities commonly used.In the third chapter,the exponential stability of the Timoshenko beam system with internal viscous damping with dynamic boundaries is proved by using operator semigroup theory,multiplier technique and counter-evidence method:Firstly,the original system is rewritten as an abstract Cauchy problem,and the operator semigroup theory is used to obtain the system's well-posedness.Further,the multiplier technique and the counter-evidence method are used to prove that the system under study is consistent and exponentially stable.In the fourth chapter,the stability of the following Timoshenko beam system with dynamic boundary feedback damping is proved by the similar method in Chapter3:The fifth chapter summarizes the content of the article and looks forward to the direction of future exploration.