Stability And Controllability Of Timoshenko Beam System With Memory

Over the past half century,the flexible structures are widely used in the space science and robotics.The research work of system control has become a hot topic,where the Timoshenko beam model is the most complete thick beam on the physical model and has important application in structure engineering,which can better meet the needs of practical applications.Therefore,it is very meaningful to study the uniform exponential stability and exact controllability of Timoshenko beam systems.In this thesis,the uniform stability and L2-exact controllability of Timoshenko beam systems with memory damping are studied.On the one hand,for the purpose of obtaining the stability of Timoshenko beam systems,the associated methods such as the operator semigroup theory,multiplier techniques and the contradiction of frequency domain method are applied in this work.On the other hand,with regard to the controllability of Timoshenko beam systems,we apply some approaches including Hilbert unique method,Fourier expansion and multiplier techniques to establish and prove the observation inequality,and the L2-exact controllability of Timoshenko beam systems with initial values.This thesis is divided into five chapters:In the first chapter,the research background of elastic systems and the research status of the uniform exponential stability and exact controllability are briefly introduced.Finally,the content of this paper is briefly summarized.The the second chapter introduces,the basic concepts,basic theories and commonly used inequalities involved in this thesis are introduced,in order to prepare for the study of the uniform stability and exact controllability of the systems.In the third chapter,the uniform stability of the following heterogeneous Timoshenko with viscous damping beam equation is considered:(?)Firstly,by using functional analysis method and linear operator semigroup theory,Timoshenko beam equation is written as the abstract Cauchy problem.Then,the equivalence and well-posedness of the systems are proved by using the semigroup theory of linear operators,and the spectral properties of the A operators are given.Finally,the uniform exponential stability of the systems is proved by using the multiplier technique and the contradiction of the frequency-domain methodIn the fourth chapter,the L2-exact controllability of the following homogeneous Timoshenko beam systems with viscous damping is considered:(?)Firstly,the existence and regularity of Timoshenko beam equation solutions are studied;Then,the observation inequalities are established and proved by using Hilbert uniqueness method,Fourier expansion and multiplier techniques;Finally,the L2-exact controllability of Timoshenko beam systems with initial values is considered.In the fifth chapter,a brief summary of the research in this paper is given,and the future research direction is prospected.