Optimality And Stability Of A Class Of Coupled Equations

In recent decades,the research on the optimal control of elastic systems and the stability analysis of nonlinear systems have received extensive attention.By using the optimal control theory to obtain the corresponding optimal results under the given cost function(the shortest time,the minimum energy and so on).For the stability of the nonlinear systems,in order to achieve the desired objectives by using the approximation theory.In this thesis,the optimal control problem of Timoshenko beam and the stability problem of a class of nonlinear coupled systems with nonlinear damping is mainly discussed.The optimality of the Timoshenko beam is studied and the maximum principle is obtained by using the method of adjoint systems.The stability of coupled nonlinear system with nonlinear damping is studied by the Galerkin approximation method combined with multiplier technique.This thesis is divided into five parts:In the first part,at first,the development process of the two parts of the optimal and the stability of the distributed parameter system is introduced.Then,the research background and current situation is described.Finally,the main research contents of this thesis and the methods and the skills used are illustrated.In the second part,some related definitions,the maximum principle of optimal control and the inequalities common used in this thesis are introduced which provide a theoretical basis for the study of the optimization and stability of the latter.In the third part,the optimality of the Timoshenko beam is discussed by using the method of adjoint systems,the spectral analysis and so on.In the fourth part,the exponential stability of a class of coupled nonlinear systems with nonlinear damping is studied by using the Faedo-Galerkin method and the energy perturbation method and the multiplier technique.In the fifth part,the study of this thesis is summarized,and the future development direction is also studied.