Control Problems For Several Classes Of Nonlinear Second Order Differential Equations

Cybernetics has grown rapidly in recent years,the control problem related to Hemivariational inequality and the study of differential Hemi-variational inequality have become the mainstream and hot topic in the world.Based on this,the control problems of some nonlinear second order differential equations are studied in this thesis.This thesis is divided into four chapters.In Chapter 1,we introduce the background of feedback control problem,Hemi-variational inequality and differential variational inequality,research status and research development trend,and expound the main work content of this thesis.In Chapter 2,we introduce the necessary preparatory knowledge,including the set-valued mapping,Bochner integral,operator semigroup theory,Clarke’s subdifferential and some other basic definitions,properties and relevant lemmas.In Chapter 3,we mainly study the feedback control of a class of Clarke’s subdifferential evolution equations containing non-autonomous second order Hemi-variational inequality.Under appropriate conditions,we first give the existence of feasible pairs of feedback control systems.In addition,under some reasonable assumptions,we also prove the existence of the optimal control pair.Finally,we give a concrete example.In Chapter 4,we study a class of differential variational inequalities,which consists of a second order autonomous differential equation with history dependent operator and a mixed quasi-variational-Hemi-variational inequality.By using KKM theorem,monotone theory of operators,Minty technique,the existence of solutions to differential Hemi-variational inequalities is established based on strong continuous cosine operator theory and fixed point theorem of aggregate value operators.At the end,the present research work is summarized and the future research ideas are put forward.