Optimal Control Problems For Two Kinds Of Elliptic Variational Inequalities

Variational inequality problems are widely used in aerospace technology,engineering optimization,finance and economy,and have important research significance.In this paper,we study the optimal control problems of two kinds of elliptic variational inequalities.Combining Lagrange multiplier method,nonlinear functional analysis and optimal control theory,we prove the existence of the solution of the optimal control problem and establish their optimal system.The full text is divided into five chapters:In chapter 1,we introduce the research background and current situation of optimal control problems with variational inequalities,analyze the existing research methods,and give the main results of this thesis.In chapter 2,we summarize the preparatory knowledge needed in this paper.It mainly includes important lemmas and inequalities such as nonlinear functional analysis,partial differential equations and monotone operator theory.In chapter 3,we consider the optimal control of a class of unilateral obstacle elliptic variational inequalities applied to mixed boundary value problems.First,the existence of optimal solution is proved,and then the optimal system of the approximation problem is obtained by introducing a family of approximation problems and using Lagrange multiplier method.Then,we let the limit approach zero.In this process,the convergence theorem between the original problem and the approximation problem is obtained.Finally,the optimality conditions of the original problem are derived.In chapter 4,we study the optimal control problem of a class of bilateral obstacle elliptic variational inequalities applied to mixed boundary value problems.Firstly,the existence of the solution of the boundary optimal control problem is proved by combining the theory of nonlinear functional analysis.Then,by using the smooth approximation technique,the optimality system for approaching the optimal control problem are established.Because the regularity of state variables is improved,the optimality condition of approximation problem is relatively complete.Finally,by analyzing the convergence of the optimality conditions of the approximation problem,the optimality system of the original problem is established.In chapter 5,we summarize the content of this paper and gives the research prospect.