Optimal Control Problems For Two Kinds Of Parabolic Variational Inequalities

Variational inequality problems are widely used in aerospace technology,engineering optimization,finance and economy,and have important research significance.This thesis mainly studies the optimal control problem of two kinds of parabolic variational inequalities,which is divided into the following five chapters:In Chapter 1,we introduce the research background of optimal control problems with variational inequalities and current situation at home and abroad,and the content and structure of this thesis are given.In Chapter 2,we summarize the necessary preparatory knowledge,including some definitions,lemmas and inequalities in partial differential equations and nonlinear functional analysis.In Chapter 3,we study the optimal control problem of semilinear parabolic variational inequalities of bilateral obstacle type with Neumann boundary conditions.By regularization method,the variational inequalities are approximated by a family of semilinear parabolic partial differential equations.Then,the existence of solutions to the original and approximate optimal control problems is proved,and the convergence relation is given.Finally,the weak directional differentiability of approximate control-state mapping is established,and the optimality system of the original optimal control problem is derived.In Chapter 4,we study the boundary optimal control problem of semilinear parabolic variational inequalities of bilateral obstacle type with Robin boundary conditions.Using regularization method,we approximate variational inequalities by a family of partial differential equations.Then,the existence of solutions to the original and approximate optimal control problems is proved,and the convergence relation is given.The Lipschitz continuity and weak directional differentiability of approximate control-state mapping are established,and the optimality system of the original optimal control problem is derived.Finally,the results are compared with those in related literature.In Chapter 5,we summarize the whole thesis and give the research prospect.