Research On A Class Of Stationary Thermistors Problem With Mixed Boundary Conditions And Its Optimal Control

A class of stationary thermistor systems is studied in this paper.The system is coupled by two nonlinear elliptic partial differential equations,in which the energy conservation equation has multivalued mixed boundary conditions for temperature u while the charge conservation equation has non-homogeneous Dirichlet boundary condition for electric potential φ.By means of Schauder’s fixed point theorem,Lax-Milgram theorem,Hemivariational inequality theory,the existence of weak solutions is proved and the uniqueness of solution under certain constraint is proved too.On this basis,combined with partial differential optimal control theory,the optimal control problem is further studied.This paper is divided into five chapters:In chapter 1,we introduce the backgrounds and applications of thermistor problem and research methods involved in this paper,including the background of variational inequality problem and its optimal control problem.Meanwhile,the research status of thermistor problem and variational inequality is introduced too.Finally,the research content and structure of this paper are given.In chapter 2,we summarizes the preparatory knowledge of this paper,including the basic definitions and important conclusions of Sobolve space,nonlinear functional analysis,partial differential equation and monotone operator theory.In Chapter 3,a class of stationary thermistors with mixed boundary conditions is considered,part of which is Clarke subdifferential boundary condition.The weak form of the problem is derived by integration by parts formula,and the existence of the weak solutions is proved by Schauder’s fixed point theorem.Finally,the solution is proved to be unique under certain constraints.In chapter 4,the optimal control of stationary thermistor system is studied.Given a target functional,taking theristor system as the state equation and heat source term as the control,combining with the optimal control theory of partial differential equations and the monotone operator theory,it is proved that there is a solution to the optimal control problem.In Chapter 5,we summarizes the content of the thesis and gives the prospect of the research work.