Optimal Control Problem For P-Laplacian Equations

In this paper,we study the optimal control problem for two types of equations.It can be concluded that the existence of the optimal control function for the initial marginal value problem of the developmental equation is obtained by taking the limit of the minimization sequence of the cost generalization function.First,the existence of the solution of this problem is studied uniquely by means of energy estimation,then the convergence of the minimization sequence of the cost generalization is analyzed by means of tightness estimation and tight embedding theorem,and finally the existence of the optimal control function is obtained.Chapter 1 introduces the background of this problem,summarizes the previous work,and describes the origin of the problem and its contents,solution ideas and final results.Chapter 2 investigates the optimal control problem for elliptic equations with Robin’s boundary conditions.Firstly,the optimal control problem with the control function acting inside the region is studied,and secondly,the optimal control problem with the control function acting at the boundary of the region is studied.The study is carried out by first proving the uniqueness of the existence of weak solutions by means of energy estimation,and then proving the existence of optimal control by taking the limit of the minimization sequence.In the third chapter,the problem of initial values of parabolic equations with Robin’s boundary conditions is studied,and the optimal control problem of the control function acting inside the region and the optimal control problem of t he control function acting at the region boundary are investigated.Similar to the study in Chapter 2,the suitability of weak solutions of the third initial marginal value problem for parabolic equations and the existence of optimal control are obtained.Chapter 4 of this paper summarizes the important results and research methods obtained in the paper.This paper proves the fitness and existence of optimal control for two types of elliptic equations and two types of parabolic equations,details the proof method,and introduces the new methods and ideas proposed in this paper.