Spectral Methods For Elliptic And Fractional Diffusion Optimal Control Problems

Optimal control problems governed by partial differential equations(referred to as control problems)are widely used in real life,such as economics,engineering and other fields.In recent decades,the theoretical and numerical methods of optimal control problems have been rapidly developed;In recent years,the research and de-velopment of control problems governed by fractional differential equations(referred to as fractional control problem)is relatively late.The theoretical research and nu-merical simulation of fractional control problem have become a hot issue.In view of the advantages of spectral methods,the development of spectral element methods and adaptive spectral element methods for optimal control problems and fractional control problems requires further consideration and discussion.In order to analyze and improve efficient numerical methods,we studies this paper spectral approxima-tion and spectral element discretization for elliptic control problem with state and control constraints,and spectral discretization for fractional control problem.This paper mainly discusses the following aspects:Firstly,we investigate spectral discretization for elliptic control problem with L~2-norm state constraint,and derived the optimality conditions rigorously.Then spectral discretization for the problem is constructed.Next,a prior error analysis of spectral approximation are investigated in detail.Moreover,L~2-H~1a posterior error analysis are established.Furthermore,L~2-L~2a posterior error analysis are investigated for the problem.Finally,numerical experiments are completed to confirm the effective of spectral methods;Secondly,we consider spectral discretization for elliptic control problem with L~2-norm control constraint,and derived the optimality conditions.Then we established spectral approximation of the problem.Next,a prior error analysis are established for the state and control.Moreover,L~2-H~1a posterior error analysis and L~2-L~2a posterior error analysis are investigated.Finally,we carry out numerical experiments to confirm the effectiveness of spectral methods;Thirdly,we develop hp spectral element discretization for elliptic control prob-lem with state constraint and control constraint,respectively.According to the optimality conditions,a posterior error estimates of spectral element discretization are derived,and the numerical experiments verify the effective of spectral element methods,that the convergence rate can be improved by increasing the number of elements or increasing the polynomial order;Furthermore,the spectral approximation for the control constrained optimal control problem with space-time fractional diffusion equation is analyzed.First,we consider fractional optimal control problem without control constraints,and analyze a priori error estimates.Next,we investigate fractional optimal control problem with control constraints,and derive the optimality conditions.Then a prior error analysis are derived for the control problem,as well as a posterior error analysis are proved rigorously.Both the time direction and space directions can achieve spectral accuracy;Finally,spectral approximation for optimal control problems governed by space-time fractional diffusion equations with state constraint is considered,First,the op-timality conditions are derived and spectral approximation of the control problem is constructed.Then a prior error analysis are established for the control problem,and a posterior error analysis are proved,the analysis results show that this method can achieve spectral accuracy in both time and space directions.