Finite Element Method For Optimal Control Problem Governed By Time Fractional Diffusion Equation

Optimal control problem governed by fractional differential equations has wide applications in real\xworld problem,for example,the groundwater pollution problem.It has important theoretical significance and application value to develop numerical algorithm for this kind of problem.This paper mainly studies the finite element method for solving the following time fractional optimal control problems:Here u is the state variable,q is the control variable.R0??tu is the left Riemann-Liouville fractional derivative of order ?(0<?<1)about u.ud denotes the observed state,Uad denotes the control set,? is regularization constant.Firstly,we deduce the continuous first-order optimality condition and ana-lyze the regularity of the solution of the control problem.For the state equation,piecewise eonstant discontinuous Galerkin finite element method is used for timedirectization and piecewise linear finite element method is used for space directiza-tion,variational discretization method is applied to discretize the control variables.Then we build up the time diacontinuous Galerkin finite element scheme for time fractional optimal control problem.Based on the strategy of first discretize,then optimize,the discrete first-order optimality condition of the optimal control prob?lem is derived.The a prior error estimate of state variable and control variable is established by means of the finite element analysis techniques such as interpolate estimate and dual argument.Secondly,due to the nonlocal property of time fractional devivative the com?putational cost for solving the discrete algebraic system consisting of discrete state equation,adjoint state equation and optimal inequality is mach bigger than that in integer order optimal control problem.In order to improve the efficiency of finite el-ement method,we analyze the structure of discrete state equation and adjoint state equation,and then develop a fast projection gradient algorithm for time fractional optimal control problem on time uniform grid and time block uniform grid base on the block Toeplitz matrix.Finally,numerical examples are presented to illustrate the theoretical finding and effectiveness of fasc projection gradient algorithm.