Finite Element Error Estimation for Parabolic Optimal Control Problems with Pointwise Observations  

作　　者：Dongdong Liang Wei Gong Xiaoping Xie 

机构地区：[1]School of Mathematics,Sichuan University,Chengdu 610064,China [2]LSEC,Institute of Computational Mathematics,Academy of Mathematics and Systems Science,Chinese Academy of Sciences&School of Mathematical Sciences,University of Chinese Academy of Sciences,Beijing 100190,China

出　　处：《Numerical Mathematics(Theory,Methods and Applications)》2022年第1期165-199,共35页高等学校计算数学学报（英文版）

基　　金：supported in part by the Strategic Priority Research Program of Chi-nese Academy of Sciences(Grant No.XDB 41000000);the National Key Basic Research Program(Grant No.2018YFB0704304);the National Natural Science Foundation of China(Grants No.12071468,11671391);Xiaoping Xie was supported in part by the National Natural Science Foundation of China(Grants No.12171340,11771312).

摘　　要：In this paper,we consider parabolic distributed control problems with cost functional of pointwise observation type either in space or in time.First,we show the well-posedness of the optimization problems and derive the first order optimality systems,where the adjoint state can be expressed as the linear combination of solutions to two backward parabolic equations that involve the Dirac delta distribution as source either in space or in time.Second,we use a space-time finite element method to discretize the control problems,where the state variable is approximated by piecewise constant functions in time and continuous piecewise linear polynomials in space,and the control variable is discretized by following the variational discretization concept.We obtain a priori error estimates for the control and state variables with order O(k 12+h)up to a logarithmic factor under the L 2-norm.Finally,we perform several numerical experiments to support our theoretical results.

关 键 词：Parabolic optimal control problem pointwise observation space-time finite element method parabolic PDE with Dirac measure error estimate 

分 类 号：O241.82[理学—计算数学]