### Crank-Nicolson finite difference method for parabolic optimal control problem

YANG Cai-jie, SUN Tong-jun*

1. School of Mathematics, Shandong University, Jinan 250100, Shandong, China
• Published:2020-06-01

Abstract: A one dimensional parabolic optimal control problem with Neumann boundary condition is considered. The co-state equations and optimality conditions are presented and the optimality system is obtained. By applying a ghost-point based central difference approximation to the boundary condition, Crank-Nicolson finite difference discrete schemes are established for the optimality system. The maximum norm error estimates are proved to be of second-order convergence in both time and space for the state, co-state and control variables. Finally, numerical example is presented. In order to avoid solving large coupled algebraic equations, the iterative method is used. The numerical results validate the theoretical conclusion.

CLC Number:

• O241.82
 [1] LIONS J L. Optimal control of systems governed by partial differential equations[M]. Berlin: Springer-Verlag, 1971.[2] NEITTAANMÄKI P, TIBA D. Optimal control of nonlinear parabolic systems, theory, algorithms and applications[M]. New York: CRC Press, 1994.[3] FU Hongfei, RUI Hongxing. A priori error estimates for optimal control problem governed by transient advection-diffusion equations[J]. J Sci Comput, 2009, 38(3):290-315.[4] FU Hongfei, RUI Hongxing. A characteristic-mixed finite element method for time-dependent convection-diffusion optimal control problem[J]. Appl Math Comput, 2011, 218(7):3430-3440.[5] 郑瑞瑞,孙同军. 一类捕食与被捕食模型最优控制的有限元方法的先验误差估计[J]. 山东大学学报(理学版), 2020, 55(1):23-32. ZHENG Ruirui, SUN Tongjun. A priori error estimates of finite element methods for an optimal control problem governed by a one prey and one predator model[J]. Journal of Shandong University(Natural Science), 2020, 55(1):23-32.[6] 华冬英,李祥贵. 微分方程的数值解法与程序实现[M]. 北京: 电子工业出版社,2016. HUA Dongying, LI Xianggui. Numerical method and program realization for differential equation[M]. Beijing: Publishing House of Electronics Industry, 2016.[7] APEL T, FLAIG T G. Crank-Nicolson schemes for optimal control problems with evolution equations[J]. SIAM J Numer Anal, 2012, 50(3):1484-1512.[8] LIU Jun. Two fast finite difference schemes for elliptic Dirichlet boundary control problems[J]. J Appl Math Comput, 2019, 61(1/2):481-503.[9] ADAMS R. Sobolev spaces[M]. New York: Academic, 1975.[10] LIU Wenbin, YAN Ningning. Adaptive finite element method for optimal control governed by PDEs[M]. Beijing: Science Press, 2008.[11] THOMAS J W. Numerical partial differential equations, finite difference methods[M]. Beijing: World Publishing Corporation, 1997.[12] 刘文月,孙同军. 椭圆方程约束的最优边界控制问题的非重叠型区域分解迭代方法[J]. 山东大学学报(理学版), 2016, 51(2):21-28. LIU Wenyue, SUN Tongjun. Iterative non-overlapping domain decomposition method for optimal boundary control problems governed by elliptic equations[J]. Journal of Shandong University(Natural Science), 2016, 51(2):21-28.
 [1] LI Juan. Error analysis of a linearized Crank-Nicolson scheme for the phase field crystal equation [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(6): 118-126. [2] ZHANG Fei-ran. The lumped mass nonconforming finite element method of the Crank-Nicolson scheme for nonstationary Stokes problem [J]. J4, 2011, 46(12): 33-38. [3] ZHANG Xin-Dong, HU Yu-Hong. On the exact solution of a diffusion equation with boundary conditions by the homotopy analysis method [J]. J4, 2008, 43(12): 73-76.
Viewed
Full text

Abstract

Cited

Shared
Discussed