JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2020, Vol. 55 ›› Issue (6): 115-121.doi: 10.6040/j.issn.1671-9352.0.2020.101

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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

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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.

Key words: parabolic optimal control problem, Crank-Nicolson scheme, co-state equation, optimality system, maximum norm error estimate

CLC Number: 

  • O241.82
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