抛物型最优控制问题的全离散Crank-Nicolson有限元法

Fully Discrete Finite Element Method Based on the Crank-Nicolson Scheme for Parabolic Optimal Control Problem

针对具有积分控制约束的抛物型最优控制问题,提出了一种基于Crank-Nicolson格式的全离散有限元法.使用分段线性有限元对状态进行空间离散,采用Crank-Nicolson格式进行时间离散,对控制变量采用分段线性近似,从而得到离散的最优性系统.证明了状态变量、伴随状态变量和控制变量的误差估计,并通过数值算例验证理论结果.

Aiming at parabolic optimal control problem with integral control constraints,a fully discrete finite element method based on Crank-Nicolson scheme is proposed.The state is discretized by piecewise linear finite elements for the space discretization,Crank-Nicolson scheme for time discretization,and the control variables are approximated by piecewise linear function approximation,so as to obtain a discrete optimal system.The error estimates of state variables,adjoint state variables and control variables are proved,and the theoretical results are verified by numerical examples.