一类抛物最优控制问题的有限元误差估计

Error estimates of the finite element method for a class of parabolic optimal control problems

对一类抛物最优控制问题给出了有限元逼近格式,其中控制约束集为积分受限的形式K{=u (t)∈L^2(Ω):a≤∫_Ω u(t)≤}b。对问题的状态变量和伴随状态变量用线性连续函数离散,而控制变量使用分片常数近似;最后得到控制和状态变量逼近的先验误差估计O(h^2+k)。

In this paper,we study a finite element approximation scheme for a class of parabolic optimal control problems. The control constraint is given in an integral sense： K{= u（t） ∈L^2（Ω） ： a≤ ∫Ω u（t） ≤}b,where the state and co-state variables are discretized by piecewise linear continuous functions and the control variable is approximated by piecewise constant functions. Some error estimates are derived for both control and state approximations. It is proven that these approximations have convergence order O（h^2＋ k）.