一类非线性抛物最优控制问题的有限元误差估计

Error Estimates for the Finite Element Approximation of Nonlinear Parabolic Optimal Control Problems

该文对一类非线性抛物最优控制问题给出了有限元逼近格式,并讨论了两种不同类型的控制约束集.文中对状态和伴随状态变量采用了线性连续函数离散,而控制变量则由分片常函数近似.得到了控制和状态逼近的先验误差估计■(h_U+h+k),这里h_U与h分别表示控制和状态的空间网格步长,k表示时间步长.数值试验表明了算法的有效性.

In this paper,the finite element approximation to a class of nonlinear optimal control problems with two different kinds of control constrained sets is investigated,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 a priori error estimates are derived for both control and state approximations.It is proven that these approximations have convergence order■（h_U＋h＋k）,where h_U and h are the spatial mesh-sizes for control and state,respectively,and k is the time increment.Numerical examples are given to show the efficiency of the present scheme.