In this paper, we establish error estimates for the numerical approximation of the parabolic optimal control problem with measure data in a two-dimensional nonconvex polygonal domain. Due to the presence of measure data in the state equation and the nonconvex nature of the domain, the finite element error analysis is not straightforward. Regularity results for the control problem based on the first-order optimality system are discussed. The state variable and co-state variable are approximated by continuous piecewise linear finite element, and the control variable is approximated by piecewise constant functions. A priori error estimates for the state and control variable are derived for spatially discrete control problem and fully discrete control problem in $L^2(L^2)$-norm. A numerical experiment is performed to illustrate our theoretical findings.
翻译:在本文中, 我们为抛物线最佳控制问题的数字近似值设定了误差估计值, 在二维非convex多边形域中测量数据。 由于在状态方程式中存在测量数据, 且域的非convex性质, 有限元素误差分析并不直截了当。 讨论基于一阶优化系统的控制问题的规律性结果。 状态变量和共同状态变量以连续的小片线性线性有限元素为近似值, 控制变量则以小片常数函数为近似值。 状态变量和控制变量的先验误估计值是针对空间离散控制问题和完全离散控制问题得出的, 以$L2 (L)2美元- 诺尔姆为单位。 进行了一个数字实验, 以说明我们的理论结论 。