This paper considers the finite element approximation to parabolic optimal control problems with measure data in a nonconvex polygonal domain. Such problems usually possess low regularity in the state variable due to the presence of measure data and the nonconvex nature of the domain. The low regularity of the solution allows the finite element approximations to converge at lower orders. We prove the existence, uniqueness and regularity results for the solution to the control problem satisfying the first order optimality condition. For our error analysis we have used piecewise linear elements for the approximation of the state and co-state variables, whereas piecewise constant functions are employed to approximate the control variable. The temporal discretization is based on the implicit Euler scheme. We derive both a priori and a posteriori error bounds for the state, control and co-state variables. Numerical experiments are performed to validate the theoretical rates of convergence.

翻译：本文考虑非凸多边形域中含测度数据的抛物型最优控制问题的有限元逼近。由于测度数据的存在及域的非凸性，此类问题通常导致状态变量具有低正则性。解的低正则性使得有限元逼近的收敛阶较低。我们证明满足一阶最优性条件的控制问题解的存在性、唯一性及正则性结果。在误差分析中，我们采用分片线性元逼近状态变量和伴随状态变量，采用分片常数函数逼近控制变量。时间离散基于隐式欧拉格式。我们推导了状态、控制及伴随变量的先验和后验误差界。通过数值实验验证了理论收敛阶。