Finite element error estimation for parabolic optimal control problems with time delay

Abstract

In this paper, we develop a priori error estimates for the finite element approximations of parabolic optimal control problems with time delay and pointwise control constraints. At first, we derive the first-order optimality systems for the control problems and the corresponding regularity results. Then, to approximate the problem we use the piecewise linear and continuous finite elements for the space discretization of the state, while the piecewise constant discontinuous Galerkin method is used for the time discretization. For the control discretization, we consider variational discretization. We show $O(k+h^2)$ order of convergence rate for the control in the $L^2$ norm, which is new to the best of our knowledge. Finally, some numerical examples are provided to confirm our theoretical results.