Optimal a priori error estimates of parabolic optimal control problems with pointwise control (Q515857)

A parabolic optimal control problem with a pointwise (Dirac type) control in space, but variable in time, is solved using continuous piecewise linear approximation in space and the piecewise constant discontinuous Galerkin method in time. The adjoint state is more regular than the state equation, opposed to the optimal control problems with state constraints. The authors show that the error estimates for the control can be improved to the almost optimal order \({\mathcal {O}}(k + h^2)\) in the \(L^2\) norm in contrast to the suboptimal error estimates of order \({\mathcal {O}}(k^{1/2} + h)\) in [\textit{W. Gong} et al., ``A Priori Error Analysis for Finite Element Approximation of Parabolic Optimal Control Problems with Pointwise Control'', SIAM J. Control Optim. 52, No. 1, 97--119 (2014)]. Numerical examples for two dimensional problems in space illustrate the error estimates.