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In this paper, we establish error estimates for the numerical approximationof the parabolic optimal control problem with measure data in a two-dimensionalnonconvex polygonal domain. Due to the presence of measure data in the stateequation and the nonconvex nature of the domain, the finite element erroranalysis is not straightforward. Regularity results for the control problembased on the first-order optimality system are discussed. The state variableand co-state variable are approximated by continuous piecewise linear finiteelement, and the control variable is approximated by piecewise constantfunctions. A priori error estimates for the state and control variable arederived for spatially discrete control problem and fully discrete controlproblem in $L^2(L^2)$-norm. A numerical experiment is performed to illustrateour theoretical findings.

Error analysis for finite element approximations to the parabolic optimal control problem with measure data in a nonconvex polygon