  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.
