  In this paper we consider a parabolic optimal control problem with a Dirac
type control with moving point source in two space dimensions. We discretize
the problem with piecewise constant functions in time and continuous piecewise
linear finite elements in space. For this discretization we show optimal order
of convergence with respect to the time and the space discretization parameters
modulo some logarithmic terms. Error analysis for the same problem was carried
out in the recent paper [17], however, the analysis there contains a serious
flaw. One of the main goals of this paper is to provide the correct proof. The
main ingredients of our analysis are the global and local error estimates on a
curve, that have an independent interest.
