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We consider variational discretization of a parabolic optimal control problemgoverned by space-time measure controls. For the state discretization we use aPetrov-Galerkin method employing piecewise constant states and piecewise linearand continuous test functions in time. For the space discretization we usepiecewise linear and continuous functions. As a result the controls arecomposed of Dirac measures in space-time, centered at points on the discretespace-time grid. We prove that the optimal discrete states and controlsconverge strongly in $L^q$ and weakly-$*$ in $\mathcal{M}$, respectively, totheir smooth counterparts, where $q \in (1,\min\{2,1+2/d\}]$ is the spatialdimension. Furthermore, we compare our approach to a approach by Casas, E. andKunisch, K., where the corresponding control problem is discretized employing adiscontinuous Galerkin method for the state discretization and where thediscrete controls are piecewise constant in time and Dirac measures in space.Numerical experiments highlight the features of our discrete approach.

Maximal discrete sparsity in parabolic optimal control with measures