This thesis addresses the numerical analysis of sparse control problems for hyperbolic and parabolic state equations. The focus lies on functions of bounded Variation in time which affect fixed spatial functions. An optimal control framework using a sparsity preserving regularization and a semismooth Newton method is developed and analyzed. In parallel, a sparse Bayesian approach for the wave equation is developed and compared to the corresponding control problem. For a specific optimal control problem and a sparse Bayesian problem a priori error estimates, for a finite element discretization, are derived.