  In this paper, we investigate an optimal control problem governed by
parabolic equations with measure-valued controls over time. We establish the
well-posedness of the optimal control problem and derive the first-order
optimality condition using Clarke's subgradients, revealing a sparsity
structure in time for the optimal control. Consequently, these optimal control
problems represent a generalization of impulse control for evolution equations.
To discretize the optimal control problem, we employ the space-time finite
element method. Here, the state equation is approximated using piecewise linear
and continuous finite elements in space, alongside a Petrov-Galerkin method
utilizing piecewise constant trial functions and piecewise linear and
continuous test functions in time. The control variable is discretized using
the variational discretization concept. For error estimation, we initially
derive a priori error estimates and stabilities for the finite element
discretizations of the state and adjoint equations. Subsequently, we establish
weak-* convergence for the control under the norm $\mathcal{M}(\bar
I_c;L^2(\omega))$, with a convergence order of
$O(h^\frac{1}{2}+\tau^\frac{1}{4})$ for the state.
