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.

翻译：本文研究由带时间测度值控制的抛物型方程支配的最优控制问题。我们建立了最优控制问题的适定性，并利用Clarke次梯度推导出一阶最优性条件，揭示了最优控制的时间稀疏结构。因此，这些最优控制问题代表了发展方程脉冲控制的一种推广。为离散该最优控制问题，我们采用时空有限元方法。其中，状态方程在空间上采用分片线性连续有限元逼近，同时在时间上采用以分片常数试验函数和分片线性连续检验函数的Petrov-Galerkin方法进行求解。控制变量采用变分离散概念进行离散化。在误差估计方面，我们首先推导了状态方程和伴随方程有限元离散的先验误差估计和稳定性。随后，建立了控制变量在范数$\mathcal{M}(\bar I_c;L^2(\omega))$下的弱*收敛性，状态变量收敛阶为$O(h^\frac{1}{2}+\tau^\frac{1}{4})$。