We consider variational discretization [18] of a parabolic optimal control problem governed by space-time measure controls. For the state discretization we use a Petrov-Galerkin method employing piecewise constant states and piecewise linear and continuous test functions in time. For the space discretization we use piecewise linear and continuous functions. As a result the controls are composed of Dirac measures in space-time, centered at points on the discrete space-time grid. We prove that the optimal discrete states and controls converge strongly in $ L^q $ and weakly-$ * $ in $ \mathcal{M} $, respectively, to their smooth counterparts, where $ q \in (1,\min\{2,1+2/d\}] $ is the spatial dimension. Furthermore, we compare our approach to [8], where the corresponding control problem is discretized employing a discontinuous Galerkin method for the state discretization and where the discrete controls are piecewise constant in time and Dirac measures in space. Numerical experiments highlight the features of our discrete approach.

中文翻译：

---

测度抛物最优控制的最大离散稀疏性

我们考虑变分离散[18岁时空测度控制控制的抛物线最优控制问题。对于状态离散化，我们使用Petrov-Galerkin方法，该方法采用分段恒定状态以及分段线性和连续测试函数。对于空间离散化，我们使用分段线性和连续函数。结果，控件由时空Dirac度量组成，并以离散时空网格上的点为中心。我们证明最优离散状态和控制分别在$ L ^ q $和$ \ mathcal {M} $中分别弱收敛至平滑对等体，其中$ q \ in（1，\ min \ {2,1 + 2 / d \}] $是空间维度。此外，我们比较了[8]，其中使用状态不离散化的不连续Galerkin方法离散化相应的控制问题，以及离散控制的时间分段性和Dirac测度空间离散。数值实验突出了我们离散方法的特征。