<p style='text-indent:20px;'>We consider one-dimensional distributed optimal control problems with the state equation being given by the viscous Burgers equation. We discretize using a space-time discontinuous Galerkin approach. We use upwind flux in time and the symmetric interior penalty approach for discretizing the viscous term. Our focus is on the discretization of the convection terms. We aim for using conservative discretizations for the convection terms in both the state and the adjoint equation, while ensuring that the approaches of discretize-then-optimize and optimize-then-discretize commute. We show that this is possible if the arising source term in the adjoint equation is discretized properly, following the ideas of well-balanced discretizations for balance laws. We support our findings by numerical results.</p>

中文翻译：

---

涉及粘性 Burgers 方程的最优控制问题的保守时空 DG 离散化的交换性质

<p style='text-indent:20px;'>我们考虑一维分布式最优控制问题，状态方程由粘性 Burgers 方程给出。我们使用时空不连续 Galerkin 方法进行离散化。我们使用逆风通量及时和对称内部惩罚方法来离散粘性项。我们的重点是对流项的离散化。我们的目标是对状态和伴随方程中的对流项使用保守离散化，同时确保离散后优化和优化后离散化的方法可以通勤。我们表明，如果伴随方程中出现的源项被适当地离散化，遵循平衡定律的良好平衡离散化的思想，这是可能的。我们通过数值结果支持我们的发现。</p>