论文标题
一种用于最佳控制问题的时空变分方法：适应性，稳定性和数值解决方案
A Space-Time Variational Method for Optimal Control Problems: Well-posedness, stability and numerical solution
论文作者
论文摘要
我们考虑了一个具有罗宾边界条件的抛物线偏微分方程（PDE）的最佳控制问题。我们在Lebesgue（较少规律性的Bochner空间）中使用了良好的时空变分配方。最佳控制问题的抽象公式产生了Lagrange功能，Karush-Kuhn--tucker（KKT）条件自然。这会导致Lebesgue的伴随和梯度方程的时空变异公式，而Bochner空间具有最小的规律性。制定了必要和足够的最佳条件，并且最佳系统被证明是良好的。 接下来，我们在这些Lebesgue-Boch \ -ner空间中引入了一个均匀稳定稳定的同时时空（张量）离散化。使用在试验和测试空间的时空和时间上适当订单的有限元素，该设置已知与抛物线寄生虫问题的曲柄 - Nicolson时间步变方案相当。详细介绍了与现有方法的差异。 我们显示了与时间步长方法的数值比较。时空方法显示出良好的稳定性，并且需要更少的时间自由度才能达到相同的精度。
We consider an optimal control problem constrained by a parabolic partial differential equation (PDE) with Robin boundary conditions. We use a well-posed space-time variational formulation in Lebesgue--Bochner spaces with minimal regularity. The abstract formulation of the optimal control problem yields the Lagrange function and Karush--Kuhn--Tucker (KKT) conditions in a natural manner. This results in space-time variational formulations of the adjoint and gradient equation in Lebesgue--Bochner spaces with minimal regularity. Necessary and sufficient optimality conditions are formulated and the optimality system is shown to be well-posed. Next, we introduce a conforming uniformly stable simultaneous space-time (tensorproduct) discretization of the optimality system in these Lebesgue--Boch\-ner spaces. Using finite elements of appropriate orders in space and time for trial and test spaces, this setting is known to be equivalent to a Crank--Nicolson time-stepping scheme for parabolic problems. Differences to existing methods are detailed. We show numerical comparisons with time-stepping methods. The space-time method shows good stability properties and requires fewer degrees of freedom in time to reach the same accuracy.
