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A smooth regularization of the projection formula for constrained parabolic optimal control problems

机译：约束抛物线最优控制问题的投影公式的光滑正则化

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摘要

We present a smooth, that is, differentiable regularization of the projection formula that occurs in constrained parabolic optimal control problems. We summarize the optimality conditions in function spaces for unconstrained and control-constrained problems subject to a class of parabolic partial differential equations. The optimality conditions are then given by coupled systems of parabolic PDEs. For constrained problems, a non-smooth projection operator occurs in the optimality conditions. For this projection operator, we present in detail a regularization method based on smoothed sign, minimum and maximum functions. For all three cases, that is, (1) the unconstrained problem, (2) the constrained problem including the projection, and (3) the regularized projection, we verify that the optimality conditions can be equivalently expressed by an elliptic boundary value problem in the space-time domain. For this problem and all three cases we discuss existence and uniqueness issues. Motivated by this elliptic problem, we use a simultaneous space-time discretization for numerical tests. Here, we show how a standard finite element software environment allows to solve the problem and, thus, to verify the applicability of this approach without much implementation effort. We present numerical results for an example problem.
机译：我们提出了在受约束的抛物线形最优控制问题中出现的投影公式的平滑，即可微正则化。我们总结了服从一类抛物型偏微分方程的无约束和控制约束问题在函数空间中的最优条件。然后由抛物线形偏微分方程的耦合系统给出最佳条件。对于受约束的问题，在最佳条件下会出现不平滑的投影算子。对于此投影算子，我们详细介绍了一种基于平滑符号，最小和最大函数的正则化方法。对于这三种情况，即（1）无约束问题，（2）包括投影的约束问题，和（3）正则化投影，我们验证了最优条件可以等效地由椭圆边值问题表示。时空域。对于这个问题和所有这三种情况，我们讨论存在性和唯一性问题。受此椭圆问题的影响，我们将同时时空离散化用于数值测试。在这里，我们展示了标准的有限元软件环境如何解决该问题，从而无需过多的实现工作即可验证该方法的适用性。我们给出一个示例问题的数值结果。

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