This paper is concerned with second-order optimality conditions for Tikhonov regularized optimal control problems governed by the obstacle problem. Using a simple observation that allows to characterize the structure of optimal controls on the active set, we derive various conditions that guarantee the local/global optimality of first-order stationary points and/or the local/global quadratic growth of the reduced objective function. Our analysis extends and refines existing results from the literature and also covers those situations where the problem at hand involves additional box-constraints on the control. As a byproduct, our approach shows in particular that Tikhonov regularized optimal control problems for the obstacle problem can be reformulated as state-constrained optimal control problems for the Poisson equation and that problems involving a subharmonic obstacle and a convex objective function are uniquely solvable. The paper concludes with three counterexamples which illustrate that rather peculiar effects can occur in the analysis of second-order optimality conditions for optimal control problems governed by the obstacle problem and that necessary second-order conditions for such problems may be hard to derive.

本文关注由障碍问题控制的 Tikhonov 正则化最优控制问题的二阶最优性条件。使用允许表征活动集上最优控制结构的简单观察，我们推导出保证一阶驻点的局部/全局最优性和/或简化目标函数的局部/全局二次增长的各种条件。我们的分析扩展和改进了文献中的现有结果，还涵盖了手头问题涉及控件上额外框约束的情况。作为副产品，我们的方法特别表明，障碍问题的 Tikhonov 正则化最优控制问题可以重新表述为 Poisson 方程的状态约束最优控制问题，并且涉及次谐波障碍和凸目标函数的问题是唯一可解的。论文最后给出了三个反例，说明在分析受障碍问题支配的最优控制问题的二阶最优性条件时会出现相当奇特的效果，而且此类问题的必要二阶条件可能难以推导出。