关于双边障碍问题解算子的广义导数特征
On the Characterization of Generalized Derivatives for the Solution Operator of the Bilateral Obstacle Problem
摘要：
我们考虑一类双边障碍问题的最优控制问题，其中控制出现在一个可能的非线性源项。求解算子的不可微性是有效最优化方法应用的主要挑战，而求解算子的Bouligand广义导数的刻画是其理论基础和数值实现的关键.本文在控制算子满足自然单调性的条件下，导出了Bouligand广义微分的特殊元素.我们构造单调序列的控制，其中的解决方案运营商是Gâteaux可微和特征相应的极限元素的Bouligand广义微分作为一个狄利克雷问题的解决方案运营商的准开域。与单边障碍问题[RU 19]的类似结果相反，我们必须处理对应于上障碍和下障碍的活动集和严格活动集的相反单调行为。此外，残差不再是H^{-1}上的非负泛函，其表示为两个非负Radon测度之差需要特别注意。这需要新的证明技术，产生两个元素的Bouligand广义微分。同样对于单边情形，我们得到了[RU 19]中导出的元素的附加元素。
英文摘要：
We consider optimal control problems for a wide class of bilateral obstacle problems where the control appears in a possibly nonlinear source term. The non-differentiability of the solution operator poses the main challenge for the application of efficient optimization methods and the characterization of Bouligand generalized derivatives of the solution operator is essential for their theoretical foundation and numerical realization. In this paper, we derive specific elements of the Bouligand generalized differential if the control operator satisfies natural monotonicity properties. We construct monotone sequences of controls where the solution operator is Gâteaux differentiable and characterize the corresponding limit element of the Bouligand generalized differential as being the solution operator of a Dirichlet problem on a quasi-open domain. In contrast to a similar recent result for the unilateral obstacle problem [RU19], we have to deal with an opposite monotonic behavior of the active and strictly active sets corresponding to the upper and lower obstacle. Moreover, the residual is no longer a nonnegative functional on H^{-1} and its representation as the difference of two nonnegative Radon measures requires special care. This necessitates new proof techniques that yield two elements of the Bouligand generalized differential. Also for the unilateral case we obtain an additional element to that derived in [RU19].
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
