论文标题
关于双边障碍物问题的溶液操作员的广义衍生物的表征
On the Characterization of Generalized Derivatives for the Solution Operator of the Bilateral Obstacle Problem
论文作者
论文摘要
我们考虑了控制控制以非线性源术语出现的各种双边障碍问题的最佳控制问题。解决方案运算符的非差异性对应用有效优化方法的应用构成了主要挑战，而解决方案操作员的Bouligand广义衍生物的表征对于其理论基础和数值实现至关重要。在本文中，如果对照操作员满足自然单调性能，我们将得出Bouligand广义差异的特定元素。我们构建了对控件的单调序列，其中解决方案操作员是可差异的，并表征了Bouligand广义差异的相应限制元素，是在准打开域上dirichlet问题的解决方案操作员。与单侧障碍物问题的最新结果相比[RU19]，我们必须处理与上下障碍物相对应的主动和严格活动集的相反单调行为。此外，残留不再是H^{ - 1}的非负功能，其表示为两种非负ra量措施的差异需要特殊护理。这需要新的证明技术产生布利格的两个元素广义差异。同样对于单方面的情况，我们获得了[RU19]中得出的附加元素。
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].
