论文标题
椭圆形变体不平等的最佳控制和定向可不同性
Optimal control and directional differentiability for elliptic quasi-variational inequalities
论文作者
论文摘要
我们专注于障碍类型的椭圆形变体不平等（QVI），并证明了有关解决方案的存在，定向可区分性和对此类QVI的最佳控制的许多结果。我们根据订单方法，迭代方案和通过部分微分方程的顺序正则化提供了三个存在定理。我们表明，将源术语纳入QVI的解决方案集的解决方案图在通用数据和局部hadamard可区分的障碍物映射方向上是可以差异的，从而尤其扩展了我们先前工作的结果，这为无限维度提供了QVI的第一个不同性结果。还考虑了具有QVI约束的最佳控制问题，我们为控制问题提供了各种形式的平稳性条件，因此在该领域的第一个结果中提供了。
We focus on elliptic quasi-variational inequalities (QVIs) of obstacle type and prove a number of results on the existence of solutions, directional differentiability and optimal control of such QVIs. We give three existence theorems based on an order approach, an iteration scheme and a sequential regularisation through partial differential equations. We show that the solution map taking the source term into the set of solutions of the QVI is directionally differentiable for general data and locally Hadamard differentiable obstacle mappings, thereby extending in particular the results of our previous work which provided the first differentiability result for QVIs in infinite dimensions. Optimal control problems with QVI constraints are also considered and we derive various forms of stationarity conditions for control problems, thus supplying among the first such results in this area.
