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首页> 外文期刊>Jahresbericht der Deutschen Mathematiker-Vereinigung >Second Order Optimality Conditions and Their Role in PDE Control
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Second Order Optimality Conditions and Their Role in PDE Control

机译：二阶最优条件及其在PDE控制中的作用

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If f : R~n→ R is twice continuously differentiable, f'(u)= 0 and f"(u) is positive definite, then u is a local minimizer of f. This paper surveys the extension of this well known second order sufficient optimality condition to the case f: U → R, where U is an infinite-dimensional linear normed space. The reader will be guided from the case of finite dimensions via a brief discussion of the calculus of variations and the optimal control of ordinary differential equations to the control of nonlinear partial differential equations, where U is a function space. In particular, the following questions will be addressed: Is the extension to infinite dimensions straightforward or will unexpected difficulties occur? How have second order sufficient optimality conditions to be modified, if simple inequality constraints are imposed on u? Why do we need second order conditions and how can they be applied? If they are important, are we able to check if they are fulfilled? It turns out that infinite dimensions cause new difficulties that do not occur in finite dimensions. We will be faced with the surprising fact that the space, where f"(u) exists can be useless to ensure positive definiteness of the quadratic form v→ f"(u)v~2. In this context, the famous two-norm discrepancy, its consequences, and techniques for overcoming this difficulty are explained. To keep the presentation simple, the the-ory is developed for problems in function spaces with simple box constraints of the form α ≤ u ≤ β. The theory of second order conditions in the control of partial differential equations is presented exemplarily for the nonlinear heat equation. Different types of critical cones are introduced, where the positivity of f"(u) must be required. Their form depends on whether a so-called Tikhonov regularization term is part of the functional f or not. In this context, the paper contains also new results that lead to quadratic growth conditions in the strong sense. As a first application of second order sufficient conditions, the stability of optimal solutions with respect to perturbations of the data of the control problem is discussed. Second, their use in analyzing the discretization of control problems by finite elements is studied. A survey on further related topics, open questions, and relevant literature concludes the paper.
机译：如果f：R〜n→R是两次连续可微的，f'（u）= 0且f“（u）是正定的，则​​u是f的局部极小值。对于情形f的充分最优条件：U→R，其中U是一个无穷维线性范数空间，将通过对变量演算和常微分的最优控制的简要讨论，从有限维的情况下引导读者非线性偏微分方程的控制方程，其中U是一个函数空间，尤其要解决以下问题：扩展到无限维是直接的还是会发生意料之外的困难？如何修改二阶足够的最优条件，如果对u施加简单的不等式约束？为什么我们需要二阶条件以及如何应用它们？如果它们很重要，我们是否能够检查它们是否满足？证明无限尺寸会导致新的困难，而这些困难在有限的尺寸中不会发生。我们将面对令人惊讶的事实，即存在f“（u）的空间对于确保二次型v→f”（u）v〜2的正定性可能是无用的。在这种情况下，解释了著名的两范数差异，其后果以及克服此困难的技术。为了使表示更简单，该理论针对函数空间中具有α≤u≤β的简单框约束的问题而开发。非线性热方程示例性地提出了偏微分方程控制中的二阶条件理论。引入了不同类型的临界锥，其中必须要求f“（u）的正性。它们的形式取决于所谓的Tikhonov正则化项是否为函数f的一部分。在这种情况下，本文还包含导致强增长的二次增长条件的新结果。作为二阶充分条件的第一个应用，讨论了关于控制问题数据扰动的最优解的稳定性。第二，它们在离散化分析中的应用对有限元控制问题进行了研究，并对其他相关主题，未解决问题和相关文献进行了总结。

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