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Preface Issue 1-2015

机译：前言第1-2015期

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摘要

A typical example from optimal control with partial differential equations is as follows. Consider a domain where one wants to achieve a desired temperature distribution y_d- For this purpose one may use heat sources u which, at a given time, will result in a real temperature distribution y_u. In order to compute y_u from u one has to solve a typically nonlinear heat equation. In most cases y_d and y_u will be different. The goal will be to find a control u such that y_d - y_u becomes as small as possible in a suitable sense while at the same time the costs for the heating u stay as low as possible. Both requirements are combined in a functional u→ J(u) and the task is to decide whether a given control u minimises J. As in first year calculus, one will have to take a look at the equation J'(u) = 0. Unless J is convex it will be very difficult to detect global minima, but one would also be glad to find local minima. To this end, one will have to check the positive definiteness of J"(u). Unlike first year calculus, the space of admissible controls is now infinite dimensional, and one has to consider different norms in order to ensure differentiability and positive definiteness. Starting with this observation, Eduardo Casas and Fredi Tröltzsch explain in a very comprehensible way "Second order optimality conditions and their role in PDE control."
机译：带有偏微分方程的最优控制的典型示例如下。考虑一个想要获得所需温度分布y_d的领域。为此，可以使用热源u，该热源在给定的时间将产生实际的温度分布y_u。为了从u计算y_u，必须求解一个典型的非线性热方程。在大多数情况下，y_d和y_u将不同。目标是找到一种控制u，以使y_d-y_u在适当的意义上变得尽可能小，同时加热u的成本保持尽可能低。将这两个要求组合在一个函数u→J（u）中，任务是确定给定的控制u是否最小化J。与第一年微积分一样，必须考虑方程J'（u）= 0除非J是凸的，否则将很难检测全局最小值，但是也很高兴找到局部最小值。为此，必须检查J“（u）的正定性。与第一年的演算不同，现在允许的控制空间是无限维的，并且必须考虑不同的规范以确保可微性和正定性。从这一观察开始，爱德华多·卡萨斯（Eduardo Casas）和弗雷迪·特洛茨奇（FrediTröltzsch）以一种非常容易理解的方式解释了“二阶最优条件及其在PDE控制中的作用”。

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