Necessary Conditions for Optimality
Necessary Conditions for Optimality
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DOI：
10.1007/978-1-4612-0737-5_12
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发表时间：
1996
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影响因子：
--
通讯作者：
J. Troutman
中科院分区：
文献类型：
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作者：
J. Troutman
在我们的问题中，什么样的条件是最佳性能的必要条件？在第10章中我们看到，如果一个控制问题可以在一个固定的区间上表述，并且它的定义函数是适当凸的，那么变分法可以用来给出最优控制的充分条件。特别地，§10.3和§10.4的最小值原理可以保证问题的解的最优性。在§11.1中，我们将发现，无论凸性是否存在，这个原则对于最优性都是必要的，即使当基础区间不固定时也是如此（定理11.10）。然后在§11.2中，我们研究了一类简单但重要的线性时间最优问题，对于这类问题，时间间隔本身被最小化，伴随方程（一个必要条件）可以用来给出最优性的充分条件。最后，在§11.3中，我们将控制论方法推广到涉及拉格朗日不等式约束的更一般的问题，在定理11.20中，我们得到了Kuhn-Tucker型的拉格朗日乘子规则。
What conditions are necessary for optimal performance in our problems? In Chapter 10 we saw that if a control problem can be formulated on a fixed interval and its defining functions are suitably convex, then the methods of variational calculus can be adapted to suggest sufficient conditions for an optimal control. In particular, the minimum principle of §10.3 and §10.4 can guarantee optimality of a solution to the problem. In §11.1 we will discover that this principle is necessary for optimality whether or not convexity is present, even when the underlying interval is not fixed (Theorem 11.10). Then in §11.2, we examine the simple but important class of linear time-optimal problems for which the time interval itself is being minimized and the adjoint equation (a necessary condition) can be used to suggest sufficient conditions for optimality. Finally, in §11.3, we extend our control-theory approach to more general problems involving Lagrangian inequality constraints, and in Theorem 11.20 we obtain a Lagrangian multiplier rule of the Kuhn-Tucker type.