Conjugate convex functions in optimal stochastic control
Conjugate convex functions in optimal stochastic control
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DOI：
10.1016/0022-247x(73)90066-8
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发表时间：
1973-11
影响因子：
1.3
通讯作者：
J. Bismut
中科院分区：
文献类型：
--
作者：
J. Bismut
本文讨论一般凸分析方法在最优随机控制问题中的应用。特别地，我们将定义什么是最优随机控制中的对偶问题，以及对偶最优的共极值条件是什么。我们在这里解决的问题比纯粹的确定性问题更一般，所给出的结果包括确定性控制的结果。这些方法和结果的阐述与Rockafella在[13]中使用的相应方法非常相似，我们将经常参考该方法。该方法的一个明显缺点是，使用严格的变分方法，它必须假设信息u-场是固定的。在某些情况下，其中信息U场是由状态变量生成的，可以将对偶方法应用于修改的问题。但它们不会给我们带来通过研究更专门的问题所可能获得的强有力的结果，例如最优马尔可夫控制的存在性。我们在文献[2]中发展了其他方法来解决这类问题。显而易见的原因是，通过发展一种适用于纯确定性情况的形式，对于随机情况，它在一些纯随机情况下没有使用问题的随机特征。
This paper is concerned with the applications of general methods of convex analysis to problems of optimal stochastic control. In particular we will define what dual problems are in optimal stochastic control, and what the coextremality conditions for dual optimums are. The problem that we solve here being more general than a purely deterministic one, the results which are given include the results of deterministic control. The methods and the exposition of the results are very similar to the corresponding methods used by Rockafellar in [13], to which we will refer constantly.One of the apparent shortcomings of the method is that, using strictly variational methods, it must suppose that the information u-fields are fixed. In some cases, where the information u-fields are generated by the state variable, it is possible to apply the duality methods to a modified problem. But they will not give us the strong results it is possible to obtain by studying more specialized problems, as existence of optimal Markov controls. We develop other methods in [2] for this type of problem. The obvious reason is that, by developing a formalism applicable to purely deterministic cases, as to stochastic cases, it does not use the stochastic features of the problem in some purely stochastic cases.