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Dynamic programming equations for discounted constrained stochastic control

机译：折扣约束随机控制的动态规划方程

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摘要

In this paper, the application of the dynamic programming approach to constrained stochastic control problems with expected value constraints is demonstrated. Specifically, two such problems are analyzed using this approach. The problems analyzed are the problem of minimizing a discounted cost infinite horizon expectation objective subject to an identically structured constraint, and the problem of minimizing a discounted cost infinite horizon minimax objective subject to a discounted expectation constraint. Using the dynamic programming approach, optimality equations, which are the chief contribution of this paper, are obtained for these problems. In particular, the dynamic programming operators for problems with expectation constraints differ significantly from those of standard dynamic programming and problems with worst-case constraints. For the discounted cost infinite horizon cases, existence and uniqueness of solutions to the dynamic programming equations are explicitly shown by using the Banach fixed point theorem to show that the corresponding dynamic programming operators are contractions. The theory developed is illustrated by numerically solving the constrained stochastic control dynamic programming equations derived for simple example problems. The example problems are based on a two-state Markov model that represents an error prone system that is to be maintained.
机译：本文证明了动态规划方法在具有期望值约束的约束随机控制问题中的应用。具体而言，使用这种方法分析了两个此类问题。分析的问题是在相同结构约束下使折价无限期期望最小目标最小化的问题，以及在期望折算约束下无限度无限最大化目标最小化的问题。使用动态规划方法，针对这些问题获得了最优方程，这是本文的主要贡献。特别是，具有期望约束的问题的动态规划算子与标准动态规划和具有最坏情况约束的问题的显着不同。对于打折的成本无限期情况，利用Banach不动点定理明确表明动态规划方程解的存在性和唯一性，表明相应的动态规划算子是紧缩性。通过数值求解为简单示例问题得出的约束随机控制动态规划方程，可以说明所开发的理论。示例问题基于两状态马尔可夫模型，该模型表示将要维护的易于出错的系统。

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