Abstract

Problems of optimal control of linear and nonlinear stochastic systems with quadratic criterion qualities are studied. For such problems the existence of optimal control in feedback control form is proved by method of dynamic programming. The work consists of two parts. The first part deals with the linear problem. In the first part, the existence of a solution to the Cauchy problem for the generalized Riccati equation is proved by a method based on the idea of the Bellman linearization scheme. The proof consists in the direct application of existence theorems for ordinary differential equations to the generalized Riccati equation.

The main part of the article is its second part, which concerns the study of a nonlinear problem. Meaningful result is obtained only when the control is included with a small parameter in the stochastic part. Existence for small \(\varepsilon\) of the solution of the Bellman equation corresponding to the nonlinear problem are proved using the abstract implicit function theorem.

中文翻译：

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随机系统的最优控制问题

摘要

研究了具有二次准则质量的线性和非线性随机系统的最优控制问题。针对此类问题，通过动态规划的方法证明了最优控制存在反馈控制形式。这项工作包括两个部分。第一部分处理线性问题。在第一部分中，通过基于Bellman线性化方案思想的方法证明了广义Riccati方程Cauchy问题的解的存在。证明在于将常微分方程存在性定理直接应用于广义Riccati方程。

本文的主要部分是第二部分，它涉及对非线性问题的研究。只有在随机部分中包含较小参数的控件时，才能获得有意义的结果。使用抽象隐函数定理证明了与非线性问题相对应的Bellman方程解的小\（\ varepsilon \）的存在性。