This paper studies two types of linear quadratic (LQ) optimal control models for multistage uncertain random systems. The first model is an LQ model with additive noises, while the second model is an LQ model with both multiplicative noises and additive noises. Chance theory is a useful tool to deal with the analysis of indeterminacy including both uncertainty and randomness. Based on Bellman's principle and chance theory, the recurrence equations are presented for settling such optimal control problems. Then, by applying recurrence equations, the analytical expressions are derived for solving both the optimal objective function and optimal control strategies of the LQ models. Meanwhile, an application and a numerical example are provided to show the effectiveness of our results.

中文翻译：

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具有乘性和加性噪声的不确定随机线性二次控制

本文研究了多级不确定随机系统的两种线性二次（LQ）最优控制模型。第一个模型是具有加性噪声的 LQ 模型，而第二个模型是具有乘性噪声和加性噪声的 LQ 模型。机会理论是处理不确定性分析的有用工具，包括不确定性和随机性。基于贝尔曼原理和机会理论，提出了用于解决此类最优控制问题的递推方程。然后，通过应用递推方程，推导出用于求解 LQ 模型的最优目标函数和最优控制策略的解析表达式。同时，提供了一个应用程序和一个数值例子来展示我们结果的有效性。