Studies on Mathematical Structure of Dynamic Programming

在这个项目中，我们的目标是发展动态规划（简称DP）的结构研究，并建立更鲁棒或更灵活的DP方法，因为它在粗略近似中相当有效，并允许顺序决策过程中的波动因素。为此目的。我们尝试对各种数学决策模型进行分析研究。(1)马尔可夫集链模型作为马尔可夫决策过程中转移矩阵粗糙逼近的鲁棒模型，引入了一种控制马尔可夫集链决策模型，推导了一个DP方程，并利用该方程构造了Pareto最优策略。文中还包括了一些计算结果。(2)模糊动力系统及其停止问题研究了具有模糊状态和模糊过渡的动力系统的遍历定理，并证明了相应DP方程解的存在唯一性。提出了动态模糊系统的停车问题，并用扩展的UP方法求解。(3)在一般效用下，分析了通用效用模型A的停止马尔可夫决策过程。相应的DP方程用一组分布来描述，在DP的应用中更有用。

In this project, our objective is to develope the structural study of Dynamic Programming(DP in short) and establish DP method which is more robust or more flexible in the sense that it is reasonably efficient in rough approximation and allows for fluctuating factors in sequential decision processes.For this purpose. we tried to develope analytical studies on various mathematical decion model.(1) Markov set-chain model As a model which is robust for rough approximation of the transition matrix in Markov decision processes, we introduced a decision model, called a controlled Markov set-chain, and derived a DP equation by which Pareto optimal policies was constructed. Some computational results are included.(2) Fuzzy dynamic systems and stopping problem The ergodic theorem for the dynamic system with fuzzy state and fuzzy transition is developed and the existence and uniqueness of solutions of the corresponding DP equations is proved. Also, a stopping problem for dynamic fuzzy system is formulated and solved by an extended UP method.(3) General utility model A stopped Markov decision process is analysed under general utility. The corresponding DP equation is described by a family of distributions and more usefull in application of DP.