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的结构研究，并建立更健壮或更灵活的动态规划方法，因为它在粗略逼近方面是合理有效的，并允许顺序决策过程中的波动因素。(1)马尔可夫集链模型作为马尔可夫决策过程中转移矩阵的一种健壮性模型，我们引入了一种称为受控马尔可夫集链的决策模型，并推导了一个DP方程，利用该方程构造了Pareto最优策略。(2)模糊动态系统与停止问题建立了具有模糊状态和模糊转移的动态系统的遍历定理，并证明了相应DP方程解的存在唯一性。建立了动态模糊系统的停止问题的数学模型，并用扩展的UP方法进行了求解。(3)一般效用模型下的停止马尔可夫决策过程的分析。相应的DP方程用一族分布族来描述，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.