【作者】 豆敬霞；

【导师】 周钢；

【作者基本信息】 上海交通大学 ， 计算数学， 2011， 硕士

【摘要】 Issacs博士于1956年出版了世界上第一部微分对策专著《微分对策》,标志着微分对策的正式诞生。此后微分对策的研究引起了世界各国研究者的广泛兴趣,然而很长一段时间都是围绕着线性二次型,特别是二人线性二次型微分对策。非线性微分对策由于理论研究非常困难,这方面的研究比较少,也比较缓慢,然而实际问题却往往都是非线性的,因此,基于非线性微分对策的数值方法和算法的研究逐渐受到人们的关注,并成为了热点方向。对于二人非线性微分对策可以采用极大极小值方法(H-J-B方法)。但是多人非线性微分对策问题,由于局中人的增多,对策情况变得非常复杂,因此需要寻找新的方法来研究和分析多人微分对策。本文一、二章主要研究了如何将非线性微分对策问题转化为线性微分对策问题、同时又不失原问题全局非线性性质的T-S模糊模型,然后将得到的线性微分对策问题归结为Hamilton系统,最后借助可以保持Hamilton流的整体特征和系统守恒律的辛算法来求解。本文三、四章分别研究了二人非线性微分对策和多人非线性微分对策,并通过算例证实了:辛算法在微分对策求解方面的可行性,相应的数值结果令人满意。本文第四章推广了二人线性微分对策的情况,并证明了多人线性二次型微分对策也可以归结为Hamilton系统,从而为多人微分对策的求解奠定了理论基础。本文的主要创新点有:1.研究并提出了T-S模糊模型所涉及的隶属函数的确定问题,并给出一般的求解方法;2.证明了多人线性二次型微分对策可以归结为Hamilton系统,从而可以应用辛算法求解;3.研究多人非线性微分对策问题的求解:通过T-S模糊模型线性化、应用辛差分方法求出数值解,验证了该方法的可行性。本文主要研究的是支付函数为二次型的微分对策,对于非二次型的微分对策,则可以采用近似化的方法得到。本文的第五章,是作者按照“上海交通大学数学系硕士研究生毕业要求”的条例完成的,是在阅读、理解大量科技文献后,经思考、提炼、创新而撰写的综合报告。(主要综述了辛算法在分子经典轨迹、微分对策、振动系统中的应用以及辛Runge-Kutta方法、多辛算法。主要工作是将实际问题转化成线性Hamilton系统,然后借助于辛差分格式来求解。)更多还原

【Abstract】 In 1956,Dr.Issacs published the first book about differential ga-mes,which was named Differential Games and marked the official b-irth of differential games. It attracted many researchers from all th-e world, however,It’s all about the linear quadratic for a long time,e-specially two-person linear quadratic differential games. Due to thedifficulty in theory,few did study nonlinear differential games,but weknow,practical problems often are nonlinear,so,research on numericalmethods and algorithm of non-linear differential game gradually bec-omes a hot direction.For two-person nonlinear quadratic differential games,we can us-e H-J-B.but many-person nonlinear differential games,we need find t-he new method for the complexity. Chapter I、II of this article is about transforming nonlinear diff-erential games into a linear one, then with the help of T-S fuzzy model,keeping the global non-linear nature,and the Hamilton system,we can solve it symplectic algorithm which keeps the overall flow characteristics and system conservation laws. Chapter III and IV areon two-person nonlinear and many-person nonlinear differential gam-es,and confirmed by examples: It’s possible to solve differential ga- mes by symplectic algorithm ,and the numerical results are satisfactory.Chapter IV generalizes the case of two-person linear differentialgames,proves that many-person linear quadratic differential game ca-n also be attributed to the hamilton system,and it will be a theoreti-cal foundation on solving differential games.Main innovation:1. Propose how to make sure the membership function of T-S fuzzy model and give out the general solution;2. Prove many-person linear quadratic differential games is a h-amilton system, and symplectic algorithm can solve it.3. Examples on nonlinear differential games: T-S fuzzy model,symplectic difference method,and the results show this meth-od feasible.This paper is mainly about differential games whose payment f-unction is quadratic,for non-quadratic differential game, we can getby the approximation method.Chapter V of this article, according to the rule"Graduate gradu-ation requirements Department of Mathematics Shanghai Jiaotong University ",is a general report based on thinking, refined, innovativ-e after reading, understanding a lot of technology literature. (Mainl-y about the classical trajectories of molecules, Differential Games, vibration systems and symplectic Runge-Kutta methods, multi-sympl-ectic algorithms.)更多还原