【摘要】：在本論文中,我們研究領(lǐng)導(dǎo)者和多個(gè)追隨者之間的非合作動(dòng)態(tài)博弈。這是動(dòng)態(tài)博弈一個(gè)新的研究方向,超越了傳統(tǒng)控制理論和博弈理論的框架。假設(shè)在對(duì)稱信息框架下,微分博弈問(wèn)題的納什均衡點(diǎn)存在,我們將領(lǐng)導(dǎo)者視為第三方或其他非營(yíng)利組織,這樣可以忽略領(lǐng)導(dǎo)者的收益功能。考慮領(lǐng)導(dǎo)者給定策略時(shí),追隨者們的非合作動(dòng)態(tài)博弈。在此框架下,我們重點(diǎn)關(guān)注領(lǐng)導(dǎo)者的調(diào)控系統(tǒng)的能力,在某種意義上,它反映了領(lǐng)導(dǎo)者對(duì)非合作博弈系統(tǒng)的影響。首先,本文研究了在系統(tǒng)信息均衡的條件下追隨者們的最大收益問(wèn)題。這是一個(gè)隨機(jī)最優(yōu)控制問(wèn)題。在實(shí)際中,優(yōu)化控制問(wèn)題越來(lái)越多的受人關(guān)注并且深入研究,比如金融市場(chǎng)、能源系統(tǒng)等。本文通過(guò)引入了正倒向隨機(jī)微分方程(FBSDE)控制理論�？紤]了對(duì)稱信息下的線性二次非零和微分博弈問(wèn)題,研究追隨者們最大收益解的構(gòu)造方法,并基于極大值原理給出了隨機(jī)切換系統(tǒng)最優(yōu)解的解析表達(dá)式。其次,本文討論了系統(tǒng)狀態(tài)和控制器都受到隨機(jī)擾動(dòng)的線性切換系統(tǒng)。在隨機(jī)Nash均衡狀態(tài)存在的條件下,對(duì)線性切換系統(tǒng)中的隨機(jī)控制問(wèn)題進(jìn)行了研究。隨機(jī)控制系統(tǒng)在證明過(guò)程中,引入黎卡提(Riccati)方程和正倒向隨機(jī)微分方程(FBSDE)理論,證明了系統(tǒng)方程解的存在和唯一性。再次,本文研究了 Nash均衡存在條件下,領(lǐng)導(dǎo)者宏觀調(diào)控的可行性問(wèn)題。借鑒Peng提出的BSDE理論知識(shí),綜合運(yùn)用FBSDE理論,對(duì)于含有切換參數(shù)的一類隨機(jī)系統(tǒng),給出了納什均衡終端精確能控的充要條件。進(jìn)一步,給出了線性隨機(jī)切換系統(tǒng)的納什均衡精確能控的充要條件。同時(shí)還給出了 Nash均衡精確可控性的代數(shù)判據(jù)。最后,為體現(xiàn)倒向隨機(jī)控制系統(tǒng)的實(shí)際應(yīng)用價(jià)值,本文給出了一個(gè)市場(chǎng)上最優(yōu)投資組合調(diào)控的例子。領(lǐng)導(dǎo)者的決策是一個(gè)控制過(guò)程,跟隨者的最優(yōu)收益可以被調(diào)控,說(shuō)明我們研究的問(wèn)題是有實(shí)際意義的。同時(shí),通過(guò)Matlab數(shù)值仿真驗(yàn)證了提出的控制器可以精確控制系統(tǒng)模型。
[Abstract]:In this thesis, we study the non-cooperative dynamic game between leaders and many followers. This is a new research direction of dynamic game, which surpasses the frame of traditional control theory and game theory. Assuming that the Nash equilibrium exists in the framework of symmetric information, we treat the leader as a third party or other non-profit organization, so that the leader's income function can be ignored. When considering the leader's given strategy, the followers of the non-cooperative dynamic game. In this framework, we focus on the ability of the leader's regulatory system, which in a sense reflects the leader's influence on the non-cooperative game system. Firstly, this paper studies the maximum profit of followers under the condition of system information equilibrium. This is a stochastic optimal control problem. In practice, optimization control problems have attracted more and more attention, such as financial markets, energy systems and so on. In this paper, the (FBSDE) control theory of forward backward stochastic differential equation is introduced. The problem of linear quadratic nonzero sum differential game with symmetric information is considered. The method of constructing the maximum return solution of followers is studied and the analytical expression of the optimal solution of stochastic switched system is given based on the maximum principle. Secondly, we discuss the linear switched systems where both the system state and the controller are stochastic perturbed. Under the condition of the existence of stochastic Nash equilibrium, the stochastic control problem in linear switched systems is studied. In the process of proving stochastic control system, the existence and uniqueness of the solution of the system equation are proved by introducing the Rikati (Riccati) equation and the (FBSDE) theory of forward backward stochastic differential equation. Thirdly, this paper studies the feasibility of macro-control under the condition of Nash equilibrium. Based on the BSDE theory proposed by Peng and FBSDE theory, a necessary and sufficient condition for the precise controllability of Nash equilibrium terminal is given for a class of stochastic systems with switching parameters. Furthermore, a necessary and sufficient condition for the exact controllability of Nash equilibrium for linear stochastic switched systems is given. The algebraic criterion for the exact controllability of Nash equalization is also given. Finally, in order to reflect the practical application value of the backward stochastic control system, this paper gives an example of optimal portfolio control in the market. The leader's decision is a controlling process, and the optimal return of the follower can be regulated, which shows that the problem we study is of practical significance. At the same time, the Matlab numerical simulation shows that the proposed controller can accurately control the system model.
【學(xué)位授予單位】：北京交通大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O232

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