初中數(shù)學考案.doc

Global Optimal Strategies of a Class of Finite-horizon Continuous-time Nonaffine Nonlinear Zero-sum Game Using a New Iteration Algorithm一類全局優(yōu)化策略水平有限非仿射非線性連續(xù)時間零和博弈使用一種新的迭代算法Xin Zhang School of Information Science and Engineering,Northeastern University,Shenyang, 110004 ChinaEmail:Huaguang Zhang School of Information Science and Engineering,Northeastern University,Shenyang, 110004 China Email:Lili Cui School of Information Science and Engineering,Northeastern University,Shenyang, 110004 China Email:Yanhong Luo School of Information Science and Engineering,Northeastern University,Shenyang, 110004 China Email: 張 星 信息與科學與工程學院，東北大學，中國沈陽，110004，郵箱：J張華光 信息與科學與工程學院，東北大學，中國沈陽，110004，郵箱：桂莉莉 信息與科學與工程學院，東北大學，中國沈陽，110004，郵箱：羅艷紅 信息與科學與工程學院，東北大學，中國沈陽，110004，郵箱：AbstractIn this paper we ami to solve the global optimal strategies of a class of finite-horizon continuous-time nonaffine nonlinear zero-sum game. The idea is to use a iterative algorithm to obtain the saddle point. The iterative algorithm is between two sequences which are a sequence of linear quadratic zero-sum game and a sequence of Riccati differential equation. The necessary conditions of global optimal strategies are established. A simulation example is given to illustrate the perfoermace of the proposed approach.摘要：在本文中，我們解決一類連續(xù)時間有限視距非仿射非線性的零和對策的最佳方法。我們的想法是使用一個迭代算法獲得馬鞍點。迭代算法是兩個序列之間，這是一個序列的線性二次型的零和博弈和Riccati微分方程的序列。建立全局最優(yōu)戰(zhàn)略的必要條件。一個仿真例子說明了該方法的性能。I. INTRODUCTION1、簡介Nowadays, game theory has been widely applied in management, military battles, power networks and different types of contest, which is concerned with the study of decision making in situations where two or more rational opponents are involved under conditions of conflicting interests 3-9,10. The two-player zero-sum game with a general quadratic performance index function plays an important role in the gametheory. Two players work on the performance index function together and minimax it如今，博弈論被廣泛地應用于管理，軍事，電力網(wǎng)絡和不同類型的情況，關注的是研究決策的情況下，理性的對象是涉及兩個或兩個以上的條件下的利益沖突3-9，10。這兩個對象的的一般二次型性能指標函數(shù)的零和博弈，在博弈中起著重要的作用的理論。兩個對象的工作性能指標函數(shù)和極大極小它。The optimal strategies of linear zero-sum game and affine nonlinear zero-sum game have received a great deal of attention in the literature 1,2,7-10,19. In 1, Al-Tamimi et al. applied the heuristic dynamic programming and dual heuristicdynamic programming structures to solve a discrete-time linear quadratic zero-sum game problem in which the state and action spaces are continuous. Then, they designed the optimal strategies of the discrete-time linear quadratic zero-sum game without knowing the system dynamical matrices by the model-free Q-learning approach 2. A Class of continuoustime affine nonlinear quadratic zero-sum game problem was researched by Wei et al. in 11. Abu-Khalaf et al. studied the affine nonlinear zero-sum game problem in 7 and used neural networks to solve it in 8.零和博弈的線性仿射的最優(yōu)策略非線性零和博弈都獲得了極大的關注在文獻中1,2,7-10,19。 1中的Al-塔米米等。采用啟發(fā)式動態(tài)規(guī)劃和雙啟發(fā)動態(tài)編程結構來解決一個離散時間線性二次零和游戲的問題，其中狀態(tài)和行動空間是連續(xù)的。然后，他們設計的最優(yōu)策略的離散時間線性二次型零和不知道系統(tǒng)動力學矩陣的游戲無模型Q-學習方法2。一類連續(xù)時間仿射非線性二次零和博弈問題Wei等人研究。在11。阿布 - 哈拉夫等。研究仿射非線性的零和博弈的問題7和使用神經(jīng)網(wǎng)絡來解決8。It is worthy of mentioning that most of the above discussions are focused on the linear or affine nonlinear zero-sum game problems. However, many applications of practical zero-sum game have the nonlinear structure.這是值得提的是，最上面的討論主要集中在線性或仿射非線性的零和博弈的問題。然而，許多應用程序的實際零和博弈的非線性結構這是在控制輸入非仿射和。在19，我們提出了基于迭代算法之間的序列的線性二次型零和博弈，一個序列的Riccati零和差分方程處理仿射非線性博弈的問題。這種方法是改變非仿射非線性成一個序列的線性二次型的隨時間變化的零和博弈。 HJI方程轉(zhuǎn)化為Riccati方程的不變系數(shù)的一個序列。在本文中，建立全局的最優(yōu)的仿射非線性零和博弈的是基于迭代算法。The paper is organized as follows. Section 2 introduces a class of finite-horizon continuous-time nonaffine nonlinear zero-sum game that we want to solve in this paper, and the iterative algorithm presented in 19 is reviewed. In Section 3, the necessary conditions for global optimality of nonaffine nonlinear zero-sum game are established. In Section 4, a numerical example is given to demonstrate the convergence and effectiveness of the proposed.本文的結構如下。第2節(jié)介紹一類基于水平有限仿射非線性連續(xù)時間的零和博弈，我們在本文中要解決迭代算法19中提出的審查。在第3節(jié)中基于非仿射全局最優(yōu)的必要條件建立非線性零和博弈。在第4節(jié)，用數(shù)值例子證明收斂性和有效性。II. PROBLEM STATEMENT AND PRELIMINARIES2、問題陳述和預備知識 We consider a continuous-time nonaffine nonlinear two player zero-sum game described by the state equation我們考慮一個連續(xù)時間的非仿射非線性兩種對象零和博弈的狀態(tài)方程描述with the finite-horizon performance index function given by與有限視距性能指標函數(shù)由下式給出which is minimized by () and maximized by (). The state () takes valus in , the control vector of Player 1() takes values in convex and compact set 1 ,and the Player 2 () takes values in convex and compact