An overlapping generations stochastic differential game

The paper presents a noncooperative stochastic differential game played by an infinite number of overlapping generations of players. The number and types of players in future generations are uncertain and the state dynamics are given by a stochastic differential equation. A Markovian Nash equilibrium is characterized by a verification theorem of the Hamilton-Jacobi-Bellman type. A resource extraction game is offered as an illustration.     

Uniqueness conditions for the affine open-loop linear quadratic differential game

In this note we consider the open-loop Nash linear quadratic differential game with an infinite-planning horizon. The performance function is assumed to be indefinite and the underlying system affine. We derive both necessary and sufficient conditions under which this game has a unique Nash equilibrium.     

Linear−quadratic optimal control and nonzero‐sum differential game of forward−backward stochastic system

An existence and uniqueness result for one kind of forward–backward stochastic differential equations with double dimensions was obtained under some monotonicity conditions. Then this result was applied to the linear‐quadratic stochastic optimal control and nonzero‐sum differential game of forward–backward stochastic system. The explicit forms of the optimal control and the Nash equilibrium point are obtained respectively. We note that our method is effective in studying the uniqueness of Nash equilibrium point. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

The Nash solution of an advertising differential game: Generalization of a model by Leitmann and Schmitendorf

In this paper the differential game between two profit maximizing firms considered in [15] is extended to the case of general (concave) effectiveness functions of advertising. For this model, a phase portrait analysis of Nash equilibrium solutions is carried out providing qualitative insights into the structure of optimal advertising rates. Applying this analysis to the Leitmann-Schmitendorf advertising model, we obtain more detailed information on the structure of the optimal controls. Moreover, Nash-optimal solutions are derived for infinite duration of the game.     

基于模糊微分对策多机空战组内协同的U解

鉴于协商微分对策多具有强非线性和不确定性特点,为避免非线性问题等带来求解上的困难和能处理不确定信息,基于T-S模糊微分对策的思想,对非线性的基于Utilitarian解(简称U解)的协商微分对策的状态方程和性能分别进行了模糊化和二次型化,构造出面向U解的协商微分对策的模糊线性化模型,研究了协商U解模型中加权系数的确定,并进一步探讨了在模糊线性协商微分对策系统下相应于U解的控制器的设计方法.研究工作和仿真结果可以说明,相对于Nash协商解,协商理论的U解更能反映整体效果,更易于推广.     

Nash equilibrium in differential games and the quasi-strategy formalism

We consider a two-player nonzero-sum differential game in the case where players use nonanticipative strategies. We define the Nash equilibrium in this case and obtain a characterization of Nash equilibrium strategies. We show that a Nash equilibrium solution can be approximately realized by control-with-guide strategies.     

Analytical solution for a class of linear quadratic open-loop Nash game with multiple players

In this paper, the Nash equilibria for differential games with multiple players is studied. A method for solving the Riccati-type matrix differential equations for open-loop Nash strategy in linear quadratic game with multiple players is presented and analytical solution is given for a type of differential games in which the system matrix can be diagonalizable. As the special cases, the Nash equilibria for some type of differential games with particular structure is studied also, and some results in previous literatures are extended. Finally, a numerical example is given to illustrate the effectiveness of the solution procedure.     

Analytical solution for a class of linear quadratic open-loop Nash game with multiple players

In this paper, the Nash equilibria for differential games with multiple players is studied. A method for solving the Riccati-type matrix differential equations for open-loop Nash strategy in linear quadratic game with multiple players is presented and analytical solution is given for a type of differential games in which the system matrix can be diagonalizable. As the special cases, the Nash equilibria for some type of differential games with particular structure is studied also, and some results in previous literatures are extended. Finally, a numerical example is given to illustrate the effectiveness of the solution procedure.     

Optimal coordination in logistics warehousing with sensing and communication limits: A distributed differential game approach

Optimal coordination is essential for multi-automated guided vehicle (AGV) systems, particularly in logistic transportation cases, where the system task completion time needs to be minimized, with the guarantee of safe operation. This is because an optimal coordination strategy (OCS), if achieved, can significantly improve the transportation system's efficiency. In this paper, to deal with the dynamic interaction process among AGVs, and sensing and communication range limits, we formulate the optimal coordination problem into a distributed differential game (DDG) framework, where individual AGVs only use information communicated from nearby AGVs to design their optimal operation trajectories. This helps to significantly reduce the computational and communication requirements for the multi-AGV logistic transportation systems. Targeting operation safety and working efficiency requirements, we incorporate collision avoidance and trajectory optimization objectives into the proposed framework. It is shown that local OCS, obtained by solving the DDG problem for each AGV, will converge to the global Nash equilibrium, which represents the most efficient operating condition for the entire logistic transportation system. Finally, the efficacy of the proposed method is demonstrated, based on simulations and experiments, benchmarked with existing logistic warehousing planning and differential game methods. Compared with conventional methods, the proposed framework successfully helps reduce the task completion time by up to 16%.     

Non‐Zero Sum Differential Games of Backward Stochastic Differential Delay Equations Under Partial Information

In this paper, we study a new type of differential game problems of backward stochastic differential delay equations under partial information. A class of time‐advanced stochastic differential equations (ASDEs) is introduced as the adjoint process via duality relation. By means of ASDEs, we suggest the necessary and sufficient conditions called maximum principle for an equilibrium point of non‐zero sum games. As an application, an economic problem is putted into our framework to illustrate the theoretical results. In terms of the maximum principle and some auxiliary filtering results, an equilibrium point is obtained.     

A differential game among sectors in a macroeconomy

The purpose of this paper is primarily to model the Danish economy as a differential game among the sectors, and to solve this game using an algorithm (A) published in full detail elsewhere.This numerical algorithm (A) is first described briefly. It is used for the approximation of open-loop Nash-Cournot equilibrium controls in a differential game of fixed duration and initial state; it is based upon a hierarchical decomposition of the differential game into optimal control problems, with a fictive referee for the Nash-Cournot playing rule. Each constrained optimal control problem is solved by means of the generalized reduced gradient using constraint co-ordination.The algorithm has been applied to a nonlinear dynamic sectoral model of the Danish economy, which is described in detail. The control functions are: investments, labour, write-offs, marginal tax rates in each sector. The state variables are: foreign debt, state budget excess. Some results for the 1947–1952 period are given and they show that the open-loop Nash equilibrium controls obtained are somehow closer to the actual historical controls than those yielded by maximizing classical welfare criterions.     

基于博弈论的H2/H∞混合控制及其在汽车主动悬架中的应用

文章提出把H₂/H_∞混合控制问题抽象为两个对局者信息不完全情况下的非零和博弈模型.在构造2×2非零和博弈模型中把反映系统鲁棒性能通道和动态性能通道作为参加博弈的两方,以H_∞和H₂控制方案作为两种纯策略,基于纳什谈判解原理设计出求解H₂/H_∞混合控制问题纳什均衡点的一般算法.把该算法应用于汽车主动悬架设计出基于纳什均衡点的H₂/H_∞输出反馈控制器.使用MATLAB进行仿真,仿真结果表明主动悬架系统在保持鲁棒稳定性与获得优化的动态性能指标之间取得平衡.     

针对窃听问题的马尔可夫博弈路由模型的研究

在随机路由的基础上,给出一种针对窃听问题的马尔可夫博弈路由模型(Markov Game Theory-based Routing , MUBR)。给出的模型以发送者和窃听者为马尔可夫博弈双方,发送者通过概率进行数据传输,增加了窃听者窃听信息的难度。模型通过收益函数计算纳什均衡点,找出最优路径。使用PRISM工具进行仿真,结果表明MGBR中存在纳什均衡点,在纳什均衡点处信息被窃听的概率最小;给出信息在纳什均衡点处被窃听的概率变化趋势,与基于最小跳数算法的路由协议相比,它降低了信息被窃听的概率。     

Iterative algorithms for computing the feedback Nash equilibrium point for positive systems

The paper studies N-player linear quadratic differential games on an infinite time horizon with deterministic feedback information structure. It introduces two iterative methods (the Newton method as well as its accelerated modification) in order to compute the stabilising solution of a set of generalised algebraic Riccati equations. The latter is related to the Nash equilibrium point of the considered game model. Moreover, we derive the sufficient conditions for convergence of the proposed methods. Finally, we discuss two numerical examples so as to illustrate the performance of both of the algorithms.     

Adaptive dynamic programming for online solution of a zero-sum differential game

This paper will present an approximate/adaptive dynamic programming(ADP) algorithm,that uses the idea of integral reinforcement learning(IRL),to determine online the Nash equilibrium solution for the two-player zerosum differential game with linear dynamics and infinite horizon quadratic cost.The algorithm is built around an iterative method that has been developed in the control engineering community for solving the continuous-time game algebraic Riccati equation(CT-GARE),which underlies the game problem.We here show how the ADP techniques will enhance the capabilities of the offline method allowing an online solution without the requirement of complete knowledge of the system dynamics.The feasibility of the ADP scheme is demonstrated in simulation for a power system control application.The adaptation goal is the best control policy that will face in an optimal manner the highest load disturbance.     

Sampled-data Nash controls in non-zero-sum differential games†

Non-Zero-sum differential games where measurements of the state vector are possible only at discrete instants of time during the course of play are considered, and necessary conditions for the existence of a pair of sampled-data Nash controls are obtained. These conditions are different from those corresponding to the open-loop or closed-loop solutions. Linear quadratic games are then treated and a simple illustrative example which reduces to a pursuit-evasion game is presented.     

Robust trajectory tracking: differential game/cheap control approach

A robust trajectory tracking problem is treated in the framework of a zero-sum linear-quadratic differential game of a general type. For the cheap control version of this game, a novel solvability condition is derived. The sufficient condition, guaranteeing that the tracking problem is solved by the optimal strategy of the minimiser in the cheap control game, is established. The boundedness of the time realisations of this strategy is analysed. An illustrative example is presented.