Terror and Counterterror Operations: Differential Game with Cyclical Nash Solution

The purpose of this paper is to study efficient measures to combat terror. To choose efficient actions against terror organizations, the response of terrorists has to be taken into consideration. Thus, an appropriate framework to study the intertemporal strategic interactions of Western governments and terror organizations is dynamic game theory. Using the open-loop Nash solution concept, it turns out that the system may exhibit long-run persistent oscillations. However, transitory behavior is nonunique.     

Nash Equilibria in a Differential Game of Economic Growth

The present work studies a dynamic game of economic growth with two phases or periods. In the first phase, a political process exists where an institutional framework is fixed, that is, limits to the workers share in the national income, minimum limits of consumption by capitalists, and discretionality in investment are specified. This framework conditions the actions of the players (capitalists and workers). In the second phase, the Nash equilibria of the game are calculated. Lastly, the sensitivity of the results with respect to the institutional parameters is analyzed for a given scenario.     

Sequential Linear-Quadratic Method for Differential Games with Air Combat Applications

We present a numerical method for computing a local Nash (saddle-point) solution to a zero-sum differential game for a nonlinear system. Given a solution estimate to the game, we define a subproblem, which is obtained from the original problem by linearizing its system dynamics around the solution estimate and expanding its payoff function to quadratic terms around the same solution estimate. We then apply the standard Riccati equation method to the linear-quadratic subproblem and compute its saddle solution. We then update the current solution estimate by adding the computed saddle solution of the subproblem multiplied by a small positive constant (a step size) to the current solution estimate for the original game. We repeat this process and successively generate better solution estimates. Our applications of this sequential method to air combat simulations demonstrate experimentally that the solution estimates converge to a local Nash (saddle) solution of the original game.     

Weak and Strong Time Consistency in a Differential Oligopoly Game with Capital Accumulation

We illustrate a differential oligopoly game with capital accumulation where the accumulation dynamics of productive capacity is modelled à la Ramsey. The model is solved under the open-loop information structure, to show that it admits an open-loop Nash equilibrium which is indeed a degenerate feedback one and therefore strongly time consistent, even if, by construction, the problem under consideration is not a linear state game. We thank George Leitmann, Massimo Marinacci, Daniele Ritelli, Arsen Palestini and two anonymous referees for very useful comments and suggestions.     

Upper Bounds of the Pursuer Control Based on a Linear-Quadratic Differential Game

Differential game formulations provide an adequate basis for a guidance law synthesis against highly maneuvering targets. This paper deals with a guidance law based on a linear-quadratic differential game formulation. This guidance law has many attractive properties: it is continuous, linear with respect to the state variables, and its gain coefficients can be precalculated offline. Nevertheless, due to the lack of hard control constraints in the formulation, the magnitude of the control can exceed the admissible level imposed by the nature of the problem. In this paper, the upper bound of the interceptor control is obtained depending on the system parameters and the penalty coefficients of the game performance index. It is shown that the interceptor can guarantee an arbitrarily small miss distance without exceeding the control constraints if it has sufficient maneuverability and if the penalty coefficients are chosen properly. By manipulating the penalty coefficients, it is possible to reduce significantly the maneuverability requirements compared to the case of zero interceptor penalty coefficient.     

Homicidal Chauffeur Game with Target Set in the Shape of a Circular Angular Sector: Conditions for Existence of a Closed Barrier

A planar constant-speed pursuit-evasion problem with dynamic model similar to the one of the homicidal chauffeur game and with prescribed angular constraints in the capture criterion is analyzed as a differential game of kind. Because of the angular constraints, the target set of the game has the shape of a circular angular sector. Conditions for the existence of the game barrier (closed) are obtained. Using these conditions, a necessary and sufficient condition for capture of a slower evader from any initial state of the game is established. This condition is represented by an expression for the minimal nondimensional capture radius, normalized by the pursuer minimal turning radius, which guarantees capture of all slower evaders. This minimal capture radius depends on the angular constraints. Capture from any initial state implies that the barrier of the game does not exist and vice versa. In this game, two types of barrier are derived, with termination at either points of smoothness or points of nonsmoothness (corner points) of the boundary of the target set. The results are illustrated by numerical examples.     

常微分方程的级数解法探源

级数法是求解常微分方程最有效的方法之一.牛顿是第一位真正开始求解微分方程的数学家,级数法是其采用的第一种求解方法.在研读牛顿的微积分论文《流数法与无穷级数》基础上,探讨级数法形成的根源,揭示其思想方法对今日微分方程课程教与学的启迪作用以及对创立和发展微分方程学科的重要理论意义.     

Geometry of Optimal Paths around Focal Singular Surfaces in Differential Games

We investigate a special type of singularity in non-smooth solutions of first-order partial differential equations, with emphasis on Isaacs’ equation. This type, called focal manifold, is characterized by the incoming trajectory fields on the two sides and a discontinuous gradient. We provide a complete set of constructive equations under various hypotheses on the singularity, culminating with the case where no a priori hypothesis on its geometry is known, and where the extremal trajectory fields need not be collinear. We show two examples of differential games exhibiting non-collinear fields of extremal trajectories on the focal manifold, one with a transversal approach and one with a tangential approach.     

求解一阶隐微分方程的微分法与参数法及奇解注记

主要指出了微分法与参数法的实质及二者的本质区别,以及求奇解的一个注意事项.     

Differential games results for military expenditures of unequal antagonists in the third world

In developing countries, resources are scarce and military expenditure can hinder development. Thus, allocation of resources between defense and civilian sectors is an important decision. Also, in developing countries, most defense spending is related to regional conflicts where one frequently encounters unequal participants (e.g., India—Pakistan, Iran—Iraq, China—Vietnam). In such situations, the bigger power aspires to be the regional leader, and allocates resources to the defense sector accordingly.

A model incorporating the two above mentioned stylized facts has been examined by Deger and Sen [1] in two different ways. They first look at the problem from a single country's point of view, which takes the other's reaction function as given, and show that in the long run, military expenditures and armament stock levels will exhibit stability. Next, they formulate the problem as a noncooperative differential game in which each country assumes knowledge of the other country's welfare function, and conjecture that stability will once again be exhibited. The present paper attempts to verify that conjecture using a simplifying assumption which is necessary to be able to qualitatively solve the problem. This simplifying assumption does not change the results for the optimal control models, however, the conjecture is not valid for the simplified differential game model.     

Feedback Solution of a Class of Differential Games with Endogenous Horizons

In this paper, we consider a class of differential games in which the game ends when a subset of its state variables reaches a certain target at the terminal time. A special feature of the game is that its horizon is not fixed at the outset, but is determined endogenously by the actions of the players; conditions characterizing a feedback Nash equilibrium (FNE) solution of the game are derived for the first time. Extensions and illustrations of the derivation of FNE solutions of the game are provided.     

关于一类微分方程的周期解问题

本文借助不动点的定理,建立了一类微分方程的周期解存在性判据,并推广了一些重要的已知结果。     

Subgame Consistent Solutions of a Cooperative Stochastic Differential Game with Nontransferable Payoffs

Subgame consistency is a fundamental element in the solution of cooperative stochastic differential games. In particular, it ensures that the extension of the solution policy to a later starting time and any possible state brought about by the prior optimal behavior of the players would remain optimal. Recently, mechanisms for the derivation of subgame consistent solutions in stochastic cooperative differential games with transferable payoffs have been found. In this paper, subgame consistent solutions are derived for a class of cooperative stochastic differential games with nontransferable payoffs. The previously intractable subgame consistent solution for games with nontransferable payoffs is rendered tractable.This research was supported by the Research Grant Council of Hong Kong, Grant HKBU2056/99H and by Hong Kong Baptist University, Grant FRG/02-03/II16.Communicated by G. Leitmann     

Representation Formulas for Solutions of the HJI Equations with Discontinuous Coefficients and Existence of Value in Differential Games

In this paper, we study the Hamilton-Jacobi-Isaacs equation of zerosum differential games with discontinuous running cost. For such class of equations, the uniqueness of the solutions is not guaranteed in general. We prove principles of optimality for viscosity solutions where one of the players can play either causal strategies or only a subset of continuous strategies. This allows us to obtain nonstandard representation formulas for the minimal and maximal viscosity solutions and prove that a weak form of the existence of value is always satisfied. We state also an explicit uniqueness result for the HJI equations for piecewise continuous coefficients, in which case the usual statement on the existence of value holds.     

求解一类模糊线性微分系统的结构元方法

基于模糊结构元方法,研究了由对称模糊结构元线性生成的一般模糊线性微分系统和双重的一般模糊线性微分系统,给出了模糊线性微分系统解存在的充要条件,得到了结构元线性生成的齐次、非齐次以及双重一般模糊线性微分系统求解方法。最后,给出了这类系统的实际应用。     

二维非线性对流扩散方程特征有限元的两重网络算法

针对二维非线性对流扩散方程,构造了特征有限元两重网格算法．该算法只需要在粗网格上进行非线性迭代运算,而在所需要求解的细网格上进行一次线性运算即可．对于非线性对流占优扩散方程,不仅可以消除因对流占优项引起的数值振荡现象,还可以加快收敛速度、提高计算效率．误差估计表明只要选取粗细网格步长满足一定的关系式,就可以使两重网格解与有限元解保持同样的计算精度．算例显示:两重网格算法比特征有限元算法的收敛速度明显加快．     

非齐常系数线性微分方程的特殊解法

应用变换给出了常系数线性微分方程的一种解法.     

Differential game model of the dynastic cycle: 3D-canonical system with a stable limit cycle

Ancient Chinese history reveals many examples of a cyclical pattern of social development connected with the rise and the decline of dynasties. In this paper, a possible explanation of the periodic alternation between despotism and anarchy by a dynamic game between the rulers and the bandits is offered. The third part of the society, the farmers, are dealt with as a renewable resource which is exploited by both players in a different manner. It is shown that the Nash solution of this one-state differential game may be a persistent cycle. Although we restrict the analysis to open-loop solutions, this result is of interest for at least two reasons. First, it provides one of the few existing dynamic economic games with periodic solutions. Second, and more important, the model is an example of a three-dimensional canonical system (one state, two costates) with a stable limit cycle as solution. As far as we see, our model provides up to now the simplest (i.e., lowest dimensional) case of a persistent periodic solution of an intertemporal decision problem.This research was partially supported by the Austrian Science Foundation under Contract No. P7783-PHY.Helpful comments of T. Basar, E. J. Dockner, R. F. Hartl, A. Mehlmann, G. Sorger, and F. Wirl as well as the generous help of George Leitmann are gratefully acknowledged.     

关于可拓集合的运算

根据蔡文提出的可拓集合的新定义,给出了可拓集合的包含、并、交、非运算的新定义,并讨论了有关运算性质,进而获得可拓域与稳定域的几个交并运算结果.     

Solution of Two-Point Boundary-Value Problems Using the Differential Transformation Method

This paper considers two-point boundary-value problems using the differential transformation method. An iterative procedure is proposed for both the linear and nonlinear cases. Using the proposed approach, an analytic solution of the two-point boundary-value problem, represented by an mth-order Taylor series expansion, can be obtained throughout the prescribed range.