Characterization of Barriers of Differential Games

In pursuit-evasion games, when a barrier occurs, splitting the state space into capture and evasion areas, in order to characterize this manifold, the study of the minimum time function requires discontinuous generalized solutions of the Isaacs equation. Thanks to the minimal oriented distance from the target, we obtain a characterization by approximation with continuous functions. The barrier is characterized by the largest upper semicontinuous viscosity subsolution of a variational inequality. This result extends the Isaacs semipermeability property.     

Simple Singularities for Hamilton-Jacobi Equations with Max-Concave Hamiltonians and Generalized Characteristics

We consider piecewise smooth viscosity solutions to a Hamilton-Jacobi equation, for which the Hamiltonian is the maximum of a finite number of smooth concave functions. We describe the possible types of “basic” singularities, namely jump discontinuities in either the derivative of the solution or the Hamiltonian, which occur across a smooth hypersurface. Each such type of singularity is described in terms of the properties of the Hamiltonian and the classical characteristics in the regions on either side of the singularity. Where appropriate, singular characteristic equations of the types developed by Melikyan are formulated which can be used in constructions.     

Classification of Singular Fibers on Rational Elliptic Surfaces in Characteristic Three

ABSTRACT

We list all possible configurations of singular fibers on rational elliptic surfaces in characteristic three and prove that no others exist. In total, we find 267 distinct configurations. Configurations of purely multiplicative fibers in characteristic three are compared to those in characteristics zero and two.     

Existence of Value and Saddle Point in Infinite-Dimensional Differential Games

We study two-player zero-sum differential games of finite duration in a Hilbert space. Following the Berkovitz notion of strategies, we prove the existence of value and saddle-point equilibrium. We characterize the value as the unique viscosity solution of the associated Hamilton–Jacobi–Isaacs equation using dynamic programming inequalities.     

Direct Method Comparing Efficient and Nonefficient Payoffs in Differential Games

In this note, we present a method that allows us to decide when a Markov-perfect Nash equilibrium is not Pareto optimum, without the explicit knowledge of the respective solutions. For that purpose, we establish a sufficient condition in terms of an algebraic inequality where the gradient of the value functions of the cooperative and noncooperative games as well as the state and control variables are involved.     

非线性微分方程边值问题的奇摄动

本文研究了非线性边值问题: εy″-f(x,y,y′)=0,0相似文献   

Agreeability and Time Consistency in Linear-State Differential Games

The paper identifies conditions under which time consistency and agreeability, two intertemporal individual rationality concepts, can be verified in linear-state differential games. An illustrative example drawn from environmental economics is provided.     

Asymptotic Analysis of Linear Feedback Nash Equilibria in Nonzero-Sum Linear-Quadratic Differential Games

In this paper, we discuss nonzero-sum linear-quadratic differential games. For this kind of games, the Nash equilibria for different kinds of information structures were first studied by Starr and Ho. Most of the literature on the topic of nonzero-sum linear-quadratic differential games is concerned with games of fixed, finite duration; i.e., games are studied over a finite time horizon t _f. In this paper, we study the behavior of feedback Nash equilibria for t _f.In the case of memoryless perfect-state information, we study the so-called feedback Nash equilibrium. Contrary to the open-loop case, we note that the coupled Riccati equations for the feedback Nash equilibrium are inherently nonlinear. Therefore, we limit the dynamic analysis to the scalar case. For the special case that all parameters are scalar, a detailed dynamical analysis is given for the quadratic system of coupled Riccati equations. We show that the asymptotic behavior of the solutions of the Riccati equations depends strongly on the specified terminal values. Finally, we show that, although the feedback Nash equilibrium over any fixed finite horizon is generically unique, there can exist several different feedback Nash equilibria in stationary strategies for the infinite-horizon problem, even when we restrict our attention to Nash equilibria that are stable in the dynamical sense.     

Construction of Approximate Saddle-Point Strategies for Differential Games in a Hilbert Space

Two-person zero-sum infinite-dimensional differential games with strategies and payoff as in Berkovitz (SIAM J. Control Optim. 23: 173–196, 1985) are studied. Using Yosida type approximations of the infinitesimal generator (of the unbounded dynamics) by bounded linear operators, we prove convergence theorems for the approximate value functions. This is used to construct approximate saddle-point strategies in feedback form. A.J. Shaiju: NBHM Postdoctoral Fellow and the financial support from NBHM is gratefully acknowledged.     

Identification of Efficient Subgame-Perfect Nash Equilibria in a Class of Differential Games1

We present a method for the characterization of subgame-perfect Nash equilibria being Pareto efficient in a class of differential games. For that purpose, we propose a new approach based on new necessary and sufficient conditions for computing subgame-perfect Nash equilibria.     

New Method to Characterize Subgame Perfect Nash Equilibria in Differential Games

In this paper, we present a method for computing Nash equilibria in feedback strategies. This method gives necessary and sufficient conditions to characterize subgame perfect equilibria by means of a system of quasilinear partial differential equations. This characterization allows one to know explicitly the solution of the game in some cases. In other cases, this approach makes a qualitative study easier. We apply this method to nonrenewable resource games.     

On Coincidence of Feedback Nash Equilibria and Stackelberg Equilibria in Economic Applications of Differential Games

The scope of the applicability of the feedback Stackelberg equilibrium concept in differential games is investigated. First, conditions for obtaining the coincidence between the stationary feedback Nash equilibrium and the stationary feedback Stackelberg equilibrium are given in terms of the instantaneous payoff functions of the players and the state equations of the game. Second, a class of differential games representing the underlying structure of a good number of economic applications of differential games is defined; for this class of differential games, it is shown that the stationary feedback Stackelberg equilibrium coincides with the stationary feedback Nash equilibrium. The conclusion is that the feedback Stackelberg solution is generally not useful to investigate leadership in the framework of a differential game, at least for a good number of economic applications This paper was presented at the 8th Viennese Workshop on Optimal Control, Dynamic Games, and Nonlinear Dynamics: Theory and Applications in Economics and OR/MS, Vienna, Austria, May 14–16, 2003, at the Seminar of the Instituto Complutense de Analisis Economico, Madrid, Spain, June 20, 2003, and at the Sevilla Workshop on Dynamic Economics and the Environment, Sevilla, Spain, July 2–3, 2003. The author is grateful to the participants in these sessions, in particular F.J. Andre and J. Ruiz, for their comments. Five referees provided particularly helpful suggestions. Financial support from the Ministerio de Ciencia y Tecnologia under Grant BEC2000-1432 is gratefully acknowledged.     

Banach空间中一类带奇异性的脉冲微分方程边值问题的正解

利用锥理论和Ｍ¨ｏｎｃｈ不动点定理，讨论了Ｂａｎａｃｈ空间中一类带奇异性的脉冲微分方程边值问题正解的存在性．作为其应用，给出了一个例子．     

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.     

Point capture of two evaders in succession

A constant-speed coplanar model with unlimited turn-rates leads to a rather simple geometrical solution of the problem of point capture of two successive evaders in minimum total time: the pursuer concentrates first on the nearer evader who runs in an appropriate direction; the second evader runs directly away from the predictable point of capture of the first evader. This simple solution is valid only if the second evader remains thereby the further of the two evaders. Otherwise, the solution must be modified to include a phase involving curved motion by all three players, during which the pursuer remains equidistant from both evaders.     

Games of Simple Pursuit and Approach on a Two-Dimensional Cone

The complete solution of two differential games with simple motion on a two-dimensional cone is presented. The method of generalized (singular) characteristics is used for the analysis of singular paths. The level lines of the game value and optimal phase portrait are constructed, and the latter is shown to include several types of singular surfaces (dispersal, equivocal, universal). An algorithm is developed for the construction of the synthesis for both players and is used in computer simulation.     

Solution of a Differential Game Formulation of Military Air Operations by the Method of Characteristics

In this paper, we describe a zero-sum differential game formulation for the control of military air operations. The model consists of a system of nonlinear ordinary differential equations for the dynamics of the operations and a suitably chosen quadratic payoff function. The control variables are the engagement intensities and velocities, and there are constraints on the controls. The method of characteristics (based on the Pontryagin maximum principle) is used to solve the associated Hamilton-Jacobi equation. In this nonlinear formulation, the Hamiltonian can be optimized explicity with respect to the controls. Numerical simulations study the enforcement of constraints (a) by means of penalties in the payoff function or (b) explicitly. The numerical results show robustness with respect to various parameters.Effort sponsored by the Defense Advanced Research Projects Agency (DARPA) and Air Force Research Laboratory, Air Force Material Command, USAF, under Agreement F30602-99-2-0551. A shorter version of this paper appeared in the Proceedings of the 2001 American Control Conference, pp. 168--175, 2001.     

Existence and Uniqueness of Open-Loop Stackelberg Equilibria in Linear-Quadratic Differential Games

We present existence and uniqueness results for a hierarchical or Stackelberg equilibrium in a two-player differential game with open-loop information structure. There is a known convexity condition ensuring the existence of a Stackelberg equilibrium, which was derived by Simaan and Cruz (Ref. 1). This condition applies to games with a rather nonconflicting structure of their cost criteria. By another approach, we obtain here new sufficient existence conditions for an open-loop equilibrium in terms of the solvability of a terminal-value problem of two symmetric Riccati differential equations and a coupled system of Riccati matrix differential equations. The latter coupled system appears also in the necessary conditions, but contrary to the above as a boundary-value problem. In case that the convexity condition holds, both symmetric equations are of standard type and admit globally a positive-semidefinite solution. But the conditions apply also to more conflicting situations. Then, the corresponding Riccati differential equations may be of H-type. We obtain also different uniqueness conditions using a Lyapunov-type approach. The case of time-invariant parameters is discussed in more detail and we present a numerical example.     

一类二阶拟线性微分方程边值问题的奇摄动(英文)

本文讨论了一类二阶拟线性微分方程的奇摄动问题 .在适当的条件下 ,本文用一种新的方法分析了原问题解的存在性、唯一性及渐近性态 .