Conditional Viability for Impulse Differential Games

We introduce in this paper the concept of “impulse evolutionary game”. Examples of evolutionary games are usual differential games, differentiable games with history (path-dependent differential games), mutational differential games, etc. Impulse evolutionary systems and games cover in particular “hybrid systems” as well as “qualitative systems”. The conditional viability kernel of a constrained set (with a target) is the set of initial states such that for all strategies (regarded as continuous feedbacks) played by the second player, there exists a strategy of the first player such that the associated run starting from this initial state satisfies the constraints until it hits the target. This paper characterizes the concept of conditional viability kernel for “qualitative games” and of conditional valuation function for “qualitative games” maximinimizing an intertemporal criterion. The theorems obtained so far about viability/capturability issues for evolutionary systems, conditional viability for differential games and about impulse and hybrid systems are used to provide characterizations of conditional viability under impulse evolutionary games.     

On the Numerical Solution of Differential Games for Neutral-Type Linear Systems

The paper deals with a zero-sum differential game, in which the dynamics of a conflict-controlled system is described by linear functional differential equations of neutral type and the quality index is the sum of two terms: the first term evaluates the history of motion of the system realized up to the terminal time, and the second term is an integral–quadratic evaluation of the corresponding control realizations of the players. To calculate the value and construct optimal control laws in this differential game, we propose an approach based on solving a suitable auxiliary differential game, in which the motion of a conflict-controlled system is described by ordinary differential equations and the quality index evaluates the motion at the terminal time only. To find the value and the saddle point in the auxiliary differential game, we apply the so-called method of upper convex hulls, which leads to an effective solution in the case under consideration due to the specific structure of the quality index and the geometric constraints on the control actions of the players. The efficiency of the approach is illustrated by an example, and the results of numerical simulations are presented. The constructed optimal control laws are compared with the optimal control procedures with finitedimensional approximating guides, which were developed by the authors earlier.     

带随机跳跃的线性二次非零和微分对策问题

对于一类以布朗运动和泊松过程为噪声源的正倒向随机微分方程,在单调性假设下,给出了解的存在性和唯一性的结果．然后将这些结果应用于带随机跳跃的线性二次非零和微分对策问题之中,由上述正倒向随机微分方程的解得到了开环Nash均衡点的显式形式．     

Open Loop Saddle Point on Linear Quadratic Stochastic Differential Games

In this paper, we deal with one kind of two-player zero-sum linear quadratic stochastic differential game problem. We give the existence of an open loop saddle point if and only if the lower and upper values exist.     

Viability for Differential Inclusions on Graphs

Let X be a Banach space, I a nonempty, bounded interval and let $K:I\leadsto X$ be a given multi-valued function. We rephrase the concept of tangent set introduced by Cârj? et al. (Trans Amer Math Soc, 2008; Viability, Invariance and Applications. North-Holland Mathematics Studies, vol. 207. Elsevier, 2007) by saying that a bounded set $E\subseteq X$ is right tangent to at if $\liminf_{h\downarrow0}{1}/{h}\,\text{\rm dist}(\xi+hE;K(\tau+h))=0$ . Next, by using a tangency condition expressed in the terms of this concept, we establish several necessary and sufficient conditions for viability referring to differential inclusions on graphs.     

Viability Theorems for Higher-Order Differential Inclusions

The aim of this paper is to establish both necessary and sufficient conditions for the existence of viable solutions of a higher-order differential inclusion. Higher-order tangent sets are introduced and studied to get such conditions. These tangent sets are analysed in the case of the viable set defined by equality or inequality constraints and viability conditions are rewritten in terms of these constraints.     

Linear Differential Equations as a Data Structure

Foundations of Computational Mathematics - A lot of information concerning solutions of linear differential equations can be computed directly from the equation. It is therefore natural to consider...     

Games,Kernels, and Antitone Operations

In this note we want to point out the usefulness of antitone operations in the study of games on graphs. We demonstrate this by proving, among others, the generalizations from games to antitone operations of theorems stated by Fraenkel. This perspective makes also clear the general nature of some problems by separating what is general and what is particular.     

Some Remarks on Kernels

Kernel smoothing provides a simple way of finding a structure in data. Oneof the most popular settings where kernel smoothing ideas can be applied isthe simple regression model. In the context of kernel estimates of aregression function, the choice of a kernel from the different points ofview can be investigated. The aim of this paper is to present constructionsof minimum variance kernels and smooth kernels by means of the Legendrepolynomials and the Gegenbauer polynomials as well. Some of these kernelshave been introduced, e.g., in [2], [3], and [5], but here another approachby using the variational calculus is presented.     

Oscillation Result for Some Non-homogeneous Linear Differential Equations and Its Application

This paper investigates solutions of some non-homogeneous linear differential equations, which have non-homogeneous term as the small function of solution. Using the similar method, we can generalize the result of G.Gundersen and L.Z.Yang.     

Some Oscillation Problems for a Second Order Linear Delay Differential Equation

For the delay differential equation[formula]a connection between the following properties is established: non-oscillation of the differential equation and the corresponding differential inequality, positiveness of the fundamental function, and the existence of a nonnegative solution of a generalized Riccati inequality. Explicit conditions for non-oscillation and oscillation and comparison theorems are presented.     

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.     

Some Isoperimetric Inequalities for Kernels of Free Extensions

If G is a hyperbolic group (resp. synchronously or asynchronously automatic group) which can be expressed as an extension of a finitely presented group H by a finitely generated free group, then the normal subgroup H satisfies a polynomial isoperimetric inequality (resp. exponential isoperimetric inequality).     

关于线性微分方程组的几点注记

给出了线性微分方程组为可化组的两个充分条件,并由此得到关于解的有界性与稳定性的若干结果.     

Local Linear Smoothers Using Asymmetric Kernels

This paper considers using asymmetric kernels in local linear smoothing to estimate a regression curve with bounded support. The asymmetric kernels are either beta kernels if the curve has a compact support or gamma kernels if the curve is bounded from one end only. While possessing the standard benefits of local linear smoothing, the local linear smoother using the beta or gamma kernels offers some extra advantages in aspects of having finite variance and resistance to sparse design. These are due to their flexible kernel shape and the support of the kernel matching the support of the regression curve.     

一些特殊类型的变系数二阶线性微分方程解法的研究

运用变量变换的方法将一些特殊类型的变系数二阶线性微分方程化为常系数二阶线性微分方程,或已知齐次方程的一个解来求出齐次方程的另一个线性无关解,从而达到按照常系数二阶线性微分方程的特殊方法和利用常数变易法来求方程的通解的目的,同时纠正了文献[3]的结论和例子2的错误.