A construction of the value function in some differential games of approach with two pursuers and one evader

2006 ◽  
pp. 38-47
Author(s):  
A. Y. Levchenkov ◽  
A. G. Pashkov ◽  
S. D. Terekhov
Keyword(s):  
Differential Games ◽  
Value Function ◽  
The Value Function

Level Sets of the Value Function in Linear Differential Games with Elliptical Vectograms

2000 ◽  
Vol 33 (16) ◽  
pp. 579-584 ◽  
Author(s):  
S.S. Kumkov ◽  
V.S. Patsko ◽  
J. Shinar
Keyword(s):  
Differential Games ◽  
Level Sets ◽  
Value Function ◽  
Linear Differential Games ◽  
Linear Differential ◽  
The Value Function

Verification Theorems for Optimal Feedback Strategies in Differential Games

2003 ◽  
Vol 05 (02) ◽  
pp. 167-189 ◽  
Author(s):  
Ştefan Mirică
Keyword(s):  
Differential Games ◽  
Differential Game ◽  
Value Function ◽  
Generalized Derivatives ◽  
Differential Inequalities ◽  
Value Functions ◽  
Optimal Solutions ◽  
Optimal Feedback ◽  
Feedback Strategies ◽  
The Value Function

We give complete proofs to the verification theorems announced recently by the author for the "pairs of relatively optimal feedback strategies" of an autonomous differential game. These concepts are considered to describe the possibly optimal solutions of a differential game while the corresponding value functions are used as "instruments" for proving the relative optimality and also as "auxiliary characteristics" of the differential game. The 6 verification theorems in the paper are proved under different regularity assumptions accompanied by suitable differential inequalities verified by the generalized derivatives, mainly of contingent type, of the value function.

scholarly journals Revisiting a Three-Player Pursuit-Evasion Game

2021 ◽  
Author(s):  
János Szőts ◽  
Andrey V. Savkin ◽  
István Harmati
Keyword(s):  
Differential Games ◽  
Value Function ◽  
Optimal Trajectories ◽  
Computationally Efficient ◽  
Partial Derivatives ◽  
Pursuit Evasion ◽  
Detailed Explanation ◽  
Evasion Game ◽  
Derivatives Of ◽  
The Value Function

AbstractWe consider the game of a holonomic evader passing between two holonomic pursuers. The optimal trajectories of this game are known. We give a detailed explanation of the game of kind’s solution and present a computationally efficient way to obtain trajectories numerically by integrating the retrograde path equations. Additionally, we propose a method for calculating the partial derivatives of the Value function in the game of degree. This latter result applies to differential games with homogeneous Value.

scholarly journals ON LEVEL SETS WITH "NARROW THROATS" IN LINEAR DIFFERENTIAL GAMES

2005 ◽  
Vol 07 (03) ◽  
pp. 285-311 ◽  
Author(s):  
SERGEY S. KUMKOV ◽  
VALERY S. PATSKO ◽  
JOSEF SHINAR
Keyword(s):  
Differential Games ◽  
Level Sets ◽  
Value Function ◽  
Payoff Function ◽  
Terminal Time ◽  
Linear Differential Games ◽  
Neighboring Level ◽  
Linear Differential ◽  
Zero Sum ◽  
The Value Function

Examples with zero-sum linear differential games of fixed terminal time and a convex terminal payoff function depending on two components of the phase vector are considered. Such games can have an indifferent zone with constant value function. The level set of the value function associated with the indifferent zone is called the "critical" tube. In the selected examples, the critical tube and the neighboring level sets exhibit "narrow throats". Presence of such throats requires extremely precise computations for constructing the level sets. The paper presents different forms of critical tubes with narrow throats and indicates the combinations of problem parameters that can produce them.

scholarly journals Stochastic differential games and viscosity solutions of Isaacs equations

1988 ◽  
Vol 110 ◽  
pp. 163-184 ◽  
Author(s):  
Makiko Nisio
Keyword(s):  
Differential Games ◽  
Differential Game ◽  
Value Function ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Function ◽  
Stochastic Differential Games ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Representation Formulas ◽  
The Value Function

Recently P. L. Lions has demonstrated the connection between the value function of stochastic optimal control and a viscosity solution of Hamilton-Jacobi-Bellman equation [cf. 10, 11, 12]. The purpose of this paper is to extend partially his results to stochastic differential games, where two players conflict each other. If the value function of stochatic differential game is smooth enough, then it satisfies a second order partial differential equation with max-min or min-max type nonlinearity, called Isaacs equation [cf. 5]. Since we can write a nonlinear function as min-max of appropriate affine functions, under some mild conditions, the stochastic differential game theory provides some convenient representation formulas for solutions of nonlinear partial differential equations [cf. 1, 2, 3].

Level Sets of the Value Function in Differential Games with Two Pursuers and One Evader. Interval Analysis Interpretation

2014 ◽  
Vol 8 (3-4) ◽  
pp. 443-454 ◽  
Author(s):  
Sergey S. Kumkov ◽  
Stéphane Le Ménec ◽  
Valerii S. Patsko
Keyword(s):  
Differential Games ◽  
Interval Analysis ◽  
Level Sets ◽  
Value Function ◽  
The Value Function

scholarly journals Markov approximations of nonzero-sum differential games

2020 ◽  
Vol 30 (1) ◽  
pp. 3-17
Author(s):  
Yu.V. Averboukh
Keyword(s):  
Differential Games ◽  
Differential Inclusion ◽  
Continuous Time ◽  
Value Function ◽  
Dynamic Game ◽  
Approximate Solutions ◽  
Time Dynamic ◽  
Markov Game ◽  
Approximate Nash Equilibrium ◽  
The Value Function

The paper is concerned with approximate solutions of nonzero-sum differential games. An approximate Nash equilibrium can be designed by a given solution of an auxiliary continuous-time dynamic game. We consider the case when dynamics is determined by a Markov chain. For this game the value function is determined by an ordinary differential inclusion. Thus, we obtain a construction of approximate equilibria with the players' outcome close to the solution of the differential inclusion. Additionally, we propose a way of designing a continuous-time Markov game approximating the original dynamics.

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