On Exact Construction of Solvability Set for Differential Games with Simple Motion and Non-convex Terminal Set

2017 ◽  
pp. 237-258
Author(s):  
Liudmila Kamneva ◽  
Valerii Patsko
Keyword(s):  
Differential Games ◽  
Simple Motion ◽  
Terminal Set ◽  
Solvability Set

DIFFERENTIAL GAMES OF FRACTIONAL ORDER WITH DISTRIBUTED PARAMETERS

2021 ◽  
Vol 4 ◽  
pp. 38-47
Author(s):  
Mashrabzhan Mamatov ◽  
◽  
Jalolkon Nuritdinov ◽  
Egamberdi Esonov ◽  
◽  
...  
Keyword(s):  
Differential Games ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Initial Position ◽  
Difference Approximation ◽  
Pursuit Problem ◽  
Spatial Variables ◽  
Terminal Set ◽  
Distributed Parameters ◽  
Discrete Game

The article deals with the problem of pursuit in differential games of fractional order with distributed parameters. Partial fractional derivatives with respect to time and space variables are understood in the sense of Riemann - Liouville, and the Grunwald-Letnikov formula is used in the approximation. The problem of getting into some positive neighborhood of the terminal set is considered. To solve this problem, the finite difference method is used. The fractional Riemann-Liouville derivatives with respect to spatial variables on a segment are approximated using the Grunwald-Letnikov formula. Using a sufficient criterion for the existence of a fractional derivative, a difference approximation of the fractional-order derivative with respect to time is obtained. By approximating a differential game to an explicit difference game, a discrete game is obtained. The corresponding pursuit problem for a discrete game is formulated, which is obtained using the approximation of a continuous game. The concept of the possibility of completing the pursuit, a discrete game in the sense of an exact capture, is defined. Sufficient conditions are obtained for the possibility of completing the pursuit. It is shown that the order of approximation in time is equal to one, and in spatial variables is equal to two. It is proved that if in a discrete game from a given initial position it is possible to complete the pursuit in the sense of exact capture, then in a continuous game from the corresponding initial position it is possible to complete the pursuit in the sense of hitting a certain neighborhood. A structure for constructing pursuit controls is proposed, which will ensure the completion of the game in a finite time. The methods used for this problem can be used to study differential games described by more general equations of fractional order.

Differential Games of Survival with Space-Like Terminal Set

1974 ◽  
Vol 12 (1) ◽  
pp. 167-177 ◽  
Author(s):  
Ronald J. Stern
Keyword(s):  
Differential Games ◽  
Terminal Set

Differential games with optional stopping

1974 ◽  
Vol 76 (1) ◽  
pp. 263-284 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Differential Games ◽  
Bellman Equations ◽  
Terminal Set ◽  
Entry Time ◽  
Optional Stopping ◽  
Existence Of Value ◽  
Image Position

Consider a differential game of survival governed by the differential equationin , with pay-offwhere tF is the entry time of the trajectory (t, x(t)) into a given terminal set F. Under suitable conditions on f, g, h and the terminal set F, it was shown in (3) that the question of existence of value of such a game can be approached by considering a certain pair of partial differential equations called the Isaacs-Bellman equations.

scholarly journals Simple Motion Pursuit and Evasion Differential Games with Many Pursuers on Manifolds with Euclidean Metric

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Atamurat Kuchkarov ◽  
Gafurjan Ibragimov ◽  
Massimiliano Ferrara
Keyword(s):  
Euclidean Space ◽  
Differential Games ◽  
Differential Game ◽  
Finite Time ◽  
The State ◽  
Simple Motion ◽  
Euclidean Metric ◽  
Pursuit Game ◽  
Pursuit And Evasion ◽  
Evasion Game

We consider pursuit and evasion differential games of a group ofmpursuers and one evader on manifolds with Euclidean metric. The motions of all players are simple, and maximal speeds of all players are equal. If the state of a pursuer coincides with that of the evader at some time, we say that pursuit is completed. We establish that each of the differential games (pursuit or evasion) is equivalent to a differential game ofmgroups of countably many pursuers and one group of countably many evaders in Euclidean space. All the players in any of these groups are controlled by one controlled parameter. We find a condition under which pursuit can be completed, and if this condition is not satisfied, then evasion is possible. We construct strategies for the pursuers in pursuit game which ensure completion the game for a finite time and give a formula for this time. In the case of evasion game, we construct a strategy for the evader.

Near Optimal Strategy for Nonlinear Stochastic Differential Games Based on the Technique of Statistical Linearization

2014 ◽  
Vol 39 (4) ◽  
pp. 390-399
Author(s):  
Ping ZHANG ◽  
Yang-Wang FANG ◽  
Xiao-Bin HUI ◽  
Xin-Ai LIU ◽  
Liang LI
Keyword(s):  
Differential Games ◽  
Optimal Strategy ◽  
Stochastic Differential Games ◽  
Statistical Linearization
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