Sufficient conditions for optimality of threat strategies in a differential game

1981 ◽  
Vol 33 (1) ◽  
pp. 99-109 ◽  
Author(s):  
A. Ray ◽  
A. Blaqui�re
Keyword(s):  
Differential Game ◽  
Sufficient Conditions ◽  
Sufficient Conditions For Optimality ◽  
Threat Strategies

On two pursuit differential game problems with state and geometric constraints in a Hilbert space

2021 ◽  
Vol 65 (3) ◽  
pp. 5-16
Author(s):  
Abbas Ja’afaru Badakaya ◽  
Keyword(s):  
Differential Game ◽  
Convex Set ◽  
Sufficient Conditions ◽  
Nonempty Closed Convex Subset ◽  
Geometric Constraints ◽  
First Order ◽  
Control Functions ◽  
Closed Convex Set ◽  
The Given ◽  
First Order Differential Equations

This paper concerns with the study of two pursuit differential game problems of many pursuers and many evaders on a nonempty closed convex subset of R^n. Throughout the period of the games, players must stay within the given closed convex set. Players’ laws of motion are defined by certain first order differential equations. Control functions of the pursuers and evaders are subject to geometric constraints. Pursuit is said to be completed if the geometric position of each of the evader coincides with that of a pursuer. We proved two theorems each of which is solution to a problem. Sufficient conditions for the completion of pursuit are provided in each of the theorems. Moreover, we constructed strategies of the pursuers that ensure completion of pursuit.

scholarly journals Invexity of supremum and infimum functions

2002 ◽  
Vol 65 (2) ◽  
pp. 289-306 ◽  
Author(s):  
Nguyen Xuan Ha ◽  
Do Van Luu
Keyword(s):  
Sufficient Conditions ◽  
Infinite Family ◽  
Lipschitz Functions ◽  
Necessary And Sufficient Conditions ◽  
Directional Derivatives ◽  
Mathematical Programs ◽  
Sufficient Conditions For Optimality ◽  
Invex Function ◽  
Necessary And Sufficient

Under suitable assumptions we establish the formulas for calculating generalised gradients and generalised directional derivatives in the Clarke sense of the supremum and the infimum of an infinite family of Lipschitz functions. From these results we derive the results ensuring such a supremum or infimum are an invex function when all functions of the invex. Applying these results to a class of mathematical programs, we obtain necessary and sufficient conditions for optimality.

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