scholarly journals An Optimal Pursuit Differential Game Problem with One Evader and Many Pursuers

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 842
Author(s):  
Idris Ahmed ◽  
Poom Kumam ◽  
Gafurjan Ibragimov ◽  
Jewaidu Rilwan ◽  
Wiyada Kumam
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Differential Game ◽  
Control Function ◽  
Optimal Strategies ◽  
Game Problem ◽  
Energy Resources ◽  
Integral Constraints ◽  
Value Of The Game ◽  
Optimal Pursuit

The objective of this paper is to study a pursuit differential game with finite or countably number of pursuers and one evader. The game is described by differential equations in l 2 -space, and integral constraints are imposed on the control function of the players. The duration of the game is fixed and the payoff functional is the greatest lower bound of distances between the pursuers and evader when the game is terminated. However, we discuss the condition for finding the value of the game and construct the optimal strategies of the players which ensure the completion of the game. An important fact to note is that we relaxed the usual conditions on the energy resources of the players. Finally, some examples are provided to illustrate our result.

scholarly journals Pursuit-Evasion Differential Game with Many Inertial Players

2009 ◽  
Vol 2009 ◽  
pp. 1-15 ◽  
Author(s):  
Gafurjan I. Ibragimov ◽  
Mehdi Salimi
Keyword(s):  
Hilbert Space ◽  
Lower Bound ◽  
Differential Game ◽  
Optimal Strategies ◽  
Countable Number ◽  
Integral Constraints ◽  
Pursuit Evasion ◽  
Value Of The Game ◽  
Control Functions

We consider pursuit-evasion differential game of countable number inertial players in Hilbert space with integral constraints on the control functions of players. Duration of the game is fixed. The payoff functional is the greatest lower bound of distances between the pursuers and evader when the game is terminated. The pursuers try to minimize the functional, and the evader tries to maximize it. In this paper, we find the value of the game and construct optimal strategies of the players.

scholarly journals Differential Game of Optimal Pursuit for an Infinite System of Differential Equations

2017 ◽  
Vol 42 (1) ◽  
pp. 391-403 ◽  
Author(s):  
Gafurjan Ibragimov ◽  
Idham Arif Alias ◽  
Usman Waziri ◽  
Abbas Badakaya Ja’afaru
Keyword(s):  
Differential Equations ◽  
Differential Game ◽  
Infinite System ◽  
System Of Differential Equations ◽  
Optimal Pursuit

scholarly journals Linear evasion differential game of one evader and several pursuers with integral constraints

2021 ◽  
Author(s):  
Gafurjan Ibragimov ◽  
Massimiliano Ferrara ◽  
Marks Ruziboev ◽  
Bruno Antonio Pansera
Keyword(s):  
Positive Integer ◽  
Differential Equations ◽  
Total Energy ◽  
Differential Game ◽  
Linear Differential Equations ◽  
State Variables ◽  
Integral Constraints ◽  
Control Functions ◽  
Evasion Strategy ◽  
Linear Differential

AbstractAn evasion differential game of one evader and many pursuers is studied. The dynamics of state variables $$x_1,\ldots , x_m$$ x 1 , … , x m are described by linear differential equations. The control functions of players are subjected to integral constraints. If $$x_i(t) \ne 0$$ x i ( t ) ≠ 0 for all $$i \in \{1,\ldots ,m\}$$ i ∈ { 1 , … , m } and $$t \ge 0$$ t ≥ 0 , then we say that evasion is possible. It is assumed that the total energy of pursuers doesn’t exceed the energy of evader. We construct an evasion strategy and prove that for any positive integer m evasion is possible.

scholarly journals A Purusit Differential Game Problem on a Closed Convex Subset of a Hilbert Space

2020 ◽  
pp. 115-119
Author(s):  
Abbas Ja'afaru Badakaya ◽  
Bilyaminu Muhammad
Keyword(s):  
Hilbert Space ◽  
Differential Equations ◽  
Finite Number ◽  
Differential Game ◽  
Convex Subset ◽  
Game Problem ◽  
Nonempty Closed Convex Subset ◽  
First Order ◽  
Control Functions ◽  
And Control

We study a pursuit differential game problem with finite number of pursuers and one evader on a nonempty closed convex subset of the Hilbert space l2. Players move according to certain first order ordinary differential equations and control functions of the pursuers and evader are subject to integral constraints. Pursuers win the game if the geometric positions of a pursuer and the evader coincide. We formulated and prove theorems that are concern with conditions that ensure win for the pursuers. Consequently, wining strategies of the pursuers are constructed. Furthermore, illustrative example is given to demonstrate the result.

scholarly journals PURSUIT-EVASION DIFFERENTIAL GAMES WITH THE GRÖNWALL TYPE CONSTRAINTS ON CONTROLS

2020 ◽  
Vol 6 (2) ◽  
pp. 95
Author(s):  
Bahrom T. Samatov ◽  
Gafurjan Ibragimov ◽  
Iroda V. Khodjibayeva
Keyword(s):  
Differential Games ◽  
Differential Game ◽  
Optimal Strategy ◽  
Optimal Strategies ◽  
The State ◽  
Differential Constraints ◽  
Quality Problem ◽  
Pursuit Evasion ◽  
Optimal Pursuit ◽  
Pursuit Strategy

A simple pursuit-evasion differential game of one pursuer and one evader is studied. The players' controls are subject to differential constraints in the form of the integral Grönwall inequality. The pursuit is considered completed if the state of the pursuer coincides with the state of the evader. The main goal of this work is to construct optimal strategies for the players and find the optimal pursuit time. A parallel approach strategy for Grönwall-type constraints is constructed and it is proved that it is the optimal strategy of the pursuer. In addition, the optimal strategy of the evader is constructed and the optimal pursuit time is obtained. The concept of a parallel pursuit strategy (\(\Pi\)-strategy for short) was introduced and used to solve the quality problem for "life-line" games by L.A.Petrosjan. This work develops and expands the works of Isaacs, Petrosjan, Pshenichnyi, and other researchers, including the authors.

scholarly journals A Study on Nash-Collative Differential Game of N -Players for Counterterrorism

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Mohamed Abd El-Hady Kassem ◽  
Abd El-Monem A. Megahed ◽  
Hebatallah K. Arafat
Keyword(s):  
Differential Game ◽  
Necessary Conditions ◽  
Optimal Strategies ◽  
Strategic Interaction ◽  
Game Problem ◽  
Terrorist Organizations ◽  
The Stability

In this work, by using the Nash-collative approach for a differential game problem between N -governments and terrorist organizations, we study governments’ cooperation and the role of each government for counterterrorism. Furthermore, we discuss the intertemporal strategic interaction of governments and terrorist organizations, where all world governments have to cooperate to fight terrorism. Also, we study the necessary conditions for finding the optimal strategies for each government to fight terrorism; we discuss the existence of the solution of the formulated problem and the stability set of the first kind of the optimal strategies.

scholarly journals On Some Pursuit and Evasion Differential Game Problems for an Infinite Number of First-Order Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Abbas Badakaya Ja'afaru ◽  
Gafurjan Ibragimov
Keyword(s):  
Differential Equations ◽  
Differential Game ◽  
Infinite Number ◽  
Control Function ◽  
Sufficient Conditions ◽  
First Order ◽  
Control Functions ◽  
Pursuit And Evasion ◽  
And Control ◽  
First Order Differential Equations

We study pursuit and evasion differential game problems described by infinite number of first-order differential equations with function coefficients in Hilbert spacel2. Problems involving integral, geometric, and mix constraints to the control functions of the players are considered. In each case, we give sufficient conditions for completion of pursuit and for which evasion is possible. Consequently, strategy of the pursuer and control function of the evader are constructed in an explicit form for every problem considered.

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

2019 ◽  
Vol 485 (2) ◽  
pp. 142-144
Author(s):  
A. A. Zevin
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Lipschitz Constant ◽  
Minimal Period ◽  
Arbitrary Vector ◽  
Vector Norm ◽  
Sharp Lower Bound

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

scholarly journals Differential Games for an Infinite 2-Systems of Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1467
Author(s):  
Muminjon Tukhtasinov ◽  
Gafurjan Ibragimov ◽  
Sarvinoz Kuchkarova ◽  
Risman Mat Hasim
Keyword(s):  
Hilbert Space ◽  
Differential Equations ◽  
Differential Games ◽  
Differential Game ◽  
Infinite System ◽  
The State ◽  
Geometric Constraints ◽  
Control Parameters ◽  
Systems Of Differential Equations ◽  
Pursuit And Evasion

A pursuit differential game described by an infinite system of 2-systems is studied in Hilbert space l2. Geometric constraints are imposed on control parameters of pursuer and evader. The purpose of pursuer is to bring the state of the system to the origin of the Hilbert space l2 and the evader tries to prevent this. Differential game is completed if the state of the system reaches the origin of l2. The problem is to find a guaranteed pursuit and evasion times. We give an equation for the guaranteed pursuit time and propose an explicit strategy for the pursuer. Additionally, a guaranteed evasion time is found.

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