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 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.

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 Simple motion pursuit differential game problem of many players with integral and geometric constraints on controls function.

2021 ◽  
pp. 12-16
Author(s):  
Jamilu Adamu ◽  
B. M. Abdulhamid ◽  
D. T. Gbande ◽  
A. S. Halliru
Keyword(s):  
Hilbert Space ◽  
Differential Game ◽  
Admissible Control ◽  
Sufficient Conditions ◽  
Game Problem ◽  
Geometric Constraints ◽  
Simple Motion ◽  
Control Functions

We study a simple motion pursuit differential game of many pursuers and one evader in a Hilbert space $l_{2}$. The control functions of the pursuers and evader are subject to integral and geometric constraints respectively. Duration of the game is denoted by positive number $\theta $. Pursuit is said to be completed if there exist strategies $u_{j}$ of the pursuers $P_{j}$ such that for any admissible control $v(\cdot)$ of the evader $E$ the inequality $\|y(\tau)-x_{j}(\tau)\|\leq l_{j}$ is satisfied for some $ j\in \{1,2, \dots\}$ and some time $\tau$. In this paper, sufficient conditions for completion of pursuit were obtained. Consequently strategies of the pursuers that ensure completion of pursuit are constructed.

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.

scholarly journals On pursuit-evasion differential game problem in a Hilbert space

2020 ◽  
Vol 5 (6) ◽  
pp. 7467-7479
Author(s):  
Jamilu Adamu ◽  
◽  
Kanikar Muangchoo ◽  
Abbas Ja’afaru Badakaya ◽  
Jewaidu Rilwan ◽  
...  
Keyword(s):  
Hilbert Space ◽  
Differential Game ◽  
Game Problem ◽  
Pursuit Evasion

Counterexamples Concerning Support Theorems for Convex Sets in Hilbert Space

1988 ◽  
Vol 31 (1) ◽  
pp. 121-128 ◽  
Author(s):  
R. R. Phelps
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Convex Function ◽  
Convex Subset ◽  
Convex Sets ◽  
Lower Semicontinuous ◽  
Nonempty Closed Convex Subset ◽  
Common Point ◽  
Support Points ◽  
Support Theorems

AbstractThe Bishop-Phelps theorem guarantees the existence of support points and support functionals for a nonempty closed convex subset of a Banach space; equivalently, it guarantees the existence of subdifferentials and points of subdifferentiability of a proper lower semicontinuous convex function on a Banach space. In this note we show that most of these results cannot be extended to pairs of convex sets or functions, even in Hilbert space. For instance, two proper lower semicontinuous convex functions need not have a common point of subdifferentiability nor need they have a subdifferential in common. Negative answers are also obtained to certain questions concerning density of support points for the closed sum of two convex subsets of Hilbert space.

scholarly journals Implicit iteration process of nonexpansive non-self-mappings

2005 ◽  
Vol 2005 (19) ◽  
pp. 3103-3110
Author(s):  
Somyot Plubtieng ◽  
Rattanaporn Punpaeng
Keyword(s):  
Hilbert Space ◽  
Convex Subset ◽  
Real Hilbert Space ◽  
Iteration Process ◽  
Nonempty Closed Convex Subset ◽  
Implicit Iteration Process ◽  
Nearest Point ◽  
Nearest Point Projection ◽  
Point Projection ◽  
Implicit Iteration

SupposeCis a nonempty closed convex subset of real Hilbert spaceH. LetT:C→Hbe a nonexpansive non-self-mapping andPis the nearest point projection ofHontoC. In this paper, we study the convergence of the sequences{xn},{yn},{zn}satisfyingxn=(1−αn)u+αnT[(1−βn)xn+βnTxn],yn=(1−αn)u+αnPT[(1−βn)yn+βnPTyn], andzn=P[(1−αn)u+αnTP[(1−βn)zn+βnTzn]], where{αn}⊆(0,1),0≤βn≤β<1andαn→1asn→∞. Our results extend and improve the recent ones announced by Xu and Yin, and many others.

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.

Ulam’s-Type Stability of First-Order Impulsive Differential Equations with Variable Delay in Quasi–Banach Spaces

2018 ◽  
Vol 19 (5) ◽  
pp. 553-560 ◽  
Author(s):  
JinRong Wang ◽  
Akbar Zada ◽  
Wajid Ali
Keyword(s):  
Differential Equations ◽  
Finite Number ◽  
Banach Spaces ◽  
Integral Inequality ◽  
Impulsive Differential Equations ◽  
Variable Delay ◽  
First Order ◽  
Variable Delays ◽  
Ulam’S Type Stability ◽  
First Time

AbstractIn this paper, Ulam’s-type stabilities are studied for a class of first-order impulsive differential equations with bounded variable delays on compact interval with finite number of impulses. Results of stability are proved via newly established integral inequality of Bellman–Grönwall–Bihari type with delay for discontinuous functions. Using this inequality for the first time and assumption of $\alpha$-H$\ddot{o}$lder’s condition instead of common Lipschitz condition is novelty of this paper. Moreover, solution is obtained in quasi–Banach spaces which is best suited for obtaining results under the assumptions of $\alpha$-H$\ddot{o}$lder’s condition.

scholarly journals EVASION FROM ONE PURSUER IN A HILBERT SPACE

2012 ◽  
Vol 09 ◽  
pp. 529-536
Author(s):  
FATEH ALLAHABI ◽  
G.I. IBRAGIMOV
Keyword(s):  
Hilbert Space ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Differential Game ◽  
Hilbert Spaces ◽  
Second Order ◽  
Geometric Constraints ◽  
Infinite Systems

We study a differential game of one pursuer and one evader described by infinite systems of second order ordinary differential equations. Controls of players are subjected to geometric constraints. Differential game is considered in Hilbert spaces. We proved one theorem on evasion. Moreover, we constructed explicitly a control of the evader.

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