scholarly journals Multiple Capture in a Group Pursuit Problem with Fractional Derivatives and Phase Restrictions

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1171
Author(s):  
Nikolay Nikandrovich Petrov
Keyword(s):  
Fractional Derivatives ◽  
Convex Compact ◽  
Dimensional Euclidean Space ◽  
Initial State ◽  
Pursuit Problem ◽  
Finite Dimensional ◽  
Convex Polyhedral Cone ◽  
Conflict Interaction ◽  
Admissible Controls ◽  
Target Sets

The problem of conflict interaction between a group of pursuers and an evader in a finite-dimensional Euclidean space is considered. All participants have equal opportunities. The dynamics of all players are described by a system of differential equations with fractional derivatives in the form D(α)zi=azi+ui−v,ui,v∈V, where D(α)f is a Caputo derivative of order α of the function f. Additionally, it is assumed that in the process of the game the evader does not move out of a convex polyhedral cone. The set of admissible controls V is a strictly convex compact and a is a real number. The goal of the group of pursuers is to capture of the evader by no less than m different pursuers (the instants of capture may or may not coincide). The target sets are the origin. For such a conflict-controlled process, we derive conditions on its parameters and initial state, which are sufficient for the trajectories of the players to meet at a certain instant of time for any counteractions of the evader. The method of resolving functions is used to solve the problem, which is used in differential games of pursuit by a group of pursuers of one evader.

scholarly journals Capture of two coordinated evaders in a problem with fractional derivatives, phase restrictions and a simple matrix

2020 ◽  
Vol 56 ◽  
pp. 50-62
Author(s):  
N.N. Petrov ◽  
A.I. Machtakova
Keyword(s):  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Basic Research ◽  
Dimensional Euclidean Space ◽  
Finite Dimensional ◽  
History Of ◽  
Simple Matrix ◽  
Admissible Controls ◽  
Approach Problem ◽  
Target Sets

In the finite-dimensional Euclidean space, a task of pursuing two evaders by a group of pursuers is considered, described by a system of the form D(α)zij=azij+ui−v, where D(α)f is the Caputo fractional derivative of order α∈(0,1) of the function f, and a is a real number. It is assumed that all evaders use the same control and that the evaders do not leave a convex cone with vertex at the origin. The aim of the group of pursuers is to capture two evaders. The pursuers use program counterstrategies based on information about the initial positions and the control history of the evaders. The set of admissible controls is a unit ball centered at zero, the target sets are the origins. In terms of initial positions and game parameters, sufficient conditions for the capture are obtained. Using the method of resolving functions as a basic research tool, we derive sufficient conditions for the solvability of the approach problem in some guaranteed time

Matrix Resolving Function in the Nonstationary Linear Group Pursuit Problem Concepting Multiple Capture

2021 ◽  
pp. 2150016
Author(s):  
N. N. Petrov
Keyword(s):  
Euclidean Space ◽  
Linear Group ◽  
Sufficient Conditions ◽  
Dimensional Euclidean Space ◽  
Pursuit Problem ◽  
Mathematical Basis ◽  
Group Pursuit ◽  
Finite Dimensional

In finite-dimensional Euclidean space, an analysis is made of the problem of pursuit of a single evader by a group of pursuers, which is described by a system of the form [Formula: see text] The goal of the group of pursuers is the capture of the evader by no less than [Formula: see text] different pursuers (the instants of capture may or may not coincide). Matrix resolving functions, which are a generalization of scalar resolving functions, are used as a mathematical basis of this study. Sufficient conditions are obtained for multiple capture of a single evader in the class of quasi-strategies. Examples illustrating the results obtained are given.

EVALUATION OF THE PURSUIT TIME IN DIFFERENTIAL GAMES OF MANY PLAYERS ON A CONVEX COMPACT

2021 ◽  
Vol 2 ◽  
pp. 51-66
Author(s):  
Mashrabzhan Mamatov ◽  
◽  
Azizkhon Zunnunov ◽  
Egamberdi Esonov ◽  
◽  
...  
Keyword(s):  
Differential Games ◽  
Finite Time ◽  
Convex Set ◽  
Auxiliary Problem ◽  
Unit Cube ◽  
Convex Compact ◽  
Dimensional Euclidean Space ◽  
Compact Set ◽  
Pursuit Problem ◽  
Pursuit Strategy

The paper is devoted to the study of the problem of constructing a pursuit strategy in simple differential games of many persons with phase constraints in the state of the players, in the sense of getting into a certain neighborhood of the evader. The game takes place in -dimensional Euclidean space on a convex compact set. The pursuit problem is considered when the number of pursuing players is , that is, less than , in the sense of — captures. A structure for constructing pursuit controls is proposed, which will ensure the completion of the game in a finite time. An upper bound is obtained for the game time for the completion of the pursuit. An auxiliary problem of simple pursuit on a unit cube in the first orthant is considered, and strategies of pursuing players are constructed to complete the game with special initial positions. The results obtained are used to solve differential games with arbitrary initial positions. For this task, a structure for constructing a pursuit strategy is proposed that will ensure the completion of the game in a finite time. The generalization of the problem in the sense of complicating the obstacle is also considered. A more general problem of simple pursuit on a cube of arbitrary size in the first orthant is considered. With the help of the proposed strategies, the possibilities of completing the pursuit are proved and an estimate of the time is obtained. As a consequence of this result, lower and upper bounds are obtained for the pursuit time in a game with ball-type obstacles. Estimates are obtained for the pursuit time when the compact set is an arbitrarily convex set. The concept of a convex set in a direction relative to a section, which is not necessarily convex, is defined. And in it the problem of simple pursuit in a differential game of many players is studied and the possibilities of completing the pursuit using the proposed strategy are shown. The time of completion of the pursuit of the given game is estimated from above.

On Vitali's Theorem for Groups of Motions of Euclidean Space

1999 ◽  
Vol 6 (4) ◽  
pp. 323-334
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Finite Dimensional

Abstract We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

On inequalities of the Tchebychev type

1963 ◽  
Vol 59 (1) ◽  
pp. 135-146 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Probability Theory ◽  
Euclidean Space ◽  
Random Variable ◽  
Dimensional Euclidean Space ◽  
Real Random Variable ◽  
Finite Dimensional

1. A type of problem which frequently occurs in probability theory and statistics can be formulated in the following way. We are given real-valued functions f(x), gi(x) (i = 1, 2, …, k) on a space (typically finite-dimensional Euclidean space). Then the problem is to set bounds for Ef(X), where X is a random variable taking values in , about which all we know is the values of Egi(X). For example, we might wish to set bounds for P(X > a), where X is a real random variable with some of its moments given.

scholarly journals On the construction of resolving control in the problem of getting close at a fixed time moment

2021 ◽  
Vol 58 ◽  
pp. 73-93
Author(s):  
V.N. Ushakov ◽  
A.V. Ushakov ◽  
O.A. Kuvshinov
Keyword(s):  
Euclidean Space ◽  
Compact Space ◽  
Time Moment ◽  
Fixed Time ◽  
Dimensional Euclidean Space ◽  
Controlled System ◽  
Finite Dimensional ◽  
Solvability Set

The problem of getting close of a controlled system with a compact space in a finite-dimensional Euclidean space at a fixed time is studied. A method of constructing a solution to the problem is proposed which is based on the ideology of the maximum shift of the motion of the controlled system by the solvability set of the getting close problem.

scholarly journals The completion of the space of convex, bounded sets with respect to the Demyanov metric

2013 ◽  
Vol 46 (1) ◽  
Author(s):  
Jerzy Grzybowski ◽  
Andrzej Leśniewski ◽  
Tadeusz Rzeżuchowski
Keyword(s):  
Convex Compact ◽  
Compact Sets ◽  
Finite Dimensional ◽  
The Family ◽  
Bounded Sets ◽  
Convex Compact Sets

AbstractThe Demyanov metric in the family of convex, compact sets in finite dimensional spaces has been recently extended to the family of convex, bounded sets – not necessarily closed. In this note it is shown that these spaces are not complete and a model for the completion is proposed. A full answer is given in ℝ

scholarly journals Controllability of Semilinear Systems with Multiple Variable Delays in Control

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1955
Author(s):  
Jerzy Klamka
Keyword(s):  
Control Systems ◽  
Time Variable ◽  
Multiple Time ◽  
Variable Delays ◽  
Finite Dimensional ◽  
Multiple Variable ◽  
Variable Point ◽  
Admissible Controls ◽  
Point Delays ◽  
Controllability Conditions

In the paper semilinear, finite-dimensional, control systems with multiple time variable point delays in admissible controls are considered. Using Rothe’s fixed-point theorem, sufficient controllability conditions are formulated. The results of the paper are generalization to many time variable delays in control, of the results published recently.

On the Interrelation of Two Linear NonStationary Problems with Multiple Evaders

2015 ◽  
Vol 17 (04) ◽  
pp. 1550013 ◽  
Author(s):  
N. N. Petrov ◽  
K. A. Shchelchkov
Keyword(s):  
Finite Time ◽  
Time Interval ◽  
Finite Time Interval ◽  
Infinite Time ◽  
Pursuit Problem ◽  
Infinite Time Interval ◽  
Admissible Controls

A linear nonstationary pursuit problem in which a group of pursuers and a group of evaders are involved is considered under the condition that the group of pursuers includes participants whose admissible controls set coincides with that of the evaders and participants whose admissible controls sets belong to interior of admissible controls set of the evaders. The aim of the group of pursuers is to capture all the evaders. The aim of the group of evaders is to prevent the capture, that is, to allow at least one of the evaders to avoid the rendezvous. It is shown that, if in the game in which all the participants have equal capabilities at least one of the evaders avoids the rendezvous on an infinite time interval, then as a result of the addition of any number of pursuers with less capabilities, at least one of the evaders will avoid the rendezvous on any finite time interval.

scholarly journals Product rule for vector fractional derivatives

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
Diogo Bolster ◽  
Mark Meerschaert ◽  
Alla Sikorskii
Keyword(s):  
Vector Space ◽  
Fractional Derivatives ◽  
Fourier Transforms ◽  
Product Rule ◽  
Finite Dimensional ◽  
Euclidean Vector Space ◽  
Derivatives Of

AbstractThis paper establishes a product rule for fractional derivatives of a realvalued function defined on a finite dimensional Euclidean vector space. The proof uses Fourier transforms.

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