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

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.

Constrained bilinear control problem: Application to a cancer chemotherapy model

2017 ◽  
Vol 10 (04) ◽  
pp. 1750054 ◽  
Author(s):  
El Hassan Zerrik ◽  
Nihale El Boukhari
Keyword(s):  
Optimal Control ◽  
Control Problem ◽  
Cancer Chemotherapy ◽  
Sufficient Conditions ◽  
Treatment Protocol ◽  
Bilinear Model ◽  
Bilinear Control ◽  
Finite Dimensional ◽  
Quadratic Cost ◽  
Admissible Controls

The aim of this paper is to investigate the optimal control problem for finite-dimensional bilinear systems and its application to a chemotherapy model. We characterize an optimal control that minimizes a quadratic cost functional in two cases of constrained admissible controls, then we give sufficient conditions for the uniqueness of such a control, and we derive useful algorithms for the computation of the optimal control. The established results are applied to a cancer chemotherapy bilinear model in order to simulate the optimal treatment protocol using two different approaches: one based on a limited instant toxicity, and the other on a limited cumulative toxicity along the therapy session.

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.

Smooth Finite Dimensional Embeddings

1999 ◽  
Vol 51 (3) ◽  
pp. 585-615 ◽  
Author(s):  
R. Mansfield ◽  
H. Movahedi-Lankarani ◽  
R. Wells
Keyword(s):  
Hilbert Space ◽  
Euclidean Space ◽  
Compact Subset ◽  
Structure Theorem ◽  
Sufficient Conditions ◽  
Dimensional Euclidean Space ◽  
Locally Compact ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
Compact Subsets

AbstractWe give necessary and sufficient conditions for a norm-compact subset of a Hilbert space to admit a C1 embedding into a finite dimensional Euclidean space. Using quasibundles, we prove a structure theorem saying that the stratum of n-dimensional points is contained in an n-dimensional C1 submanifold of the ambient Hilbert space. This work sharpens and extends earlier results of G. Glaeser on paratingents. As byproducts we obtain smoothing theorems for compact subsets of Hilbert space and disjunction theorems for locally compact subsets of Euclidean space.

scholarly journals Ulam–Hyers–Mittag-Leffler stability for tripled system of weighted fractional operator with TIME delay

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohammed A. Almalahi ◽  
Satish K. Panchal ◽  
Fahd Jarad ◽  
Thabet Abdeljawad
Keyword(s):  
Fixed Point ◽  
Time Delay ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Sufficient Conditions ◽  
Fractional Operator ◽  
Caputo Fractional Derivatives ◽  
Picard Operator

AbstractThis study is aimed to investigate the sufficient conditions of the existence of unique solutions and the Ulam–Hyers–Mittag-Leffler (UHML) stability for a tripled system of weighted generalized Caputo fractional derivatives investigated by Jarad et al. (Fractals 28:2040011 2020) in the frame of Chebyshev and Bielecki norms with time delay. The acquired results are obtained by using Banach fixed point theorems and the Picard operator (PO) method. Finally, a pertinent example of the results obtained is demonstrated.

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.

scholarly journals Oscillation criteria of certain fractional partial differential equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Di Xu ◽  
Fanwei Meng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Riccati Transformation ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Oscillation Criteria ◽  
Generalized Riccati Transformation

Abstract In this article, we regard the generalized Riccati transformation and Riemann–Liouville fractional derivatives as the principal instrument. In the proof, we take advantage of the fractional derivatives technique with the addition of interval segmentation techniques, which enlarge the manners to demonstrate the sufficient conditions for oscillation criteria of certain fractional partial differential equations.

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.

A Climate for Science Policy

2015 ◽  
Vol 45 (4) ◽  
pp. 577-609 ◽  
Author(s):  
Matthew L. Wallace
Keyword(s):  
Science Policy ◽  
Scientific Community ◽  
Basic Research ◽  
Internal Conflict ◽  
Atmospheric Sciences ◽  
Federal Organization ◽  
History Of ◽  
New Policies ◽  
Policies And Programs ◽  
Second World

Led by the Meteorological Service of Canada, atmospheric research in Canada underwent a period of rapid growth after the end of the Second World War. Within this federal organization, and in response to operational challenges and staff shortages, there were significant investments in basic research and in research oriented toward external users within Canada. Specifically, new policies and programs were put in place to enable the organization to gain legitimacy within the scientific community and within the federal government. New links with stakeholders and, more importantly, the development of explicit policies to guide research were a prime focus. These formalized strategies for pursuing two parallel types of research generated some internal conflict, but also helped form a common scientific identity among personnel. There were concerted efforts to disseminate research products and reinforce links both with the scientific community and with external users of meteorological and climatological research. Borne out by quantitative data, this science policy–centered history sheds light on the development of research and research specializations in the field in Canada. Most importantly, it provides insight into the global postwar expansion of the atmospheric sciences, which is strongly tied to national contexts. Indeed, the quest for legitimacy and the close connection to government priorities is central to the history of the atmospheric sciences in the twentieth century. More broadly, this case study points to a possible new conception of government science driven by political, bureaucratic, and scientific imperatives, as a means to shed light on scientific networks and practices.

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.

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