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10
Coordination of Groups of Mobile Autonomous Agents Using Nearest Neighbor Rules
, 2002
"... In a recent Physical Review Letters paper, Vicsek et. al. propose a simple but compelling discretetime model of n autonomous agents fi.e., points or particlesg all moving in the plane with the same speed but with dierent headings. Each agent's heading is updated using a local rule based on the a ..."
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Cited by 604 (44 self)
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In a recent Physical Review Letters paper, Vicsek et. al. propose a simple but compelling discretetime model of n autonomous agents fi.e., points or particlesg all moving in the plane with the same speed but with dierent headings. Each agent's heading is updated using a local rule based on the average of its own heading plus the headings of its \neighbors." In their paper, Vicsek et. al. provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models.
A positive systems model of TCPlike congestion control: Asymptotic results
 IEEE/ACM Transactions on Networking
, 2004
"... In this paper we study communication networks that employ droptail queueing and AdditiveIncrease MultiplicativeDecrease (AIMD) congestion control algorithms. We show that the theory of nonnegative matrices may be employed to model such networks. In particular, we show that important network p ..."
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Cited by 30 (8 self)
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In this paper we study communication networks that employ droptail queueing and AdditiveIncrease MultiplicativeDecrease (AIMD) congestion control algorithms. We show that the theory of nonnegative matrices may be employed to model such networks. In particular, we show that important network properties such as: (i) fairness; (ii) rate of convergence; and (iii) throughput; can be characterised by certain nonnegative matrices that arise in the study of AIMD networks. We demonstrate that these results can be used to develop tools for analysing the behaviour of AIMD communication networks. The accuracy of the models is demonstrated by means of several NSstudies.
Stability and Paracontractivity of Discrete Linear Inclusions
 Linear Algebra Appl
, 1999
"... We study stability properties of a finite set Sigma of n×nmatrices such as paracontractivity, BV and LCPstability, and their relations to each other. The conjecture on equivalence of paracontractivity and LCPstability is proved. Moreover, we prove the equivalence of the uniform BVstabilit ..."
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Cited by 11 (1 self)
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We study stability properties of a finite set Sigma of n×nmatrices such as paracontractivity, BV and LCPstability, and their relations to each other. The conjecture on equivalence of paracontractivity and LCPstability is proved. Moreover, we prove the equivalence of the uniform BVstability and the property of vanishing length of steps of any trajectory of Sigma.
Convergence of infinite products of matrices and innerouter iteration schemes,” Electron
 Trans. Numer. Anal
, 1994
"... Dedicated to Wilhelm Niethammer on the occasion of his sixtieth birthday. Abstract. We develop conditions under which a product ∏∞ i=0 Ti of matrices chosen from a possibly infinite set of matrices S = {Tjj ∈ J} converges. We obtain the following conditions which are sufficient for the convergence ..."
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Cited by 10 (0 self)
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Dedicated to Wilhelm Niethammer on the occasion of his sixtieth birthday. Abstract. We develop conditions under which a product ∏∞ i=0 Ti of matrices chosen from a possibly infinite set of matrices S = {Tjj ∈ J} converges. We obtain the following conditions which are sufficient for the convergence of the product: There exists a vector norm such that all matrices in S are nonexpansive with respect to this norm and there exists a subsequence {ik} ∞ k=0 of the sequence of the nonnegative integers such that the corresponding sequence of operators { } ∞ Tik k=0 converges to an operator which is paracontracting with respect to this norm. We deduce the continuity of the limit of the product of matrices as a function of the sequences {ik} ∞ k=0. But more importantly, we apply our results to the question of the convergence of inner–outer iteration schemes for solving singular consistent linear systems of equations, where the outer splitting is regular and the inner splitting is weak regular.
On Distributed Coordination of Mobile Agents with Changing Nearest Neighbors
, 2003
"... In a recent paper [10], we provided a formal analysis for a distributed coordination strategy proposed in [17] for coordination of a set of agents moving in the plane with the same speed but variable heading direction. Each agents heading is updated as the average of its heading and a set of its nea ..."
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Cited by 2 (0 self)
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In a recent paper [10], we provided a formal analysis for a distributed coordination strategy proposed in [17] for coordination of a set of agents moving in the plane with the same speed but variable heading direction. Each agents heading is updated as the average of its heading and a set of its nearest neighbors. As the agents move, the graph induced by the nearest neighbor relationship changes, resulting in switching. We recently demonstrated that by modelling the system as a discrete linear inclusion (in a discrete time setting) and a switched linear system (in continuous time setting), conditions for convergence of all headings to the same value can be provided. In this paper, we extend these results and demonstrate that in order for convergence to happen switching has to stop in a finite time. Moreover, we will show that a necessary and su#cient condition for convergence is that the switching stops on a connected graph. We also provide connections between this problem and Left convergent product (LCP) sets of matrices.
ASYMPTOTICS AND SEQUENTIAL CLOSURES OF CONTINUED FRACTIONS AND GENERALIZATIONS
"... We would like to dedicate this paper to our mathematical father and grandfather, ..."
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We would like to dedicate this paper to our mathematical father and grandfather,
IEEE/ACM TRANSACTIONS ON NETWORKING 1 A positive systems model of TCPlike congestion control: Asymptotic results
"... Abstract — We study communication networks that employ droptail queueing and AdditiveIncrease MultiplicativeDecrease (AIMD) congestion control algorithms. It is shown that the theory of nonnegative matrices may be employed to model such networks. In particular, important network properties such a ..."
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Abstract — We study communication networks that employ droptail queueing and AdditiveIncrease MultiplicativeDecrease (AIMD) congestion control algorithms. It is shown that the theory of nonnegative matrices may be employed to model such networks. In particular, important network properties such as: (i) fairness; (ii) rate of convergence; and (iii) throughput; can be characterised by certain nonnegative matrices. We demonstrate that these results can be used to develop tools for analysing the behaviour of AIMD communication networks. The accuracy of the models is demonstrated by several NSstudies. Index Terms — IEEEtran, journal, L ATEX, paper, template. I.
ftp ejde.math.txstate.edu ASYMPTOTIC STABILITY OF SWITCHING SYSTEMS
"... Abstract. In this article, we study the uniform asymptotic stability of the switched system u ′ = f ν(t)(u), u ∈ R n, where ν: R+ → {1, 2,..., m} is an arbitrary piecewise constant function. We find criteria for the asymptotic stability of nonlinear systems. In particular, for slow and homogeneous ..."
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Abstract. In this article, we study the uniform asymptotic stability of the switched system u ′ = f ν(t)(u), u ∈ R n, where ν: R+ → {1, 2,..., m} is an arbitrary piecewise constant function. We find criteria for the asymptotic stability of nonlinear systems. In particular, for slow and homogeneous systems, we prove that the asymptotic stability of each individual equation u ′ = fp(u) (p ∈ {1, 2,..., m}) implies the uniform asymptotic stability of the system (with respect to switched signals). For linear switched systems (i.e., fp(u) = Apu, where Ap is a linear mapping acting on E n) we establish the following result: The linear switched system is uniformly asymptotically stable if it does not admit nontrivial bounded full trajectories and at least one of the equations x ′ = Apx is asymptotically stable. We study this problem in the framework of linear nonautonomous dynamical systems (cocyles). 1.
DOI: 10.1109/TAC.2010.2054950 On the Convergence of Linear Switched Systems
, 2012
"... Abstract—This paper investigates sufficient conditions for the convergence to zero of the trajectories of linear switched systems. We provide a collection of results that use weak dwelltime, dwelltime, strong dwelltime, permanent and persistent activation hypothesis. The obtained results are shown ..."
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Abstract—This paper investigates sufficient conditions for the convergence to zero of the trajectories of linear switched systems. We provide a collection of results that use weak dwelltime, dwelltime, strong dwelltime, permanent and persistent activation hypothesis. The obtained results are shown to be tight by counterexample. Finally, we apply our result to the threecell converter. Index Terms—Switched systems, dwelltime, stability, omegalimit set, threecell converter.
OPINION DYNAMICS WITH STUBBORN VERTICES ∗
"... Abstract. Consider a social network where each person holds an opinion represented by a numerical value. Whenever a member of the social network is given a chance, the member updates his/her opinion according to a certain convex combination of the opinions of all network members. The influence digra ..."
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Abstract. Consider a social network where each person holds an opinion represented by a numerical value. Whenever a member of the social network is given a chance, the member updates his/her opinion according to a certain convex combination of the opinions of all network members. The influence digraph of the network has network members as vertices, and there is an arc from a vertex v to a vertex u if and only if, in the opinion update formula for v, the coefficient of u’s opinion is positive. The sink vertices in the influence digraph correspond to those stubborn people who never change their opinions. Assuming network members update their opinions one by one according to a given sequence, this note provides a description of the resulting opinion dynamics when every vertex can reach some sink vertex in the influence digraph.