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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 2, FEBRUARY 2005 169

Stability of Multiagent Systems With Time-DependentCommunication Links

Luc Moreau

Abstract—We study a simple but compelling model of networkof agents interacting via time-dependent communication links. Themodel finds application in a variety of fields including synchroniza-tion, swarming and distributed decision making. In the model, eachagent updates his current state based upon the current informa-tion received from neighboring agents. Necessary and/or sufficientconditions for the convergence of the individual agents’ states toa common value are presented, thereby extending recent resultsreported in the literature. The stability analysis is based upon ablend of graph-theoretic and system-theoretic tools with the no-tion of convexity playing a central role. The analysis is integratedwithin a formal framework of set-valued Lyapunov theory, whichmay be of independent interest. Among others, it is observed thatmore communication does not necessarily lead to faster conver-gence and may eventually even lead to a loss of convergence, evenfor the simple models discussed in the present paper.

Index Terms—Multiagent systems, set-valued Lyapunov theory,stability analysis, swarms, synchronization.

I. INTRODUCTION

RECENT years have witnessed an increasing interest inthe interaction between information flow and system dy-

namics. It is further recognized that information and communi-cation constraints may have a considerable impact on the per-formance of a control system. The study of these topics formsan active area of research, giving rise to new control paradigmssuch as quantized control systems, networked control systemsand multiagent (multivehicle, multirobot) systems.

An important aspect of information flow in a dynamicalsystem is the communication topology which determines whatinformation is available for which component at a given timeinstant. We study the role of the communication topology inthe context of multi-agent systems. Our interest in multi-agentsystems is motivated by the emergence of several applicationsincluding formation flying of unmanned aerial vehicles (UAVs),cooperative robotics [2]–[4] and sensor networks [5], [6].

In this paper, we consider a group of agents, not necessarilyidentical. The individual agents share a common state space andeach agent updates his current state based upon the information

Manuscript received July 8, 2003; revised February 24, 2004. Recommendedby Associate Editor Wei Kang. This paper presents research results of the Bel-gian Programme on Interuniversity Poles of Attraction, initiated by the BelgianState, Prime Minister’s Office for Science, Technology, and Culture. The sci-entific responsibility rests with its author. A preliminary version of this workhas appeared as a conference paper in [1]. L. Moreau was a Postdoctoral Fellowof the Fund for Scientific Research—Flanders (F.W.O.-Vlaanderen) associatedwith Ghent University, EeSA Department, SYSTeMS Research Group, Tech-nologiepark 914, 9052 Zwijnaarde, Belgium.

The author is with Sidmar, 9000 Ghent, Belgium (e-mail: [email protected]).

Digital Object Identifier 10.1109/TAC.2004.841888

received from other agents, according to a simple rule. Our aimis to relate the information flow and communication structurewith the stability properties of the group of agents.

The model that we consider encompasses, or is closely relatedto, several models reported in the literature. A prominent andwell-studied example concerns synchronization of coupled os-cillators, a phenomenon which is ubiquitous in the natural worldand finds several applications in physics and engineering; see [7]for a review and a historical perspective. The Kuramoto equa-tion [8]–[10], which is widely appreciated in that field, modelsa population of oscillators with (typically sinusoidal) couplingterms. The Kuramoto equation may be studied with the tools ofthis paper, at least when attention is restricted to identical os-cillators whose initial phase differences are sufficiently small.Another interesting example concerns swarming, a cooperativebehavior observed for a variety of living beings such as birds,fish, bacteria, etc. In the physics literature, swarming models areoften individual-based with each individual being representedby a particle moving at constant speed, its direction of motionbeing updated according to nearest neighbor coupling; see, forexample, [11]. In recent years, engineering applications suchas formation control have increased the interest of engineersin swarming and collective motion patterns [12]–[26]. The ap-proach of the present paper applies to the linear swarming modelof [14] as well as to the nonlinear model of [11] provided the ini-tial orientation differences are sufficiently small. A third and lastexample of multiagent systems that fall within the scope of thispaper are the consensus algorithms studied in [27]–[31]. Con-sensus protocols enable a network of agents to agree upon quan-tities of interest via a process of distributed decision making.Distributed agreement problems have a long history; see, for ex-ample, the book [32, Sec. 4.6] and [33].

We impose very weak assumptions on the communicationtopology; we allow for nonbidirectional and time-dependentcommunication patterns. Unidirectional communication is im-portant in practical applications and can easily be incorporated,for example, via broadcasting. Also, sensed information flowwhich plays a central role in schooling and flocking is typicallynot bidirectional. In addition, we do not exclude loops in thecommunication topology. This means that, typically, we areconsidering leaderless coordination rather than a leader-fol-lower approach. (Leaderless coordination is also considered,for example, in [34]–[38]). Finally, we allow for time-depen-dent communication patterns which are important if we want totake into account link failure and link creation, reconfigurablenetworks and nearest neighbor coupling.

The contribution of the paper is threefold. We presentnecessary and/or sufficient conditions on the communication

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170 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 2, FEBRUARY 2005

topology guaranteeing convergence of the individual agents’states to a common value, thereby extending some of the resultsreported in [14], [27]–[29], and [31]. A second contributionconcerns the analysis technique enabling us to obtain theseresults. We introduce a novel approach which is centeredaround the notion of convexity. Other studies that considerthe connection between communication topology and systemstability such as [14], [27]–[29], [31], [34], and [35] typicallyrely on algebraic graph theory, relating the graph topology withthe algebraic structure of associated graph matrices. In thispaper, instead of relying on algebraic graph theory, we proposea blend of graph-theoretic and system-theoretic tools to analyzestability. Our approach is of an inherently nonlinear nature withthe notion of convexity playing a central role. (Independently ofthe present research, convexity ideas are also being applied tothe distributed rendezvous problem in [20].) Third, the stabilityanalysis is integrated within a formal framework of set-valuedLyapunov theory, which may be of independent interest.

The paper is organized as follows. Section II illustrates themain results of the paper by means of a linear example. Sec-tion III introduces the more general model which is studied inthe remainder of the paper. Necessary and/or sufficient condi-tions for the convergence of the agents’ states to a commonvalue are presented in Section IV. Detailed proofs of these re-sults are postponed until Section V. Section VI summarizes themain conclusions. In the paper, considerable attention is devotedto a precise formulation of the stability and convergence proper-ties. For this purpose, we introduce in Appendix A several novelstability concepts, as well as a suitable Lyapunov characteriza-tion. Appendix B contains a graph-theoretic result that is usedin the technical proofs of this paper. A preliminary version ofthis work has appeared as a conference paper [1].

Notation and terminologyWe distinguish between the set-inclusion symbols and .

The symbol denotes strict inclusion whereas denotes non-strict inclusion.

Let be a subset of a finite-dimensional Euclidean space. The set is defined as the set of points inwhose distance to is strictly smaller than . When is a

singleton the notation is used instead of .Consider finite-dimensional Euclidean spaces and . We

denote by the collection of all closed subsets of . Aset-valued function is a (single-valued) function

. A set-valued function is calledupper semicontinuous if for every and every thereis such that whenever .

The relative interior (ri) of a convex set in finite-dimen-sional Euclidean space is the interior which results when thisset is regarded as a subset of its affine hull and is denoted by

. A polytope is a set which is the convex hull of finitelymany points. See [39] for more information.

The set of nonnegative (respectively, strictly positive) realnumbers is denoted by (respectively, ).

II. EXAMPLE

The present section illustrates the main results of the paper bymeans of a linear example. The results that we obtain for this ex-

ample generalize results that have recently been reported in theliterature. At the same time, the present section also introducesseveral notions that will be used repeatedly in the remainder ofthis paper (directed graph, neighbors, connectivity).

Definition 1 (Directed Graph): A directed graph is a pairwhere is a nonempty, finite set and is a subset ofsatisfying for all . Elements of are

referred to as nodes and an element of is referred to asan arc from to .

Definition 2 (Neighbors): Consider a directed graphand a nonempty subset . The set isthe set of those nodes for which there issuch that . When is a singleton , the notation

is used instead of .Notice that any nonempty subset satisfies

Definition 3 (Weighted Directed Graph): A weighted di-rected graph is a triple where is a directedgraph and is a map associating to each arca strictly positive weight denoted by .

Remark 1: Notice that we have excluded self-loops in thedefinition of a directed graph. This is merely a matter of con-vention. It is important to emphasize that the stochastic matrixassociated to a weighted directed graph according to (1) belowdoes, in fact, have strictly positive diagonal elements.

Consider, for example, the weighted directed graph depictedin Fig. 1. Each node in this graph corresponds to an agent andeach arc represents a communication channel. Each agent

is being attributed a real state variable .1 Sup-pose that this weighted directed graph represents the communi-cation pattern at time . Agent 1 receives information from agent2 and updates his state according to the weighted average

Notice that agent 1’s own state is taken into account withunit weight 1. Agent 2 receives information from agents 1 and3 and updates his state according to the weighted average

Agents 3 and 4 do not receive any information

We are interested in the dynamics of the -variables that ariseswhen the communication graph changes over time. In order toformulate the dynamics for general graphs, we associate to aweighted directed graph with vertex set

1Within the context of swarming models [11], [14], agent i could be a particlemoving at constant speed in two-dimensional space and x could represent itsheading. More general swarming models may be conceived, where x is notnecessarily a scalar but possibly a vector representing the agent’s configurationin space; see the following section.

MOREAU: STABILITY OF MULTIAGENT SYSTEMS WITH TIME-DEPENDENT COMMUNICATION LINKS 171

Fig. 1. Weighted directed graph.

a real -matrix whose components wedefine as shown in

if

if

otherwise.(1)

Notice that the matrix is stochastic; that is, it is a squarematrix with nonnegative entries and with the property that itsrow sums are all equal to 1. In addition, its diagonal elementsare strictly positive by construction. For the weighted directedgraph of Fig. 1, for example, the matrix becomes

Consider a sequence of weighted directed graphswith common vertex set and

where . We are interested in the following discrete-timesystem on :

(2)

In particular, we want to formulate necessary and/or sufficientconditions for the convergence of the components to acommon value as . It should be clear that the convergenceproperties of (2) depend not only on the connectivity propertiesof the sequence of directed graphs but also on the associatedweight functions. As we intend to focus on the role of the con-nectivity properties, we assume for simplicity that the weightsare well-behaved by imposing uniform bounds on the weights.Here, we state two results that follow as a consequence of thegeneral theory established in the paper (Theorems 2 and 3). Theformulation of these results relies on the notion of connectivity.

Definition 4 (Connectivity): Consider a directed graph. A node is connected to a node

if there is a path from to in the graph which respects theorientation of the arcs. Consider a sequence of directed graphs

with . A node is connected to a nodeacross an interval if is connected to for

the directed graph .Proposition 1 (Nonbidirectional): Consider a sequence

of weighted directed graphs withcommon vertex set and where . Assumethe existence of real numbers such that

for all and . If thereis such that for all there is a node connected to

all other nodes across , then the componentsof any solution of (2) converge to a common value as .

Proposition 2 (Bidirectional): Consider a sequence ofweighted directed graphs withcommon vertex set and where . Assumethe existence of real numbers such that

for all and . Assumein addition that the graphs are bidirectional2 for all

. If for all there is a node connected to all othernodes across ,3 then the components of anysolution of (2) converge to a common value as .

A. Discussion of Propositions 1 and 2

Let us mention some of the subtleties involved in the study of(2). First of all, (2) belongs to the class of linear time-varyingsystems, whose stability properties are hard to analyze in gen-eral. An eigenvalue analysis, for example, is no longer appli-cable if the system matrix depends explicitly on time. Second,the equilibrium value to which the components are shown toconverge, is not known explicitly. This equilibrium value de-pends, in general, both on the initial data and on the sequenceof weighted directed graphs. Third, it has been shown in [14]that, in general, there does not exist a time-invariant, quadraticLyapunov function for (2). Partially motivated by this negativeresult we put forward in this paper a series of graph-theoreticand system-theoretic analysis tools which are not based on thelinear structure of (2). As a consequence of our inherently non-linear approach Propositions 1 and 2 hold verbatim if we replacethe linear dynamics (2) by their nonlinear, saturated counterpart

where the saturation function sat should be interpreted compo-nentwise as .

For the special case that the weights are all equal to one, (2)reduces to the model studied in [14, Ths. 1–2]. From a controlperspective, an important contribution of [14] is that this workrelates the information flow and communication structure withthe stability properties of the group of agents. Apart from the ob-servation that the present models are more general and possiblynonlinear, it is instructive to notice that we do not assume bidi-rectional communication (Proposition 1) and, when we considerbidirectional communication, we do not assume uniformity inthe connectivity condition (Proposition 2).

By specializing Proposition 1 to the case of time-independentweighted directed graphs, we recover the following well-knowngraph theoretic result implicitly contained in [40, Secs. 7.2 and7.3]. A stochastic -matrix with positive diagonal ele-ments has all but one of its eigenvalues strictly inside the unitcircle (the only exception being the trivial eigenvalue 1) if (andonly if—see Theorem 2) the associated directed graph

and

2That is, (k; l) 2 A(t) whenever (l; k) 2 A(t). Notice, however, that we donot require the weights w (t) and w (t) to be equal.

3Since the graphs are assumed to be bidirectional this means that all nodesare connected to all other nodes across [t ;1).


has a node connected to all other nodes. Indeed, observe that theright eigenvector corresponding to the trivial eigen-value 1 of a stochastic matrix determines the equilibrium set

of (2). The convergence propertyestablished in Proposition 1 corresponds to all other eigenvaluesbeing located strictly inside the unit circle. As mentioned before,we recover this well-known graph theoretic result using inher-ently nonlinear analysis tools. Moreover, the present approachenables us to go beyond this result and consider time-dependentstochastic matrices.

As a preview of the remainder of the paper, let us announcethat our approach is based on the convexity property or contrac-tion property of (2), which says that the future, updated valueof any agent in the network is a convex combination of its cur-rent value as well as the current values of its neighbors. It is thusclear that is a nonincreasing function of time.Likewise cannot decrease. In the remainder ofthis paper, we will work out these ideas in a more general set-ting and study their consequences for the stability properties ofmulti-agent systems with time-dependent communication links.The convexity or contraction property mentioned here has actu-ally been known for a long time, notably in the context of non-negative and stochastic matrices. It serves as a starting point forthe ergodicity analysis of products of stochastic matrices in [32].See also [45], where the relationship between ergodicity theoryand distributed algorithms is further explored.

III. MULTIAGENT DYNAMICS

We introduce here the dynamical equations that will bestudied in the remainder of this paper. These equations modela network of interacting agents. The individual agents sharea common state space which is assumed to be finite-dimen-sional Euclidean. As a model for this multiagent network weconsider a nonlinear discrete-time system on

(3)

or, expressed in terms of the individual agents’ states

...

Notice that the individual -variables need not be scalars; theymay belong to a common Euclidean space of arbitrary finitedimension. For simplicity of the presentation we assume thatthe map is continuous.

The formulation in (3) is still very general and does notyet capture the constraints imposed by the limited interagentcommunication. In order to incorporate the limited interagentcommunication in the model, it is common to introduce asequence of directed graphs with common node set

and with . The directed graphcharacterizes the communication links at time . Each node ofthis graph corresponds to an agent and each arc represents a

communication channel. The communication graphdetermines what information is available for which agent attime . The following assumption relates the sequence of com-munication graphs with the discrete-time dynamics(3). It is a strict convexity assumption requiring that, for eachagent , the updated state is a strict convex combina-tion of agent ’s current state and the current statesof its neighbors .

Assumption 1 (Convexity): Associated to each directedgraph with node set , each agentand each state , there is a compact setsatisfying

1)(4)

2) whenever the states of agentand agents are all equal;

3) is contained in the relative interiorof the convex hull of the states of agent and agents

whenever the states of agent andagents are not all equal;

4) depends continuously on ; that is, the set-valued function is continuous.

The following examples illustrate that the present modelfinds wide application in a variety of fields including syn-chronization, swarming and distributed decision making. Thenecessary and/or sufficient conditions for the convergence ofthe individual agents’ states to a common value that are pre-sented in the following section apply directly to these examples.

Example 1 (Linear): The linear model of Section II fallswithin the scope of the present study. Consider a sequenceof weighted directed graphs withcommon vertex set and where . Assumethe existence of real numbers such that theweights satisfy for alland . The linear model studied in Section II corre-sponds to the case of and with thematrix defined in (1). The discrete-time map satisfiesAssumption 1 with being the set of all possible valuesof that are obtained by considering all possible weighteddirected graphs with the given arc set and with weightscontained in the compact interval .

Example 2 (Synchronization): Consider a population of os-cillators that share a common state space . Denote the state ofoscillator by and consider for each communication graph

the Kuramoto equation [8], [9]

(5)

with strictly positive coupling strengths . Let us considerthe case that the are all equal. In this case, we may assumewithout loss of generality (by introducing a rotating referenceframe) that . We restrict attention to an-gles in the interval and introduce local coordinates


. Expressed in termsof these local coordinates, the Kuramoto equation becomes4

(6)The continuous-time system (6) may be studied withinthe present framework by introducing its time-1 map

which depends, of course, on thecommunication structure . Explicitly, consider a time-se-quence and let .This discrete-time map satisfies Assumption 1 with

. The present paper allows to studysynchronization of oscillators with time-varying and nonbidi-rectional coupling.

Example 3 (Consensus Algorithm): The following nonlinearconsensus protocol is studied in [27]. Let and considerfor a given communication graph the continuous-timesystem on

(7)

where the functions are uneven, locally Lips-chitz and strictly increasing for all . Similarly asbefore (Example 2), the continuous-time system (7), as wellas time-dependent versions of it, may be studied within thepresent framework by introducing its time-1 map

. This paper allows to study nonlinear consensus protocolswith arbitrary time-varying and nonbidirectional communica-tion topologies.

Example 4 (Swarming): Reference [11] proposes a simplemodel to investigate self-ordered motion in systems of particleswith biologically motivated interaction. The model consists ofa collection of particles moving at constant speed. At each timestep a given particle assumes the average direction of motion ofthe particles in its neighborhood with some random perturbationadded. Denote the direction of motion of particle by .The following model is studied in [11]:

(8)

where we have omitted an additive disturbance term. Here,characterizes the nearest neighbor coupling at time .

The nonlinear swarming model (8) may be analyzed withthe tools of this paper by restricting attention to angles inthe interval and introducing local coordinates

, similarly as inExample 2.

4The reason for this coordinate transformation is simply that, in the presentstudy, we have chosen to assume that the common state–space of the individualagents is an unbounded Euclidean space (and not an open subset of Euclideanspace). Also, the formulation of the stability notions in Appendix A assumesthat the state–space is unbounded Euclidean. For these reasons, we perform apreliminary coordinate transformation from (��=2; �=2) to .

If the headings are close to each other, then (8) may beapproximated by the linear equation

(9)with and where denotes thenumber of neighbors of agent at time . This is the swarmingmodel studied in [14] and corresponds to the linear example ofSection II with all weights equal to 1.

Assumption 1 plays a central role in the forthcoming analysis.Its importance is already reflected in the following simple butappealing result.

Lemma 1: Consider a time sequence of directed graphswith common node set and a con-

tinuous update map satisfying Assumption1. The convex hull of the individual agents’ states, denotedby , does not grow along thesolutions of the discrete-time system (3); that is

(10)

Proof: Assumption 1 implies

(11)

from which (10) follows immediately.Within the current framework, the statement of Lemma 1 may

seem almost trivial; it arises as a straightforward consequenceof Assumption 1. However, its nontrivial contribution lies pre-cisely in the observation that many examples reported in theliterature satisfy Assumption 1 and, hence, may be studied bymeans of Lemma 1. The convex hull of the individual agents’states serves as a measure for disagreement. Lemma 1 statesthat, in terms of this measure, the level of disagreement cannotincrease with time. A consensus is reached when the convexhull reduces to a singleton, as this implies a common state valuefor all agents. In order for this to happen, additional assump-tions need to be imposed. In the forthcoming sections, we willpresent necessary and/or sufficient conditions ensuring that theconvex hull indeed approaches a singleton.

We end this section with some remarks on the limitations ofthe present model.

Remark 2 (Nonstrict Convexity): Assumption 1 is a strictconvexity assumption. For some applications it may be desirableto relax this assumption. For example, the discrete-time system

(12)does not satisfy Assumption 1, but satisfies instead a nonstrictconvexity assumption: belongs to (the boundary of)the convex hull of the state and the states of agents

. The study of dynamical equations sat-isfying a nonstrict convexity assumption is beyond the scope ofthis paper.

Remark 3 (Non-Euclidean Space): By assuming that thecommon state space for the individual agents is Euclideanand by imposing a convexity condition on the dynamics, weexclude some global phenomena arising, for example, in the


study of synchronization of periodic motions [8] and in attitudealignment problems [13]. Indeed, the natural state–space forthe individual agents in these examples is, respectively, and

and the global features of the dynamics that are relatedto the nontrivial topology of these manifolds, fall outside thescope of the present method. Nevertheless, local results maybe obtained within the present framework, for example, byintroducing suitable coordinate charts. This has already beenillustrated in Examples 2 and 4.

IV. CONVERGENCE RESULTS

Before actually studying the convergence properties, we firstpresent the following stability and boundedness result, which isincluded here for reasons of completeness. It is a formalizationof the observation that the level of disagreement can not increasewith time.

Theorem 1: Consider a time sequence of directed graphswith common node set and a

continuous update map satisfying Assump-tion 1. The discrete-time system (3) is uniformly stable anduniformly bounded with respect to the collection of equilibriumsolutions constant.

For a precise definition of the notions of uniform stability anduniform boundedness with respect to a collection of equilibriumsolutions we refer to Appendix A. Theorem 1 is a consequenceof Lemma 1. A formal proof is provided in Section V.

It is important to emphasize that Theorem 1 leaves open thequestion of convergence, which is our primary object of interest.Contrary to what might be expected from Lemma 1 or The-orem 1, the dynamics of (3) may be surprisingly complex. Forexample, the agents’ states may fail to converge to a commonvalue, even in the presence of persistent interagent communica-tion; see Section IV-C. In the remainder of the paper a blend ofgraph-theoretic and system-theoretic tools will be used in orderto establish necessary and/or sufficient conditions for conver-gence (attractivity).

A. Uniform Global Attractivity

We first restrict attention to global attractivity uniform withrespect to initial time. We provide a necessary and sufficientcondition for uniform global attractivity of (3). The conditionthat we present does not involve the actual discrete-time map

; it only involves the sequence of communication graphs.

Theorem 2 (Uniform Global Attractivity): Consider a timesequence of directed graphs with common node set

and a continuous update mapsatisfying Assumption 1. The discrete-time system (3) is

uniformly globally attractive with respect to the collection ofequilibrium solutions constant if andonly if there is such that for all there is a nodeconnected to all other nodes across .

For a precise definition of the notion of uniform global at-tractivity with respect to a collection of equilibrium solutionswe refer to Appendix A. The proof of the if-part of Theorem 2is based on the observation that the convex hull of the individualagents’ states, although not necessarily decreasing at each time

step, is in fact decreasing over time intervals of sufficient length.The proof of the only if-part is based upon the graph theoretic re-sult from Appendix B. A formal proof is provided in Section V.

B. Nonuniform Global Attractivity

The previous subsection has presented a necessary and suffi-cient condition for uniform global attractivity. Not surprisingly,this condition involves a connectivity requirement on the se-quence of directed graphs, uniform with respect to initial time.We now turn attention to global attractivity, not necessarily uni-form with respect to initial time. Unlike the previous subsectionwhere the situation is quite clear, the study of nonuniform at-tractivity turns out to be much more subtle. We start with a nec-essary condition for global attractivity, not necessarily uniformwith respect to initial time.

Proposition 3 (Global Attractivity): Consider a timesequence of directed graphs with commonnode set and a continuous update map

satisfying Assumption 1. Global attractivityof the discrete-time system (3) with respect to the collection ofequilibrium solutions constant impliesfor all the existence of a node connected to all othernodes across .

For a precise definition of the notion of (nonuniform) globalattractivity with respect to a collection of equilibrium solutionswe refer to Appendix A. The proof of Proposition 3 is very sim-ilar to the proof of the only if-part of Theorem 2 and is omitted.The necessary condition featuring in Proposition 3 may be in-terpreted as a nonuniform version of the necessary and suffi-cient condition featuring in Theorem 2. It may therefore seemtempting to conjecture that this necessary condition is also suffi-cient for global attractivity, not necessarily uniform with respectto time. However, the following counterexample shows that thisis not the case.

C. Counterexample

The counterexample is concerned with three agents sharinga common state–space . Among all possible directedgraphs on the vertex set we consider

(13)

We will introduce a sequence of directed graphs which consistsof the concatenation of finite sequences of the form

Proposition 4: Let the number of agents be given byand consider the sequence of communication graphs

with corresponding to the concatenationLet the common state space for the indi-

vidual agents be given by and consider the discrete-timemap corresponding to the linear example of Section II withall the weights equal to one. Let be the solution of (3) with


Fig. 2. Individual agents’ states fail to converge to a common value in spite ofpersistent interagent communication.

initial data and . Then, the threecomponents of do not converge to a common value as

.Proof: In order to show that the three components ofdo not converge to a common value as , we

evaluate the difference at the sequence of time-instantsdetermined by for all

and (Fig. 2). For the ease of notation, let usdenote by . It is not difficult to see that

(14)

(15)

from which it follows, among others, that for all. Clearly, is decreasing with and converges to a

limit as . The total accumulative decrease of satisfies

(16)where we have used the recursive relation (14) and the observa-tion that for all . Since , weconclude that

(17)

We end this subsection with a discussion of the coun-terexample. First of all, notice that the sequence of directedgraphs featuring in the counterexample satisfies the nec-essary condition of Proposition 3. The union of the arcsets over any interval of the form is given by

which actually corresponds to

Fig. 3. Convergence of the individual agents’ states to a common value isobtained by reducing the communication.

a connected, bidirectional graph. This counterexample thusclearly shows that the necessary condition of Proposition 3 isnot sufficient for global attractivity. In other words, persistentinteragent communication does not necessarily guarantee con-vergence of the individual agents’ states to a common value.

The counterexample points toward an interesting phenom-enon, namely that more information exchange does not neces-sarily lead to improved convergence properties, and may even-tually even lead to a loss of convergence. To be more precise,if the arc sets and are replaced by empty sets, thus ef-fectively reducing the communication, then the discrete-timesystem featuring in Proposition 4 becomes globally attractive.This follows from Theorem 3 stated below and is illustrated inFig. 3. By taking and as defined in (13) instead of emptysets, thus effectively increasing the communication, the agents’states fail to converge to a common value. In the present examplethe loss of convergence may be understood as follows. Agent 2serves as an intermediate agent facilitating communication be-tween agents 1 and 3. In other words, agent 2 is responsible forcarrying information from agent 3 to agent 1 and vice versa.However, the negative effect of the unidirectional communica-tion links, say link (1,2) in , is to (partially) replace agent3’s information with agent 1’s information before it is passedto agent 1. Notice that the loss of convergence observed in thecounterexample can only occur in the context of time-dependentand nonbidirectional communication; see Theorems 2 and 3.

D. Bidirectional Communication

The study of global attractivity of (3), not necessarily uniformwith respect to initial time, is simplified considerably when bidi-rectional communication is assumed.

Theorem 3 (Bidirectional): Consider a time sequenceof directed graphs with common node set

and a continuous update mapsatisfying Assumption 1. Assume in addition that thecommunication graphs are bidirectional for all

. The discrete-time system (3) is globally attrac-tive with respect to the collection of equilibrium solutions


constant if and only if for allthere is a node connected to all other nodes across .5

The proof of the if-part of Theorem 3 is based on an anal-ysis of -limit sets. It is, of course, well-known that -limit setsplay a central role in the stability analysis of dynamical systemsthrough the celebrated LaSalle principle, but it is quite remark-able that -limit sets prove to be useful in the present context,since we are considering time-varying difference equations, notnecessarily periodic. A proof of Theorem 3 is provided in Sec-tion V. Reference [41] provides further insight why bidirectionalcommunication simplifies the analysis and enables nonuniformattractivity results.

V. TECHNICAL PROOFS

A. Proof of Theorem 1

We provide a formal proof based upon Theorem 4 of Ap-pendix A. In order to apply Theorem 4 we introduce the set-valued function according to

The set-valued function is derived from featuring inLemma 1 following the procedure of Remark 6 in Appendix Awith , playing the role of , . The set-valued function

is easily seen to be globally Lipschitz. Theorem 1 followsimmediately from Theorem 4 and Lemma 1.

B. Proof of Theorem 2

(Only if, proof by contraposition.) Assume that for everythere is such that no node is connected to

all other nodes across . By Theorem 5 of Ap-pendix B, this implies that for every there isand there are nonempty, disjoint subsets suchthat and are bothempty for all . Pick two different elements

and consider any solution of (3) with initial data

.(18)

Assumption 1 implies that, at time , we still have

(19)

since and are bothempty for all . As the time may be chosenarbitrarily large, it is not difficult to see that this contradicts uni-form global attractivity of (3) with respect to the equilibriumsolutions constant.

(If) Let be such that for all there is a nodeconnected to all other nodes across . Consider an ar-bitrary solution of (3) and an arbitrary time and assume

5Since the graphs are assumed to be bidirectional this means that all nodesare connected to all other nodes across [t ;1).

that the ’s are not all equal. We first show thatis strictly contained in ,

where is the set-valued function introduced in Lemma 1:.

In order to show this, we introduce a number of auxiliaryfunctions. Denote the vertices of the polytope

by . Associate to each vertex theset-valued function identifying the agentslocated at that vertex

(20)

The strict convexity assumption (Assumption 1) implies that anagent which is not located at a vertex at some time cannever reach this vertex in finite time

(21)

We now establish a strict decrease property for the functions. We show that, under the connectivity assumption of the the-

orem, at least of the functions are empty-valued at . Consider any interme-diate time interval

. The assumption of the theorem guarantees theexistence of an agent which is connected to all other agentsacross . We distinguish twocases.

• If this agent is located in one of the vertices at time; that is, if there is such

that , then all the other verticeswhere agents are located at time

, that is, , satisfy thefollowing: forsome and hence, bythe strict convexity assumption (Assumption 1),

is strictly contained in .• If this agent is not located in one of the vertices at time

; that is, if for all, then all vertices where

agents are located at time , that is,, satisfy the following:

for someand hence, by the strict convexity assump-

tion (Assumption 1), is strictlycontained in .

Since the aforementioned argument holds for every time intervaland since

the maximum number of agents at each vertex at time is notgreater than , we may conclude that at least of thefunctions are empty-valued at

. This establishes a strong decrease property for the set-valuedfunction . The polytope is strictlycontained in the polytope and both sets have not morethan one vertex in common.

The remainder of the proof is based upon Theorem 4 of Ap-pendix A. In order to prepare for an application of Theorem 4,


we introduce the set-valued function accordingto

similarly as in the proof of Theorem 1. In addition we also in-troduce a real-valued function according to

(22)where denotes the diameter of a set and where the infimum istaken over all sequences in satisfying

...

(23)

for some . The may be interpreted as the states that arepossibly reachable from in time steps according to Assump-tion 1.

We establish some useful properties of . First, the collectionof all sequences in satisfying (23) for some

• is nonempty and compact for all ;• depends continuously on .

This follows from the observations that the set-valued functionsare continuous and take nonempty, compact

values (implied by Assumption 1) and that there is only a fi-nite number of possible sequences of communication graphs.Second, the expression being minimizedin (22)

• depends continuously on and : indeed, this followsfrom the global Lipschitz property of ;

• is zero whenever the components of are all equal:indeed, in this case the only sequence satisfying (23) is

;• is strictly positive whenever the components of are

not all equal: indeed, this follows from the strict decreaseproperty of that has been established above and the re-lation .

Putting all the elements together, we conclude that the functionis continuous and positive definite with respect to

.The proof is concluded with an application of Theorem 4 (see

Remark 5) which establishes uniform global asymptotic sta-bility of (3).

C. Proof of Theorem 3

(Only if) This follows from Proposition 3.(If) It suffices to prove that every solution of (3) converges

to one of the equilibrium solutionsconstant. Indeed, by continuity and compactness arguments,

convergence of all individual solutions actually implies globalattractivity in the sense of Definition 5 of Appendix A.6

In the remainder of the proof we show that every solution of(3) converges to one of the equilibrium solutions

constant. Consider arbitrary and , andlet denote the solution of (3) with initial data . Let

denote the -limit set of . We start with observing that

(24)

where is the set-valued function introduced in Lemma 1. In-deed, the existence of with would con-tradict the nonincrease property established in Lemma 1. Wedenote the constant value of on by .

Clearly, in order to establish that converges to one of theequilibrium solutions constant, it suf-fices to prove that is a singleton. We prove this by contra-diction. Assume that is not a singleton. In this case,is a polytope with vertices which we denoteby . We associate to each vertex a set-valuedfunction identifying the agents located at thatvertex

(25)Observation (24) implies that

(26)

We arrive at a contradiction with the aid of the following resultwhich holds if the connectivity condition of Theorem 3 is satis-fied and is not a singleton.

Claim 1: For every with for all, there exists with for all

and for some .Indeed, a repetitive application of this result eventually leads

to the existence of with for some, contradicting (26). We have thus shown, by contra-

diction, that is a singleton and thus that converges to oneof the equilibrium solutions constant.

We end the proof of Theorem 3 with a proof of Claim 1. Con-sider an arbitrary with for all .There is, by definition, a sequence of times tendingto infinity such that as . Based upon thesequence of times we construct another sequence of times

tending to infinity, as follows. Let for eachthe time be defined as the first time greater than or equal to

at which communication occurs between an agentand an agent for some

for some

The validity of this construction follows from the connectivitycondition of Theorem 3 that we are assuming to hold. We may

6The observation that compactness may be invoked is not trivial, since the setof equilibrium points is unbounded, but follows from the boundedness propertyof (3) established in Theorem 1.


assume without loss of generality (by considering an appro-priate subsequence if necessary) that

• the converge to a limit point as (byboundedness of the solution );

• the are all equal, say, for all(since there is only a finite number of possible communi-cation graphs);

• the converge to a limit point as (bycontinuity of the set-valued map and compactness of

implied by Assumption 1).It follows that . By construction of the sequence oftimes , we conclude that, on the one hand,

for all and, on the other hand,for some (by As-

sumption 1). This concludes the proof of claim 1 and hencealso of Theorem 3.

VI. CONCLUSION

We have considered in the paper a simple but appealingmodel for agents interacting via time-dependent communi-cation links. The model finds wide application in a variety offields including synchronization, swarming, and distributed de-cision making. In the model, each agent updates his state basedupon the information received from other agents, according to asimple rule. By considering simple dynamics for the individualagents we are able to focus on the role of the communicationtopology without having to deal with the additional complica-tions arising from complex dynamical agents.

The analysis starts with the assumption that each agent up-dates his state according to a strict convex combination of itsneighbors’ states, an assumption which is satisfied in variousexamples studied in the literature. This assumption leads to thedevelopment of a set-valued Lyapunov theory, with the convexhull of the individual agents’ states playing the role of a non-increasing Lyapunov function. Contrary to what might be ex-pected, the dynamics of the multiagent system turns out to bequite subtle and a blend of graph-theoretic and system theoretictools is used in order to establish necessary and/or sufficient con-ditions for convergence.

The strongest result is obtained for the case of bidirectionalcommunication. In that case, it is shown that convergence of theindividual agents’ states to a common value is guaranteed if,during each time interval of the form , each agent sendsinformation to each other agent, either through direct commu-nication or indirectly via intermediate agents.

The case of unidirectional communication is more subtle. Itis shown by means of a counterexample that, contrary to whatmight be expected, convergence of the individual agents’ statesto a common value is not necessarily guaranteed, even if duringeach interval of the form there is an agent who sendsinformation to all other agents, either through direct commu-nication or indirectly via intermediate agents. Convergence isproven, however, if a uniform bound is imposed on the time ittakes for the information to spread over the network.

The counterexample that is used to show that the agents’states may fail to converge to a common value, even in the pres-ence of communication, points toward an interesting phenom-enon. Namely that more information exchange does not neces-

sarily lead to improved convergence and may eventually evenlead to a loss of convergence, even for the simple models studiedin the present paper. The study of the quantitative relationshipbetween communication topology and speed of convergence re-mains an interesting area of research; see [27], [28], and [30] forsome recent results in that direction.

APPENDIX ISTABILITY DEFINITIONS

In order to enable a clear and precise formulation of the sta-bility and convergence properties of the discrete-time system (3)we extend the familiar stability concepts of Lyapunov theoryto the present framework. Notice that we are interested in theagents’ states converging to a common, constant value and thatwe expect this common value to depend continuously on the ini-tial states. In other words, we are dealing with a continuum ofequilibrium points. This means that the classical stability con-cepts developed for the study of individual, typically isolated,equilibria are not well-adapted to the present situation. Alter-natively, one may shift attention away from the individual equi-libria and consider the stability properties of the set of equilibriainstead. The stability of invariant sets has been discussed, forexample, in [42] and [43]; see also [44]. Even though set sta-bility indeed provides a valuable framework, we do not proceedalong those lines because set stability is not able to fully capturethe convergence properties that we are aiming at. For example,even though set stability may be used to assert that the individualagents’ states converge to a common value, this common valueis not guaranteed to be constant in time and may eventually evengrow unbounded; compare with Example 5. This is related tothe well-known phenomenon that a trajectory may converge toa set of equilibria without converging to any of the individualequilibria. The stability notions that we introduce below incor-porate, by definition, the requirement that all trajectories con-verge to one of the equilibria.

In the following definition, we make a conceptual distinctionbetween equilibrium solutions and equilibrium points: an equi-librium point is an element of the state space which is the con-stant value of an equilibrium solution. By referring explicitly toequilibrium solutions in the following definition we distinguishthe present stability concepts from the more familiar set stabilityconcepts [42]–[44].

Definition 5 (Stability): Let be a finite-dimensional Eu-clidean space and consider a continuous mapgiving rise to the discrete-time system

(27)

Consider a collection of equilibrium solutions of (27) and de-note the corresponding set of equilibrium points by . With re-spect to the considered collection of equilibrium solutions, thedynamical system (27) is called

1) stable if for each , for all and for allthere is such that every solution of (27)

satisfies: if then there is suchthat for all ;

2) bounded if for each , for all and for allthere is such that every solution of (27)


satisfies: if then there is suchthat for all ;

3) globally attractive if for each , for alland for all there is such that every solution

of (27) satisfies: if then there issuch that for all ;

4) globally asymptotically stable if it is stable, bounded andglobally attractive.

Remark 4: We may take without loss of generalityin items 1) and 2) of Definition 5 [but not in item 3)]. Comparewith the proof of Theorem 4. We have chosen not to incorporate

in items 1) and 2) in order to make the similarities withitem 3) more transparent.

Definition 5 may be interpreted as follows. Stability andboundedness require that any solution of (27) which is initiallyclose to remains close to one of the equilibria in , thusexcluding, for example, the possibility of drift along the set .Global attractivity implies that every solution of (27) convergesto one of the equilibria in . If the collection of equilibriumsolutions is a singleton consisting of one equilibrium solution,then the notions of stability, boundedness, global attractivityand global asymptotic stability of Definition 5 coincide with theclassical notions that have been introduced for the study of in-dividual equilibria. In general, however, the notions introducedhere are strictly stronger than their respective counterparts fromset stability theory, as is illustrated by the following example.

Example 5: Consider the discrete-time system on

(28)

(29)

with . This system has a continuum of equilibrium pointsand it is easy to see that this invariant

set is globally asymptotically stable. Nevertheless, the solutionof this system starting in at time isgiven by and for all .The -component of this solution diverges to infinity as

. Accordingly, the collection of equilibrium solutions is notglobally asymptotically stable in the sense of Definition 5.

Slightly stronger stability notions result when uniformitywith respect to initial time is introduced in Definition 5. If thenumber (respectively and ) may be chosen indepen-dently of in item 1) [respectively items 2) and 3)] then thedynamical system (27) is called uniformly stable (respectively,uniformly bounded and uniformly globally attractive) withrespect to the considered collection of equilibrium solutions.

Theorem 4 provides a sufficient condition for uniform sta-bility, uniform boundedness and uniform global asymptotic sta-bility in terms of the existence of a set-valued Lyapunov func-tion. This result is convenient since, on the one hand, set-valuedLyapunov functions arise naturally within the present contextas illustrated by Lemma 1, and on the other hand, the stabilitynotions that are asserted in Theorem 4 are precisely those thatwe are aiming to prove. Apart from its application in the presentpaper, Theorem 4 may be of independent interest.

Theorem 4 (Lyapunov Characterization): Let be a finite-dimensional Euclidean space and consider a continuous map

giving rise to the discrete-time system (27). Letbe a collection of equilibrium solutions of (27) and denote the

corresponding set of equilibrium points by . Consider an uppersemicontinuous set-valued function satisfying

1) , ;2) , , .

If for all , then the dynamical system (27)is uniformly stable with respect to . If is bounded for all

, then the dynamical system (27) is uniformly boundedwith respect to .

Consider in addition a function and alower semicontinuous function satisfying

3) is bounded on boundedsubsets of ;

4) is positive definite with respect to ; that is,for all and for all ;

5) 5. , , .If for all and is bounded for all

then the dynamical system (27) is uniformly globallyasymptotically stable with respect to .

We briefly comment on the role of the functions , andin Theorem 4. The set-valued function plays the role of a

Lyapunov function which is nonincreasing (decreasing) alongthe solutions of (27). The set-valued nature of is important:unlike a real-valued function, a set-valued function allows fora continuum of minima which are not comparable with eachother. For this reason, a set-valued Lyapunov function, unlikea real Lyapunov function, may be used to conclude that eachtrajectory converges to one equilibrium out of a continuum ofequilibria. The function serves as a measure for the size ofthe values of . In this paper, we let be the diameterof the set . The function characterizes the decrease ofalong the solutions of (27) as measured in terms of . Of course,the proof of Theorem 4 bears similarities with the classical Lya-punov proofs for real-valued Lyapunov functions.

Proof: (Uniform stability) Consider arbitrary and. If , then, by upper semicontinuity of

, there is such that for all. Consider arbitrary and and

let denote the solution of (27) with . Conditions 1)and 2) of the theorem imply that

(30)

(Uniform boundedness) Consider arbitrary and. If is bounded for all then, by upper semiconti-

nuity of , there is such that for all. Consider arbitrary and

and let denote the solution of (27) with . Condi-tions 1) and 2) of the theorem imply that

(31)

(Uniform global asymptotic stability) We have alreadyproved that, if for all and is boundedfor all , then the dynamical system (27) is uniformlystable and uniformly bounded with respect to . It remains toprove uniform global attractivity with respect to .


Consider arbitrary and . If is boundedfor all then, by upper semicontinuity of , there is acompact set such that for all .Similarly as before, Conditions 1) and 2) of the theorem implythat every solution of (27) initiated in remains in .

Consider in addition arbitrary . If for allthen, by upper semicontinuity of , there is

such that for all there is suchthat . Similarly as before, Conditions 1) and2) of the theorem imply that every solution of (27) entering

remains in a -ball around some equilibrium point.

It remains to prove the existence of such that everysolution of (27) starting in cannot remain longer than

subsequent times in without entering . Inagreement with Conditions 3) and 4) of the theorem and thelower semicontinuity of , we introduce two real numbers

Let be such that . Consider arbitraryand and let denote the solution of (27) with

. Then, Condition 5) of the theorem implies that forsome

since otherwise would be smaller than zero,contradicting that takes only nonnegative values. Putting ev-erything together, we conclude that for some

(32)

Remark 5: The strict decrease condition (Condition 5) ofTheorem 4 may be considerably relaxed. Consider, for example,the following condition which requires that decreases overtime-intervals of length .

6) There is a time such that

(33)

Theorem 4 is still true if Condition 5) is replaced byCondition 6).

Remark 6: The Lyapunov functions featuring in Theorem 4are set-valued functions , or equivalently, single-valued functions . One may be interested ingeneralizations of Theorem 4 considering more abstract Lya-punov functions with not necessarily equal to

. The determination of appropriate structures and proper-ties for the set and the function enabling Lyapunov-type ofresults is beyond the scope of this study. Nevertheless, we pointout that a partial ordering of the set seems to be an essentialingredient in order to be able to conclude that every solutionconverges to one equilibrium out of a continuum of equilibria.It turns out that the class of set-valued functions

is universal in this context, in the following sense. Consider aset and let by a strict partial ordering of .7 Consider a set

and a function . Introduce for every thesubset defined by

or

Then, we have whenever andwhenever . The proof of this

statement is elementary and, therefore, omitted. Not only doesthis result provide some motivation for restricting attention toset-valued maps in Theorem 4, it also pointstoward a constructive method for finding such set-valued Lya-punov functions; see, for example, the proofs of Theorems 1and 2.

APPENDIX IIGRAPH-THEORETIC RESULT

Theorem 5: A directed graph has a nodeconnected to all other nodes if and only if every pairof nonempty, disjoint subsets satisfies

.Independently of the current research, this result is also

proven in [22, Lemma 1].Proof: (Only if) Assume that the node is connected to all

other nodes, and consider an arbitrary pair of nonempty, disjointsubsets . We distinguish three cases.

1) If , then is nonempty.2) If , then is nonempty.3) If then and

are both nonempty.In all three cases .

(If) The proof consists of a constructive algorithm that is guar-anteed to terminate in a finite number of steps. Each step(except the last step) involves the selection of four nonemptysets and satisfying:

• and are disjoint;• each node in is connected to each other node in

and each node in is connected to each othernode in .

Step 1): Set and, where and are two arbitrary, different nodes of

the graph. (Here, we have assumed that the graph has at least2 nodes. If the graph has only one node, then the statement ofTheorem 5 is trivial.)

Step ): As we are proving the if-part of the theorem, wemay assume that

. We restrict attention to the caseand consider . (The

alternative case of can be dealtwith in a completely similar way, be interchanging the role of

and , respectively and .) We distinguish four cases.

1) If and , thenthe algorithm terminates: any agent isconnected to all other agents in the graph.

7A partial ordering� of a set Z is a transitive and antisymmetric relation onZ . If we never have z � z then � is called a strict partial ordering.


2) If and , thenset

(34)

(35)

(36)

where is an arbitrary node which does not belong to.

3) If , then set

(37)

(38)

(39)

(40)

4) If then

(41)

(42)

(43)

(44)

The algorithm is guaranteed to end in a finite number of steps,since at each step (except the last step) either the number ofagents in increases, or the number of agents in

remains unchanged and the number of agents inincreases.

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Luc Moreau was born in Ghent, Belgium, in 1973.He graduated from Ghent University and received theengineering degree in physics and the Ph.D. degreefrom the same university in 1996 and 2001, respec-tively.

For part of 2001, he held a postdoctoral position atthe University of Twente, Twente, The Netherlands,and since 2001, he has been a Postdoctoral Fellowof the Fund for Scientific Research-Flanders and aMember of the SYSTeMS Research Group, GhentUniversity. From October 2001 to September 2002,

he visited the Mechanical and Aerospace Engineering Department, PrincetonUniversity, Princeton, NJ, and from August 2003 to July 2004 he visited the Me-chanical Engineering Department at the University of Eindhoven, Eindhoven,The Netherlands. His research interests include mathematical control theory andgeometric control theory. Since October 2004, he has been with Sidmar, Ghent,Belgium.

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