Consensus in continuous-time multi-agent systems under discontinuous nonlinear protocols
aa r X i v : . [ m a t h . O C ] N ov Consensus in continuous-time multi-agent systems underdiscontinuous nonlinear protocols
Liu Bo, Lu Wenlian and Chen TianpingSchool of Mathematical Sciences, Fudan University, Shanghai, 200433, P.R.China.
Abstract
In this paper, we provide a theoretical analysis for nonlinear discontinuous consensusprotocols in networks of multiagents over weighted directed graphs. By integrating the ana-lytic tools from nonsmooth stability analysis and graph theory, we investigate networks withboth fixed topology and randomly switching topology. For networks with a fixed topology,we provide a sufficient and necessary condition for asymptotic consensus, and the consensusvalue can be explicitly calculated. As to networks with switching topologies, we providea sufficient condition for the network to realize consensus almost surely. Particularly, weconsider the case that the switching sequence is independent and identically distributed.As applications of the theoretical results, we introduce a generalized blinking model andshow that consensus can be realized almost surely under the proposed protocols. Numericalsimulations are also provided to illustrate the theoretical results.
Key words: multiagent systems, consensus, discontinuous, switching, almost sure
Introduction
In many applications involving multiagent systems, groups of agents are required to agreeupon certain quantities of interest. This is the so-called “ consensus problem ”. Due to the broadapplications of multiagent systems, consensus problem arises in various contexts such as theswarming of honeybees, flocking of birds (Olfati-Saber, 2006), formation control of autonomousvehicles (Fax & Murray, 2004), distributed sensor networks (Cort´es & Bullo, 2005) and so on.In the past decades, a considerable research effort has been devoted to this problem. Variousconsensus algorithms have been proposed and studied. For a review, see the survey Olfati-Saber,Fax & Murray (2007), Ren, Beard, & Atkins (2005) and references therein.Most existing consensus protocols are continuous protocols, i.e., the protocol are continuousfunctions of time t and the states of the agents. For example, in Olfati-Saber & Murray (2004),the authors studied the following linear consensus protocols:˙ x ( t ) = X j ∈N i a ij [ x j ( t ) − x i ( t )] , where x i ( t ) is the state of the i -th agent at time t , and N i is the set of neighbors of agent i .In Liu, Chen, & Lu (2009), the authors studied two types of nonlinear protocols over directedgraphs. The first one is as follows:˙ x i ( t ) = n X j =1 a ij φ ij ( x j , x i ) , i = 1 , , · · · , n, (1)where φ ij are nonlinear functions satisfying the following assumption: Assumption 1. φ ij are locally Lipschitz continuous;2. φ ij ( x, y ) = 0 if and only if x = y ;3. ( x − y ) φ ij ( x, y ) < ∀ x = y .They prove that this protocol can realize consensus if and only if the underlying graph has aspanning tree. The second one is as follows:˙ x i ( t ) = − n X j =1 l ij [ h ( x j ) − h ( x i )] , (2)1here h is a strictly increasing nonlinear function, and the Laplacian matrix L = [ l ij ] has theform L L L , where L , L is irreducible, and L = 0. They prove that this protocol can realize consensusvalue which is a convex combination of component states of the initial value.Previous protocols are for static networks, i.e., networks with fixed topologies. Yet many realworld networks are not static. For example, in a network of mobile agents, the topology of thenetwork is dynamical due to limited transmission range and the movement of the agents. Insome cases, the network topology changes gradually. In other cases, it changes abruptly, whichinduces discontinuity in the network topology.An important class of discontinuous dynamical network topology is the so-called switchingtopology . Let 0 = t < t < · · · < t k < t k +1 < · · · be a partition of [0 , + ∞ ), on each time interval[ t k , t k +1 ), the network has a fixed topology, while at each time point t k , the topology switches toanother one randomly or according to some given rule. Linear consensus protocols over networkswith stochastically switching topologies such as independent and identically distributed switching(Salehi & Jadbabaie, 2007), Markovian switching (Matei, Martins, & Baras, 2008), and adaptedstochastic switching (Liu, Lu, & Chen, 2011) have been studied and conditions for almost sureconsensus have been obtained, which indicates that a directed spanning tree in the expectationis sufficient for almost sure consensus.The above mentioned discontinuous consensus protocols are discontinuous in time t andcontinuous in the states of the agents. Besides, there are another important class of discontinuousconsensus protocols which are discontinuous in the states of the agents, too. Recently, suchprotocols have been discussed in several papers. In Cort´es (2006), based on normalized andsigned gradient dynamical systems associated with the Laplacian potential, the author proposedthe following two discontinuous consensus protocols:˙ p i ( t ) = P j ∈N i ( p j ( t ) − p i ( t )) k LP ( t ) k , (3)˙ p i ( t ) = sign (cid:18) X j ∈N i ( p j ( t ) − p i ( t )) (cid:19) , (4)where L is the graph Laplacian of the underlying graph, and P ( t ) = [ p ( t ) , · · · , p n ( t )] ⊤ . Finitetime convergence of both protocols on connected undirected graphs was proved, where the cen-2ralized protocol (3) can realize average consensus, while the distributed algorithm (4) can reachaverage-max-min consensus. In Cort´es (2008), the author further considered the following twodiscontinuous protocols: ˙ p i = sign + (cid:18) n X j =1 a ij ( p j − p i ) (cid:19) , (5)˙ p i = sign − (cid:18) n X j =1 a ij ( p j − p i ) (cid:19) , (6)where sign + ( x ) = 0 if x ≤ + ( x ) = 1 if x >
0, sign − ( x ) = 0 if x ≥ − ( x ) = − x <
0. Both protocols can realize finite time consensus in a strongly connected weighteddirected graph, where protocol (5) can reach max consensus, while protocol (6) can reach minconsensus. In Hui, et al. (2008), the author studied the stability of consensus under the followingdiscontinuous protocol: ˙ x i ( t ) = q X j =1 C ( i,j ) sign( x j − x i ) . Under the assumption that C is symmetric and rank( C ) = q −
1, they proved finite time conver-gence for this protocol.In this paper, we investigate a new type of nonlinear discontinuous protocols, which can beformulated as follows: ˙ x i = − n X j =1 l ij [ g ( x j ) − g ( x i )] , i = 1 , · · · , n, where L = [ l ij ] is the underlying graph Laplacian, and g ( · ) is a discontinuous function thatwill be specified later. First, we consider networks with fixed topology. Compared to existingworks which only consider connected undirected graphs or strongly connected directed graphs,we consider more general directed graphs that has spanning trees. We show that a directedspanning tree is sufficient for the network to realize asymptotic consensus. And this conditionis not only sufficient but also necessary. This is an important improvement since directionalcommunication is important in practical applications and can be easily incorporated, for example,via broadcasting. Moreover, a lot of important real world networks such as the leader-followernetworks are not strongly connected. Then, motivated by the work in synchronization analysisby Lu and Chen (2004), we locate the consensus value based on the left eigenvector correspondingto the zero eigenvalue of the graph Laplacian. Finally, we show that if the consensus value is3 discontinuous point of g , and the underlying graph is strongly connected, then finite timeconvergence can be realized.We also consider the consensus protocol over networks with switching topologies. The timeinterval between each successive switching is assumed to be an independent and identically dis-tributed random variable. And the network topology is also a random sequence. We prove asufficient condition for the network to achieve consensus almost surely in terms of the scram-blingness of the underlying graph. Based on this result, we study the special case where theswitching sequence is independent and identically distributed. We show that if the underlyinggraph has a positive probability to be scrambling, then the protocol can realize consensus almostsurely. Our results indicate that for a network with stochastically switching topology to reachconsensus almost surely, the network is unnecessary connected at each time point. This is moregeneral than the work in Hui, et. al.(2008) on network with switching topology.Finally, as applications of the theoretical results. We study consensus in a general blinkingnetwork model under the proposed consensus protocol. Numerical simulations are also providedto illustrate the theoretical results.This paper is organized as follows. In Section 2, some preliminary definitions and lemmasconcerning graph theory, matrix theory nonsmooth analysis, and probability, are provided. Con-sensus analysis under nonlinear discontinuous protocols with both fixed topology and switchingtopology, are carried out in Section 3. An application of the theoretical results to a general blink-ing network model with numerical simulations are given in Section 4. The paper is concluded inSection 5. In this section, we present some definitions and basic lemmas that will be used later. A weighted directed graph of order n is denoted by a triple {V , E , W } where V = { v , · · · , v n } isthe vertex set and E ⊆ V × V is the edge set, i.e., e ij = ( v i , v j ) ∈ E if there is an edge from v i to v j , and W = [ w ij ], i, j = 1 , · · · , n , is the weight matrix which is a nonnegative matrix such that4or i, j ∈ { , · · · , n } , w ij > i = j and e ji ∈ E . For a weighted directed graph G oforder n , the graph Laplacian L ( G ) = [ l ij ] ni,j =1 can be defined from the weight matrix W in thefollowing way: l ij = − w ij i = j n P j =1 ,j = i w ij j = i. And for a given Laplacian matrix L , the weighted directed graph corresponding to L is writtenas G ( L ).In this paper, we only consider simple graphes, i.e., there are no self links and multipleedges. A directed path of length r from v i to v j is an ordered sequence of r + 1 distinct vertices v k , · · · , v k r +1 with v k = v i and v k r +1 = v j such that ( v i s , v i s +1 ) ∈ E . A ( directed ) spanning tree is a directed graph such that there exists a vertex v r , called the root vertex, such that for anyother vertex v i ∈ V , there exists a directed path from v r to v i . We say a graph G has a spanningtree if a subgraph of G that has the same vertex set with G is a spanning tree. A graph G isstrongly connected if for any pair of vertices, say, v i , v j , there exist directed paths both from v i to v j and from v j to v i .If a graph has spanning trees, then the vertices of the graph can be divided into two disjointsets: S , S , where S contains the vertices that can be the root of some spanning tree, S contains all other vertices. We have the following lemma. Lemma 1.
If a graph G of n vertices has spanning trees, let S , S be defined as above, then(i) The subgraph of G induced by S is strongly connected.(ii) G is strongly connected if and only if S = n . Proof: (i): First, for any given vertices v , v ∈ S , since v can be the root of some spanningtree, then from definition, there is a directed path from v to v . On the other hand, v can alsobe the root of some spanning tree, so there also exists a directed path from v to v . Second,we prove that these two paths contain no vertices outside S . Otherwise, there exists a vertex v S such that v is on one of the paths. Suppose v is on the path from v to v , then thereis a directed path from v to v . Since v is a root, there exist directed paths from v to all othervertices. Thus there are directed paths from v to all other vertices, which implies v also canbe the root of some spanning tree. This contradicts the fact that v S .5ii): If S = n , then the subgraph induced by S is G itself. From (i), G is stronglyconnected. On the other hand, if G is strongly connected, from definition, each vertex can bethe root of some spanning tree. Thus, S = n . (cid:3) Remark 1.
It is known that in a leader-follower system, only the leader can influence the follower,but the follower can not influence the leader. So the final state of the system is determined onlyby the leader. In a strongly connected system, each agent can be seen as a leader. So the finalstate of the system is determined by all agents. Yet there are also many intermediate casesbetween these two extremes. In such cases, there are group of leaders, but the whole system isnot strongly connected. Lemma 1 unifies these three cases into a general framework.From the proof of Lemma 1, we can see that there exist no edges from vertices of S tovertices of S . then after a proper renumbering of its vertices, the graph Laplacian L of G canbe written in the following form: L = L ∗ L , (7)where the square submatrix L corresponds to the vertex set S . Since the subgraph induced by S is strongly connected, L is irreducible. By Perron-Frobenius theory, the left eigenvector of L corresponding to the eigenvalue 0 is positive. Thus we can define the following Definition 1. (Weighted root average) Let L = [ l ij ] ni,j =1 be the graph Laplacian of some weighteddirected graph G ( L ). Suppose that G has spanning trees and L is of the form (7). Let ξ =[ ξ , · · · , ξ n ] ⊤ be the positive left eigenvector of L corresponding to the eigenvalue 0 such that P n i =1 ξ i = 1, where n = S . Given some x = [ x , · · · , x n ] ⊤ ∈ R n , the weighted root average of x with respect to L is defined as: Wra( x, L ) = n X i =1 ξ i x i . Remark 2.
In a leader-follower system, the final state of the system is determined by the leaderonly. In the case that there are group of leaders, the final state of the system is determined bythe leader group. The weighted root average is also a generalization from the case of one leaderto the case of leader group.
Example 1.
Consider the graph in Fig. 1, it is obvious that S = { v , v } , S = { v , v } . If we6 Figure 1: Graph example 1take all the positive weight of the edges to be 1, then the graph Laplacian is L = − − − − − . Here, L = − − , and ξ = [1 / , / ⊤ . Thus, for any x = [ x , x , x , x ] ⊤ , Wra( x, L ) =( x + x ) / Metzler matrix is a matrix that has nonnegative off-diagonal entries. It is clear that − L is a Metzler matrix with zero row sum. Following Liu and Chen (2008), for a Metzler matrix M = [ m ij ], we define a function η ( M ) = max i,j {− ( m ij + m ji ) − X k = i,j min { m ik , m jk }} . and we say that M is scrambling if η ( M ) <
0. It is obvious that scramblingness is not influencedby the diagonal entries of a Metzler matrix, so L is scrambling if and only if W is scrambling.On the other hand, since there is a one to one correspondence between each weighted directedgraph and its weight matrix W (or Laplacian matix L ), we also say a graph is scrambling if W (or − L ) is scrambling. Remark 3.
It can be seen from the definition that if a graph is scrambling, then for each vertexpair ( v i , v j ), either there exists at least one directed edge between v i and v j , or there is another7ertex v k such that there are directed edges from v k to v i and v k to v j . From this, it can be seenthat the graph in Fig. 1 is scrambling. Since there exists directed edges between ( v , v ), ( v , v ),( v , v ), and ( v , v ). And there exist edges from v to v , v , and edges from v to v , v .If we incorporate a positive threshold δ on the graph G , then we get the concept of δ -graph (Moreau, 2004). The δ -graph of G is a graph that has the same vertex set and weight matrixwith G . Yet for each v i , v j , there is a directed edge from v j to v i if and only if w ij ≥ δ . We saya graph G is δ -scrambling if its δ -graph is scrambling. Remark 4.
It is obvious that if G is δ -scrambling, then η ( − L ( G )) ≥ δ . In this subsection, we will provide some concepts and lemmas concerning nonsmooth stabilityanalysis. First, we present some basic concepts and theorems from Filippov theory on differentialequations with discontinuous righthand sides. For more details, the readers are referred toFilippov (1988) directly.Consider the following differential equations:˙ x ( t ) = f ( x ( t )) (8)where x ∈ R n , and f : R n R n is a discontinuous map. Then the Filippov solution of (8) canbe defined as: Definition 2.
An absolutely continuous function ϕ : [ t , t + a ] R n is said to be a Filippovsolution to (8) on [ t , t + a ] if it is a solution of the differential inclusion:˙ x ( t ) ∈ \ δ> \ µ ( N )=0 K [ f ( B ( x, δ ) \ N )] , a.e.t ∈ [ t , t + a ] , (9)where K ( E ) is the closure of the convex hull of E , B ( x, δ ) is the open ball centered at x withradius δ >
0, and µ ( · ) denote the usual Lebesgue measure in R n .For the simplicity of notation, we denote K [ f ]( x ) = T δ> T µ ( N )=0 K [ f ( B ( x, δ ) \ N )], and (9)can be rewritten as: ˙ x ( t ) ∈ K [ f ]( x ( t )) , a.e.t ∈ [ t , t + a ] . (10)8 Filippov solution of (10) is a maximum solution if its domain of existence is maximum, i.e.,it can not be extended any further. A set S ⊆ R n is weakly invariant (resp. strongly invariant)with respect to (10) if for each x ∈ S , S contains a maximum solution (resp. all maximumsolutions) from x of (10).Let f : R n R , then the usual one-sided directional derivative of f at x in direction v isdefined as: f ′ ( x, v ) = lim t → + f ( x + tv ) − f ( x ) t . (11)The generalized directional derivative of f at x in direction v is defined as: f ◦ ( x, v ) = lim sup y → x,t → + f ( y + tv ) − f ( y ) t . (12) Definition 3. (Clarke,1983) Let f : R n R , f is said to be regular at x if for all v ∈ R n , theusual one-sided directional derivative f ′ ( x, v ) exists, and f ′ ( x, v ) = f ◦ ( x, v ).Following lemma can be used to derive regularity. Lemma 2. (Clarke,1983) Let f : R n R be Lipschitz near x , then1. If f is convex, then f is regular at x ;2. A finite linear combination (by nonnegative scalars) of functions regular at x is regular at x .From Rademacher’s Theorem (Clarke,1983), we know that locally Lipschitz functions aredifferentiable almost everywhere. Definition 4. (Clarke,1983) Let V : R n R be a locally Lipschitz continuous function. Let Ω V be the set of points where V fails to be differentiable, then the Clarke generalized gradient of V ( x ) at x is the set ∂V ( x ) , { lim i → + ∞ ∇ V ( x i ) : x i → x, x i Ω V ∪ S} (13)where S can be any set of zero measure. The set-valued Lie derivative of V with respect to(10) at x is: ˜ L f V ( x ) = { a ∈ R : ∃ v ∈ K [ f ]( x ) such that a = ζ · v, ∀ ζ ∈ ∂V ( x ) } . (14)9he following lemma shows that the evolution of the Filippov solutions can be measured bythe Lie derivative. Lemma 3.
Let x : [ t , t ] be a Filippov solution of (8). Let V : R n R be a locally Lipschitzand regular function. Then, t V ( x ( t )) is absolutely continuous, dV ( x ( t )) dt exists a.e. and dV ( x ( t )) dt ∈ ˜ L f V ( x ( t )) for a.e. t .In the following we first define a special class of discontinuous functions which will be usedthroughout this paper. Definition 5. (Function class A ) A function g : R R belongs to A , denoted by g ∈ A , if :1. g is continuous on R except for a set with zero measure, and on each finite interval, thenumber of discontinuous points of g is finite.2. On each interval where g is continuous, g is strictly increasing;3. If x is a discontinuous point of g , let g ( x +0 ) = lim x → x +0 g ( x ), g ( x − ) = lim x → x − g ( x ), then g ( x +0 ) > g ( x − ). Example 2.
Let g ( x ) = x + 1 , x > x, x < , (15)then g ∈ A with x = 0 being the only discontinuous point of g . The graph of g is shown in Fig.2. Definition 6. (shrinking condition) An absolutely continuous function x ( t ) = [ x ( t ) , · · · , x n ( t )] T : R + R n is shrinking if max i { x i ( t ) } is nonincreasing and min i { x i ( t ) } is nondecreasing withrespect to t . Furthermore, x ( t ) is completely shrinking if x ( t ) is shrinking andlim t → + ∞ max i { x i ( t ) } − min i { x i ( t ) } = 0 . Remark 5.
It is obvious that if x ( t ) is shrinking, then the limits of max i { x i ( t ) } and min i { x i ( t ) } exist as t → ∞ . 10 g ( x ) Figure 2: An example of function g ∈ A Definition 7. (Aubin & Frankowska, 1990) Let X , Y be metric spaces, A map F defined on E ⊆ X is called a set-valued map, if to each x ∈ E , there corresponds a set F ( x ) ⊆ Y . A set-valued map F is said to be upper semicontinuous at x ∈ E if for any opening set N containing F ( x ), there exists a neighborhood M of x such that F ( M ) ⊂ N . F is said to have closed(convex, compact) image, if for each x ∈ E , F ( x ) is closed (convex, compact, respectively). Definition 8. (Filippov, 1988) A set valued map F : R n R n is said to satisfy the basicconditions in a domain G ⊆ R n if for any x ∈ G , F ( x ) is non-empty, bounded, closed andconvex, and F is upper semicontinuous in x .As to the existence of Filippov solutions, we have the following Lemma 4. (Filippov,1988) If a set-valued map F ( x ) satisfies the basic conditions in the domain D ⊆ R n , then for any point x ∈ D , there exists a solution in D of the following differentialinclusion: ˙ x ( t ) ∈ F ( x ( t )) , x ( t ) = x (16)over an interval [ t , t ′ ) for some t ′ > t . Moreover, if F satisfies the basic conditions in a closedbounded domain D , then each solution of the differential inclusion (16) lying within D can becontinued either unboundedly as t increases (and decreases), i.e., as t → ∞ , or until it reachesthe boundary of the domain D . Lemma 5. (Filippov,1988) Let a set-valued map F ( x ) be upper semicontinuous on a compactum K and let for each x ∈ K the set F ( x ) be bounded, then F is bounded on K .11 emark 6. It is clear from lemma 5 that if F satisfies the basic conditions on some compactset K , then F is bounded on K . Lemma 6. (Filippov,1988) If M is a bounded closed set and if a function f is continuous,then the set f ( M ) = { f ( x ) : x ∈ M } is closed. If M is convex, f ( x ) = Ax + b , then the set f ( M ) = AM + b is convex. Remark 7.
It can be seen from lemma 6 that if a set-valued map F ( x ) satisfies the basiccondition, then for any n × n matrix T , the set-valued map T F ( x ) = { T y : y ∈ F ( x ) } alsosatisfies the basic condition.The following lemma is a generalization of LaSalle invariance principle for discontinuousdifferential equations. Lemma 7. (Cort´es, 2006) Let V : R n R be a locally Lipschitz and regular function, let x ∈ S ⊂ R n where S is compact and strongly invariant with respect to (8). Assume that eithermax ˜ L f V ( x ) ≤ L f V ( x ) = ∅ for all x ∈ S . Let Z f,V = { x ∈ R n | ∈ ˜ L f V ( x ) } . Then, anysolution x ( t ) starting from x converges to the largest invariant set M contained in Z f,V ∩ S . Let P denote the probability, and E be the mathematical expectation. The following are thesecond Borel-Cantelli Lemma concerning an independent sequence. Lemma 8. (Durrett, 2005) If the events { A n } are independent, then P P { A n } = ∞ implies P { A n i.o. } = 1, where i.o. means infinitely often. In this section, we will discuss consensus in a network under nonlinear discontinuous protocolswith both fixed topology and switching topologies.12 .1 Consensus in networks with fixed topology.
Consider the following consensus protocol in a network of multiagents with fixed graph topologies:˙ x i = − n X j =1 l ij g ( x j ) , (17)where g ∈ A and L = [ l ij ] is the graph Laplacian.Denote Φ( x ) = [Φ ( x ) , · · · , Φ n ( x )] ⊤ with Φ i ( x ) = − n P j =1 l ij g ( x j ), then we can define a set-valued map K [Φ i ]( x ) = − P nj =1 l ij γ j , with γ j ∈ K [ g ]( x j ), where K [ g ]( z ) = g ( z ) if g is continuousat z , and K [ g ]( z ) = [ g ( z − ) , g ( z + )] otherwise. Since for any x = [ x , x , · · · , x n ] ⊤ ∈ R n , the set { γ = [ γ , γ , · · · , γ n ] ⊤ : γ i ∈ K [ g ]( x i ) , i = 1 , , · · · , n. } is closed and convex, from Lemma 6, K [Φ]( x ) is a closed convex set. The Filippov solution x ( t ) to (17) is defined as the followingdifferential inclusion: ˙ x i ( t ) ∈ K [Φ i ]( x ( t )) , a.e. t. (18)First, we have the following lemma which says that all the Filippov solutions of (17) isshrinking. Lemma 9.
For any initial value x ∈ R n , the Filippov solution exists and is shrinking, thus, allthe solutions can be extended to [0 , + ∞ ). Proof:
It is clear that the set-valued map K [Φ]( x ) = P nj =1 l ij K [ g ]( x j ) satisfies the basic condi-tions on any bounded region of R n , which implies that for any initial value x ∈ R n , the Filippovsolution exists on the interval [0 , t ) for some t > V ∗ ( x ) = max i { x i } , V ∗ ( x ) = min i { x i } . It is easy to see that V ∗ ( x ) is locally Lipschitzand convex. In fact, for x = [ x , · · · , x n ] ⊤ , y = [ y , · · · , y n ] ⊤ , and λ ∈ [0 , | V ∗ ( x ) − V ∗ ( y ) | = | max i { x i } − max i { y i }| ≤ max i | x i − y i | and V ∗ ( λx + (1 − λ ) y ) = max i { λx i + (1 − λ ) y i }≤ λ max i { x i } + (1 − λ ) max i { y i } = λV ∗ ( x ) + (1 − λ ) V ∗ ( y ) , V ∗ is regular and dV ∗ ( x ( t )) dt ∈ ˜ L Φ V ∗ ( x ) , a.e. t. where ˜ L Φ V ∗ is the set-valued Lie derivative of V ∗ with respect to Φ.We will prove that V ∗ ( x ( t )) is nonincreasing and V ∗ ( x ( t )) is nondecreasing. Here, we onlyshow that V ∗ ( x ( t )) is nonincreasing, and a similar argument can apply to V ∗ ( x ( t )).Now, we will prove that for each x , either ˜ L Φ V ∗ ( x ) = ∅ or max { ˜ L Φ V ∗ ( x ) } ≤
0. Given x = [ x , · · · , x n ] ⊤ ∈ R n , let I x = { i ∈ { , · · · , n } : x i = max j { x j }} . We have ∂V ∗ ( x ) = co { e i :i ∈ I x } . If a ∈ ˜ L Φ V ∗ ( x ), then there exists some v = [ v , · · · , v n ] ⊤ ∈ K [Φ]( x ) such that a = v · ζ for each ζ ∈ ∂V ∗ ( x ). Therefore, v i = a for i ∈ I x .Noting v i = − n X j =1 l ij γ j = − n X j =1 ,j = i l ij ( γ j − γ i ) , for some γ j ∈ K [ g ]( x j ), if g is continuous at x i , then γ j = g ( x j ) = g ( x i ) = γ i for j ∈ I x , and γ j < γ i for j I x . So in this case we have v i ≤
0. Otherwise, g is discontinuous at x i . If a > i ∈ I x , v i = a >
0. Let i ∈ I x be one index satisfying γ i = max { γ i : i ∈ I x } . Thenwe obviously have v i ≤
0, which is a contradiction. So in this case we also have a ≤ dV ∗ ( x ( t )) dt ≤ , a.e. t. Thus V ∗ ( x ( t )) is nonincreasing. A similar argument can show that V ∗ ( x ( t )) is nondecreasing. So x ( t ) is shrinking. The second claim then directly follows from Lemma 4. (cid:3) Based on lemma 9, we can prove following theorem concerning the consensus of system (17).
Theorem 1.
The system (17) will achieve consensus for any initial value if and only if the graphof L has spanning trees. And the consensus value is Wra( x (0) , L ). Furthermore, if the graph of L is strongly connected, and g is discontinuous at Wra( x (0) , L ), then finite time convergence canbe achieved. Proof:
See Appendix A.It can be seen that Theorem 1 is quite similar to the result obtained in literature for continuousconsensus protocols. So the protocol (17) can be seen as natural extensions of the continuous14rotocols. Intuitively, if a networks has spanning trees, then the information from the roots canbe sent to all other nodes in the network. And the roots can exchange information with eachother. So the network can finally reach a consensus. If a network has no spanning trees, from theproof of Theorem 1, there are two possible cases. Case I: there exists an isolated subgraph thathas no connection with other parts of the network. In this case the isolated subgraph can notexchange information with other parts of the network, and consensus can not be reached. CaseII: there are no isolated subgraphs. In this case, the network has a subgraph that has spanningtrees. There are edges from nodes outside this subgraph to nodes of this subgraph which arenot roots. Fig. 5 provides an example. In this case, the roots in the subgraph can not exchangeinformation with nodes outside the subgraph, since they can neither send their information tothe nodes outside the subgraph, nor receive information from nodes outside the subgraph. Asa result, consensus also can not be reached. In the following, we will provide some examples toillustrate the theoretical results.
Example 3.
The graph shown in Fig. 3 may be called a “double-star” graph . It has spanningtrees, with { v , v } being the set of roots. Yet this graph is not strongly connected. If we takethe weight of each edge to be 1, then the graph Laplacian is L = L L I , with L = − − , L = − − − − − − − − − − ⊤ , and I being the 10 ×
10 identity matrix. For any x = [ x , x , · · · , x ] ⊤ ∈ R , Wra( x, L ) = ( x + x ) / g is given in Example 2, and the initial value x (0) is randomly chosen. The position of Wra( x (0) , L ) = ( x (0) + x (0)) / x (0) , L ). Example 4.
Fig. 5 provides an example of a graph that has no spanning trees. This graph hasno isolated subgraphs. The subgraph induced by { v , v , v , v } has spanning trees, with { v , v } being the root set. And there are edges from { v , v } to { v , v } . So this graph belongs to thesecond case discussed above. And it can not reach a consensus for arbitrary initial value. For15
12 11 1098765 4 3 2
Figure 3: A “double star” network x i ( t ) Figure 4: Consensus in a “double-star” networkeach edge, we take the weight as 1. Then the graph Laplacian is L = − − − − − −
10 0 0 0 1 −
10 0 0 0 − The simulation results are presented in Fig. 6, with g being given in Example 2. It can be seenthat no consensus is realized. 16
65 43 2
Figure 5: An example of a graph that has no spanning trees x i ( t ) Figure 6: No consensus can be reached in the graph of Fig. 5.17 .2 Consensus in networks with randomly switching topologies
In this section, we will investigate consensus in networks of multiagents under nonlinear protocolsover graphes with randomly switching topologies.Consider the following dynamical system:˙ x i ( t ) = − n X j =1 l kij g ( x j ) t ∈ [ t k , t k +1 ) , (19)where g ∈ A and L k = [ l kij ] is the graph Laplacian for the underlying graph on the time interval[ t k , t k +1 ). At each time point t k there is a switching of the network topology. We consider thecase that L k is a random sequence. Denote ∆ t k = t k +1 − t k , in this following, we make Assumption 2. { ∆ t j } is independent and identically distributed;2. the sequence { ∆ t i } and { L k } are independent;3. { L k } is uniformly bounded. Assumption 3.
There exists ε > α, β ∈ R with α = β and g is continuousat α , β , it satisfies that g ( α ) − g ( β ) α − β ≥ ε. Remark 8.
It is easy to verify that under Assumption 3, for any α, β ∈ R with α = β and v ∈ K [ g ]( α ), v ∈ K [ g ]( β ), it satisfies that v − v α − β ≥ ǫ. First, we will prove the following Theorem for almost sure consensus.
Theorem 2.
Under Assumption 2, 3, the system (19) will achieve consensus almost surely ifthere exists δ > P {G ( L k ) is δ -scrambling for infinitely k } = 1 . Proof:
See Appendix B.From Theorem 2, we can have the following corollary concerning switching sequence { L k } which is independent and identically distributed.18 orollary 1. Under Assumption 2,3, if { L k } is independent and identically distributed, thenthe system (19) will achieve consensus almost surely if E η ( − L k ) > Proof:
Denote δ = E η ( − L k ) >
0, and M = sup η ( − L k ) < ∞ . Then δ = E η ( − L k ) ≤ δ P { η ( − L k ) ≤ δ } + M P { η ( − L k ) > δ } ≤ δ M P { η ( − L k ) > δ } , which implies P { η ( − L k > δ/ } ≥ δ M .
From the second Borel-Cantelli Lemma (Lemma 8), we have P { η ( − L k ) > δ } = 1 . The conclusion follows from Theorem 2.
In this section, we will show how the theoretical results can be applied to analyze real worldnetwork models. For this purpose, we consider a generalized blinking network model.The original blinking model was proposed in Belykh, Belykh, & Hasler (2004). It is a kind ofsmall world networks that consists of a regular lattice of cells with constant 2 K nearest neighborcouplings and time dependent on-off couplings between any other pair of cells. In each timeinterval of duration τ each time dependent coupling is switched on with a probability p , and thecorresponding switching random variables are independent for different links and for differenttimes. It is a good model for many real-world dynamical networks such as computers networkedover the Internet interact by sending packets of information, and neurons in our brain interactby sending short pulses called spikes, etc.On the other hand, this model is still quite restrictive in several aspects. First, this modelis an undirected model. Second, the duration between any two successive switchings may notbe identical, nor may it be small sometimes. And it may even be not deterministic, but just arandom variable. Finally, the basic regular 2 K nearest neighbor coupling lattice may not exist,or we can say K = 0 in such case. 19
10 20 30 40 50−2.5−2−1.5−1−0.500.511.522.5 t x i ( t ) Figure 7: Consensus in a generalized blinking model.Based on the above analysis, we make the following generalizations on the original blinkingmodel. First, we assume the model to be a directed graph. For every two vertices v i , v j that haverandom switching links between them, the switching of the edge from v i to v j is independent ofthat from v j to v i . Second, we assume the duration between every two successive switchings isa random variable, and each duration is independent of others. Finally, we assume that K maybe zero in the basic 2 K nearest neighbor lattice. That is, no links exist with probability 1.It is obvious that in this generalized model, the sequence of the durations are independentand identically distributed. And the underlying graph sequence {G k } is also independent andidentically distributed. For each G k , since different links are switched on independently, it isobvious that there is a positive probability that G k is a complete graph. Since a complete graphis scrambling, if we set the weight of each link to be δ >
0, then G k is δ -scrambling for some δ > K = 0, p = 0 .
1, and the weight ofeach link to be 0 .
1. The duration between every two successive switching is a random variableuniformly distributed on (0 , g be as given in Example 2. The initial value is chosenrandomly. The simulation results are presented in Fig. 7. It can be seen that consensus can bereached almost surely. 20 Conclusion
In this paper, we investigate consensus in networks of multiagents under nonlinear discontin-uous protocols. First, we consider networks with fixed topology described by weighted directedgraphs. Compared to existing results concerning discontinuous consensus protocols, we do not re-quire the underlying graph to be strongly connected. Instead, we prove that a directed spanningtree is sufficient and necessary to realize consensus. And we can also locate the consensus value.This result can be seen as an extension of continuous protocols if we take continuous protocols asspecial case of discontinuous ones. Under this viewpoint, we establish a more generalized theo-retical framework for consensus analysis. Second, we consider networks with randomly switchingtopologies. We provide sufficient conditions for the network to achieve consensus almost surelybased on the scramblingness of the underlying graphs. Particularly, we consider the case whenthe switching sequence is independent and identically distributed. Compared to existing resultson discontinuous protocols, we do not require the network to be connected at each time point.Finally, as application of the theoretical results, we study a generalized blinking model and showthat consensus can be realized almost surely under the proposed discontinuous protocols.
References
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IEEE Transactions on Automatic Control , 49, 1465-1476.Filippov, A. (1988). Differential equations with discontinuous righthand sides, Kluwer.Hui, Q., Haddad, W., & Bhat, S. (2008). Semistability theory for differential inclusions withapplications to consensus problems in dynamical networks with switching topology, , 3981-3986.Liu, B., & Chen, T. (2008). Consensus in networks of multiagents with cooperation and com-petetion via stochastically switching topoloiges,
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A Proof of Theorem 1
Sufficiency:
Let V = V ∗ − V ∗ , where V ∗ and V ∗ are defined as in lemma 9. Then V is locallyLipschitz and regular.Given any initial value x (0) ∈ R n , denote x = max i { x i (0) } , x = min i { x i (0) } , and S = { x = [ x , · · · , x n ] ⊤ ∈ R n : x ≤ x i ≤ x } . By lemma 9, S is strongly invariant. Let Z Φ ,V = { x ∈ R n : 0 ∈ ˜ L Φ V } and M be the largest weakly invariant set contained in Z Φ ,V ∩ S . By LasalleInvariance Principle (see Lemma 7), we haveΩ( x ( t )) ⊆ M, where Ω( x ( t )) is the positive limit set of x ( t ).Let C = { x = [ x · · · , x n ] ⊤ ∈ R n , x = · · · = x n } be the consensus manifold, we claimthat ( M ⊆ C ∩ S .) Z Φ ,V ⊂ C . Otherwise, there exists x = [ x , · · · , x n ] ⊤ ∈ M such thatmax i { x i } > min i { x i } and 0 = ˙ x i ∈ n X j =1 l ij K [ g ]( x j ) , which means that for some v = [ v , · · · , v n ] ⊤ with v i ∈ K [ g ]( x i ) and max i { v i } > min i { v i } , and X j ∈ N i l ij ( v j − v i ) = 0 , Let I v = { i : v i = max j { v j }} , and I v = { i : v i = min j { v j }} . First, from the monotonicity of g , we have that I v ⊆ I x and I v ⊆ I x . For i ∈ I v , we have0 = X j ∈ N i l ij ( v j − v i ) . N i ⊆ I v for all i ∈ I v . By induction arguments it can be seen that the root setof the spanning trees is contained in I v . A similar argument reveals that the root set of spanningtrees are contained in I v . But from the assumption that max i { x i } > min i { x i } , we can obviouslyhave that I v ∩ I v = ∅ , which is a contradiction.Based on previous derivation, we proved that Ω( x ( t )) ⊆ M ⊆ C . Next, we will showthat Ω( x ( t )) only contains one point. Otherwise, there exist u = [ a, · · · , a ] ⊤ ∈ Ω( x ( t )), v =[ b, · · · , b ] ⊤ ∈ Ω( x ( t )), a = b . Assume a > b . Then there exists a sequence t n → + ∞ as n → + ∞ such that lim n → + ∞ min i { x i ( t n ) } = a . By the fact that min i { x i ( t ) } is nondecreasing, we havelim t → + ∞ min i { x i ( t ) } = a , which implies b > a . A contradiction.Summing up, we have proved that lim t → + ∞ x ( t ) = x ∞ for some x ∞ ∈ C ∩ S . This completesthe proof of the sufficiency. Necessity:
Let G ( L ) = {V , E } be the graph of L , if G ( L ) doesn’t have a spanning tree, thenthere is a subgraph G s of G ( L ) that is a maximum spanning tree, i.e., if there exists a subgraph G s ′ of G ( L ) that is a spanning tree and contains G s , then G s = G s ′ . Let V s be the vertex set of G s ,and V c = V\V s . Then, V c = ∅ . Let V sr be the set of roots of G s , and let V s ′ = V s \V sr . Obviously,the following properties hold:1. There are no edges from V s to V c .2. There are no edges from V\V sr to V sr .Here, for two vertex sets, an edge from one to the other means an edge from some vertex in theformer to some vertex in the latter.Then there are two cases to be considered. For simplicity, we denote each vertex by index,and the vertex should be renumbered if necessary.1. V s ′ = ∅ .In this case, after proper renumbering, from the above mentioned two properties, thematrix L has the following form: L = L L , where L , L correspond to V sr , V c ,respectively. Let n be the dimension of L and x = [ a, · · · , a | {z } n , b, · · · , b | {z } n − n ] ⊤ with a = b , thenobviously, x ( t ) ≡ x is a solution which can not achieve any consensus.24. V s ′ = ∅ .In this case, after proper renumbering, from the above mentioned two properties, the matrix L has the following form: L = L ∗ L ∗ L , where L , L , L correspond to V sr , V s ′ , V c , respectively, and “*” can be anything. Let n i be the dimensions of L i for i = 1 , , x = [ a, · · · , a | {z } n , b , · · · , b n , c, · · · , c | {z } n ] T for some a = c and b i ∈ R , then we have ˙ x i ≡ i ∈ V sr ∪ V c . Therefore, for any solution x ( t ) starting from x , it holds that x i ( t ) ≡ a i ∈ V sr b i ∈ V c . no consensus will be achieved.At last, we prove the consensus value is Wra( x (0) , L ). Suppose that G ( L ) has spanning trees,and L is of the following form L = L ∗ L , (20)where L corresponds to the vertex set of all the roots of the spanning trees. In such case, wehave that L = L I r .Let ξ = [ ξ , · · · , ξ I r ] ⊤ be the eigenvector corresponding to the zero eigenvalue of L . Assume P I r i =1 ξ i = 1, and let x ( t ) = ξ ⊤ x I r ( t ), then for almost all t ,˙ x = I r X i =1 ξ i I r X j =1 l ij γ j ( t ) = I r X j =1 ( I r X i =1 ξ i l ij ) γ j ( t ) = 0 , where γ j ( t ) ∈ K [ g ]( x j ( t )). This implies that x ( t ) ≡ x (0). Since lim t → + ∞ x i ( t ) = x ∞ , we havelim t → + ∞ P I r i =1 ξ i x i ( t ) = x ∞ . thus x ∞ = x (0).At last, we prove finite time convergence when g is discontinuous at Wra( x (0) , L ). Denote¯ x = Wra( x (0) , L ), and let ¯ γ ∈ K [ g ](¯ x ). For x = [ x , · · · , x n ] ⊤ , define a function V L ( x ) = n X i =1 ξ i Z x i ¯ x [ g ( s ) − ¯ γ ] ds, where ξ = [ ξ , · · · , ξ n ] ⊤ is the positive left eigenvector corresponding to the zero eigenvalue of L such that P ni =1 ξ i = 1. Then it is obvious that V L ≥ V L ( x ) = 0 if and only if x i = ¯ x for each25 . Furthermore, since g ( x ) is strictly increasing, and ξ i > i = 1 , · · · , n , V L ( x ) is convex, thusregular. Also, V L ( x ) is locally Lipschitz. So from Lemma 3, dV L ( x ( t )) dt exists for a.e. t , and dV L ( x ( t )) dt = ˜ L Φ ( x ( t )) , a.e. t. Since from definition, ∂V L ( x ) = { [ γ , · · · , γ n ] ⊤ : γ i ∈ K [ g ]( x i ) , i = 1 , · · · , n } , if ˜ L Φ ( x ) = ∅ ,then either g is continuous at each x i , or there exists γ i ∈ K [ g ]( x i ), i = 1 , · · · , n such that P nj =1 l ij γ j = 0 for each i satisfying g is discontinuous at x i . Then let γ ( t ) = [ γ ( t ) , · · · , γ n ( t )] ⊤ be such that γ i ( t ) = K [ g ]( x i ( t )), i = 1 , · · · , n , and P nj =1 l ij γ j ( t ) = 0 for each i satisfying g isdiscontinuous at x i ( t ), we have dV L ( x ( t )) dt = − n X i =1 ξ i [ γ i ( t ) − ¯ γ ] n X j =1 l ij γ j ( t )= − n X i =1 ξ i [ γ i ( t ) − ¯ γ ] n X j =1 l ij [ γ j ( t ) − ¯ γ ]= − n X i =1 n X j =1 ξ i l ij [ γ i ( t ) − ¯ γ ][ γ j ( t ) − ¯ γ ]= 12 Γ( t ) ⊤ ( − Ξ L − L ⊤ Ξ)Γ( t ) ≤ λ t ) ⊤ ¯Γ( t ) , a.e. t, where Ξ = diag[ ξ , · · · , ξ n ], λ is the second largest eigenvalue of − Ξ L − L ⊤ Ξ, Γ( t ) = [ γ ( t ) − ¯ γ, · · · , γ n ( t ) − ¯ γ ] ⊤ , and ¯Γ( t ) = [ γ ( t ) − ˜ γ ( t ) , · · · , γ n ( t ) − ˜ γ ( t )] ⊤ with ˜ γ = P ni =1 γ i ( t ) /n . The lastinequality is due to the fact that largest eigenvalue of − Ξ L − L ⊤ Ξ is 0 with the corespondingeigenspace being k [1 , , · · · , ⊤ , k ∈ R . Since the graph of L is strongly connected, L is irre-ducible, so λ <
0. Let i be the index such that x i ( t ) = max i { x i ( t ) } , and i be the index such that x i ( t ) = min i { x i ( t ) } . In the case that x i ( t ) > x i ( t ), we have x i ( t ) < ¯ x < x i ( t ). Otherwise, either x i ( t ) ≤ ¯ x or x i ( t ) ≥ ¯ x . If x i ( t ) ≤ ¯ x , then Wra( x ( t ) , L ) < ¯ x . If x i ( t ) ≥ ¯ x , then Wra( x ( t ) , L ) > ¯ x .These all contradict the fact that Wra( x ( t ) , L ) is constant. Thus γ i ( t ) − γ i ( t ) > g (¯ x + ) − g (¯ x − ) > t ) ⊤ Γ( t ) = n X i =1 [ γ i ( t ) − ˜ γ ( t )] ≥ [ γ i ( t ) − ˜ γ ( t )] + [ γ i ( t ) − ˜ γ ( t )] ≥
12 [ γ i ( t ) − γ i ( t )] > [ g (¯ x + ) − g (¯ x − )] / . dV L ( x ( t )) dt < − λ g (¯ x + ) − g (¯ x − )] , a.e. t for V L >
0. This implies that V L will converge to zero in finite time upper bounded by4 V L ( x (0)) λ [ g (¯ x + ) − g (¯ x − )] . The proof is completed. (cid:3) B Proof of Theorem 2
Let V ∗ , V ∗ and V be defined as in the previous section. Given any initial value x (0) ∈ R n andany switching sequence of time points, denoted by 0 = t < t < t < · · · , we can construct thesolution in the following way. First, with initial value x (0), there exists a Filippov solution x ( t )on some interval [0 , δ ) ⊂ [0 , t ]. By similar arguments used in the proof of Lemma 9, we canprove that x ( t ) is shrinking and can be extended to the whole interval [0 , t ]. Repeating sucharguments, we can show that a solution of (19) can be defined as follows: x ( t ) = x k ( t ) , t ∈ [ t k , t k +1 ] , where x k ( t ) is a Filippov solution successively defined from x k − ( t ) on [ t k , t k +1 ] such that x k ( t k ) = x k − ( t k ). It is obvious that x ( t ) is shrinking and absolutely continuous. Let i ∗ , i ∗ be the indicessatisfying V ∗ ( x ) = x i ∗ , V ∗ ( x ) = x i ∗ , respectively. Similar to the arguments in previous section,on each interval [ t k , t k +1 ], we have dVdt = − n X j =1 l ki ∗ j γ j ( t ) + n X j =1 l ki ∗ j γ j ( t )= − n X j =1 ,j = i ∗ l ki ∗ j [ γ j ( t ) − γ i ∗ ( t )] + n X j =1 ,j = i ∗ l ki ∗ j [ γ j ( t ) − γ i ∗ ( t )]= ( l ki ∗ i ∗ + l ki ∗ i ∗ )[ γ i ∗ ( t ) − γ i ∗ ( t )] + X j =1 ,j = i ∗ ,i ∗ { l ki ∗ j [ γ i ∗ ( t ) − γ j ( t )] + l ki ∗ j [ γ j ( t ) − γ i ∗ ( t )] }≤ ( l ki ∗ i ∗ + l i ∗ i ∗ )[ γ i ∗ ( t ) − γ i ∗ ( t )] + X j =1 ,j = i ∗ ,i ∗ max { l ki ∗ j , l ki ∗ j } [ γ i ∗ ( t ) − γ i ∗ ( t )] ≤ − ǫ ( − l ki ∗ i ∗ − l ki ∗ i ∗ + X j =1 ,j = i ∗ ,i ∗ min {− l ki ∗ j , − l ki ∗ j } ) V ≤ − ǫη ( − L k ) V, (21)where γ j ( t ) ∈ K [ g ]( x j ( t )) for each j . Therefore, we have V ( x ( t k +1 )) ≤ e − ǫη ( − L k )∆ t k V ( x ( t k )) ≤ e − ǫ P ki =0 η ( − L i )∆ t i V ( x (0)) . (22)27hus if P + ∞ k =1 η ( − L k )∆ t k = ∞ , then lim k → + ∞ V ( x ( t k )) = 0 . (23)On the other hand, let S N denote the space of strictly increasing infinite sequence of thenatural numbers, we have P (cid:8) + ∞ X k =1 η ( − L k )∆ t k = ∞ (cid:9) ≥ P (cid:8) η ( − L n k ) ≥ δ, ∞ X k =1 ∆ t n k = ∞ , { n k } ∈ S N (cid:9) = X { n k }∈ S N P { η ( − L n k ) ≥ δ (cid:12)(cid:12) ∞ X k =1 ∆ t n k = ∞} P { ∞ X k =1 ∆ t n k = ∞} (24)= X { n k }∈ S N P { η ( − L n k ) ≥ δ } P { ∞ X k =1 ∆ t n k = ∞} (25)= X { n k }∈ S N P { η ( − L n k ) ≥ δ } (26)= P { η ( − L k ) ≥ δ infinitely often } = 1 . Due to the independence of { ∆ t i } and { L k } from Assumption 2, we can have the equality from(24) to (25). Since { ∆ t k } is independent and identically distributed, the subsequence { ∆ t n k } isalso independent and identically distributed for each { n k } ∈ S N . From the strong law of largenumbers, we have lim N →∞ N N X k =1 ∆ t n k = E ∆ t > P { ∞ X k =1 ∆ t n k = ∞} = 1 . Thus we get the equality from (25) to (26).This implies P { + ∞ X k =1 η ( − L k )∆ t k = ∞} = 1 , and P { lim k → + ∞ V ( x ( t k )) = 0 } = 1 . V ( x ( t )) is nonincreasing with respect to t , we conclude P { lim t → + ∞ V ( x ( t )) = 0 } = 1 , Theorem 2 is proved completely. (cid:3)(cid:3)
Let V ∗ , V ∗ and V be defined as in the previous section. Given any initial value x (0) ∈ R n andany switching sequence of time points, denoted by 0 = t < t < t < · · · , we can construct thesolution in the following way. First, with initial value x (0), there exists a Filippov solution x ( t )on some interval [0 , δ ) ⊂ [0 , t ]. By similar arguments used in the proof of Lemma 9, we canprove that x ( t ) is shrinking and can be extended to the whole interval [0 , t ]. Repeating sucharguments, we can show that a solution of (19) can be defined as follows: x ( t ) = x k ( t ) , t ∈ [ t k , t k +1 ] , where x k ( t ) is a Filippov solution successively defined from x k − ( t ) on [ t k , t k +1 ] such that x k ( t k ) = x k − ( t k ). It is obvious that x ( t ) is shrinking and absolutely continuous. Let i ∗ , i ∗ be the indicessatisfying V ∗ ( x ) = x i ∗ , V ∗ ( x ) = x i ∗ , respectively. Similar to the arguments in previous section,on each interval [ t k , t k +1 ], we have dVdt = − n X j =1 l ki ∗ j γ j ( t ) + n X j =1 l ki ∗ j γ j ( t )= − n X j =1 ,j = i ∗ l ki ∗ j [ γ j ( t ) − γ i ∗ ( t )] + n X j =1 ,j = i ∗ l ki ∗ j [ γ j ( t ) − γ i ∗ ( t )]= ( l ki ∗ i ∗ + l ki ∗ i ∗ )[ γ i ∗ ( t ) − γ i ∗ ( t )] + X j =1 ,j = i ∗ ,i ∗ { l ki ∗ j [ γ i ∗ ( t ) − γ j ( t )] + l ki ∗ j [ γ j ( t ) − γ i ∗ ( t )] }≤ ( l ki ∗ i ∗ + l i ∗ i ∗ )[ γ i ∗ ( t ) − γ i ∗ ( t )] + X j =1 ,j = i ∗ ,i ∗ max { l ki ∗ j , l ki ∗ j } [ γ i ∗ ( t ) − γ i ∗ ( t )] ≤ − ǫ ( − l ki ∗ i ∗ − l ki ∗ i ∗ + X j =1 ,j = i ∗ ,i ∗ min {− l ki ∗ j , − l ki ∗ j } ) V ≤ − ǫη ( − L k ) V, (21)where γ j ( t ) ∈ K [ g ]( x j ( t )) for each j . Therefore, we have V ( x ( t k +1 )) ≤ e − ǫη ( − L k )∆ t k V ( x ( t k )) ≤ e − ǫ P ki =0 η ( − L i )∆ t i V ( x (0)) . (22)27hus if P + ∞ k =1 η ( − L k )∆ t k = ∞ , then lim k → + ∞ V ( x ( t k )) = 0 . (23)On the other hand, let S N denote the space of strictly increasing infinite sequence of thenatural numbers, we have P (cid:8) + ∞ X k =1 η ( − L k )∆ t k = ∞ (cid:9) ≥ P (cid:8) η ( − L n k ) ≥ δ, ∞ X k =1 ∆ t n k = ∞ , { n k } ∈ S N (cid:9) = X { n k }∈ S N P { η ( − L n k ) ≥ δ (cid:12)(cid:12) ∞ X k =1 ∆ t n k = ∞} P { ∞ X k =1 ∆ t n k = ∞} (24)= X { n k }∈ S N P { η ( − L n k ) ≥ δ } P { ∞ X k =1 ∆ t n k = ∞} (25)= X { n k }∈ S N P { η ( − L n k ) ≥ δ } (26)= P { η ( − L k ) ≥ δ infinitely often } = 1 . Due to the independence of { ∆ t i } and { L k } from Assumption 2, we can have the equality from(24) to (25). Since { ∆ t k } is independent and identically distributed, the subsequence { ∆ t n k } isalso independent and identically distributed for each { n k } ∈ S N . From the strong law of largenumbers, we have lim N →∞ N N X k =1 ∆ t n k = E ∆ t > P { ∞ X k =1 ∆ t n k = ∞} = 1 . Thus we get the equality from (25) to (26).This implies P { + ∞ X k =1 η ( − L k )∆ t k = ∞} = 1 , and P { lim k → + ∞ V ( x ( t k )) = 0 } = 1 . V ( x ( t )) is nonincreasing with respect to t , we conclude P { lim t → + ∞ V ( x ( t )) = 0 } = 1 , Theorem 2 is proved completely. (cid:3)(cid:3) r X i v : . [ m a t h . O C ] N ov In many applications involving multiagent systems, groups of agents are required to agreeupon certain quantities of interest. This is the so-called “ consensus problem ”. Due to the broadapplications of multiagent systems, consensus problem arises in various contexts such as theswarming of honeybees, flocking of birds (Olfati-Saber, 2006), formation control of autonomousvehicles (Fax & Murray, 2004), distributed sensor networks (Cort´es & Bullo, 2005) and so on.In the past decades, a considerable research effort has been devoted to this problem. Variousconsensus algorithms have been proposed and studied. For a review, see the survey Olfati-Saber,Fax & Murray (2007), Ren, Beard, & Atkins (2005) and references therein.Most existing consensus protocols are continuous protocols, i.e., the protocol are continuousfunctions of time t and the states of the agents. For example, in Olfati-Saber & Murray (2004),the authors studied the following linear consensus protocols:˙ x ( t ) = X j ∈N i a ij [ x j ( t ) − x i ( t )] , where x i ( t ) is the state of the i -th agent at time t , and N i is the set of neighbors of agent i .In Liu, Chen, & Lu (2009), the authors studied two types of nonlinear protocols over directedgraphs. The first one is as follows:˙ x i ( t ) = n X j =1 a ij φ ij ( x j , x i ) , i = 1 , , · · · , n, (1)where φ ij are nonlinear functions satisfying the following assumption: Assumption 1. φ ij are locally Lipschitz continuous;2. φ ij ( x, y ) = 0 if and only if x = y ;3. ( x − y ) φ ij ( x, y ) < ∀ x = y .They prove that this protocol can realize consensus if and only if the underlying graph has aspanning tree. The second one is as follows:˙ x i ( t ) = − n X j =1 l ij [ h ( x j ) − h ( x i )] , (2)1here h is a strictly increasing nonlinear function, and the Laplacian matrix L = [ l ij ] has theform L L L , where L , L is irreducible, and L = 0. They prove that this protocol can realize consensusvalue which is a convex combination of component states of the initial value.Previous protocols are for static networks, i.e., networks with fixed topologies. Yet many realworld networks are not static. For example, in a network of mobile agents, the topology of thenetwork is dynamical due to limited transmission range and the movement of the agents. Insome cases, the network topology changes gradually. In other cases, it changes abruptly, whichinduces discontinuity in the network topology.An important class of discontinuous dynamical network topology is the so-called switchingtopology . Let 0 = t < t < · · · < t k < t k +1 < · · · be a partition of [0 , + ∞ ), on each time interval[ t k , t k +1 ), the network has a fixed topology, while at each time point t k , the topology switches toanother one randomly or according to some given rule. Linear consensus protocols over networkswith stochastically switching topologies such as independent and identically distributed switching(Salehi & Jadbabaie, 2007), Markovian switching (Matei, Martins, & Baras, 2008), and adaptedstochastic switching (Liu, Lu, & Chen, 2011) have been studied and conditions for almost sureconsensus have been obtained, which indicates that a directed spanning tree in the expectationis sufficient for almost sure consensus.The above mentioned discontinuous consensus protocols are discontinuous in time t andcontinuous in the states of the agents. Besides, there are another important class of discontinuousconsensus protocols which are discontinuous in the states of the agents, too. Recently, suchprotocols have been discussed in several papers. In Cort´es (2006), based on normalized andsigned gradient dynamical systems associated with the Laplacian potential, the author proposedthe following two discontinuous consensus protocols:˙ p i ( t ) = P j ∈N i ( p j ( t ) − p i ( t )) k LP ( t ) k , (3)˙ p i ( t ) = sign (cid:18) X j ∈N i ( p j ( t ) − p i ( t )) (cid:19) , (4)where L is the graph Laplacian of the underlying graph, and P ( t ) = [ p ( t ) , · · · , p n ( t )] ⊤ . Finitetime convergence of both protocols on connected undirected graphs was proved, where the cen-2ralized protocol (3) can realize average consensus, while the distributed algorithm (4) can reachaverage-max-min consensus. In Cort´es (2008), the author further considered the following twodiscontinuous protocols: ˙ p i = sign + (cid:18) n X j =1 a ij ( p j − p i ) (cid:19) , (5)˙ p i = sign − (cid:18) n X j =1 a ij ( p j − p i ) (cid:19) , (6)where sign + ( x ) = 0 if x ≤ + ( x ) = 1 if x >
0, sign − ( x ) = 0 if x ≥ − ( x ) = − x <
0. Both protocols can realize finite time consensus in a strongly connected weighteddirected graph, where protocol (5) can reach max consensus, while protocol (6) can reach minconsensus. In Hui, et al. (2008), the author studied the stability of consensus under the followingdiscontinuous protocol: ˙ x i ( t ) = q X j =1 C ( i,j ) sign( x j − x i ) . Under the assumption that C is symmetric and rank( C ) = q −
1, they proved finite time conver-gence for this protocol.In this paper, we investigate a new type of nonlinear discontinuous protocols, which can beformulated as follows: ˙ x i = − n X j =1 l ij [ g ( x j ) − g ( x i )] , i = 1 , · · · , n, where L = [ l ij ] is the underlying graph Laplacian, and g ( · ) is a discontinuous function thatwill be specified later. First, we consider networks with fixed topology. Compared to existingworks which only consider connected undirected graphs or strongly connected directed graphs,we consider more general directed graphs that has spanning trees. We show that a directedspanning tree is sufficient for the network to realize asymptotic consensus. And this conditionis not only sufficient but also necessary. This is an important improvement since directionalcommunication is important in practical applications and can be easily incorporated, for example,via broadcasting. Moreover, a lot of important real world networks such as the leader-followernetworks are not strongly connected. Then, motivated by the work in synchronization analysisby Lu and Chen (2004), we locate the consensus value based on the left eigenvector correspondingto the zero eigenvalue of the graph Laplacian. Finally, we show that if the consensus value is3 discontinuous point of g , and the underlying graph is strongly connected, then finite timeconvergence can be realized.We also consider the consensus protocol over networks with switching topologies. The timeinterval between each successive switching is assumed to be an independent and identically dis-tributed random variable. And the network topology is also a random sequence. We prove asufficient condition for the network to achieve consensus almost surely in terms of the scram-blingness of the underlying graph. Based on this result, we study the special case where theswitching sequence is independent and identically distributed. We show that if the underlyinggraph has a positive probability to be scrambling, then the protocol can realize consensus almostsurely. Our results indicate that for a network with stochastically switching topology to reachconsensus almost surely, the network is unnecessary connected at each time point. This is moregeneral than the work in Hui, et. al.(2008) on network with switching topology.Finally, as applications of the theoretical results. We study consensus in a general blinkingnetwork model under the proposed consensus protocol. Numerical simulations are also providedto illustrate the theoretical results.This paper is organized as follows. In Section 2, some preliminary definitions and lemmasconcerning graph theory, matrix theory nonsmooth analysis, and probability, are provided. Con-sensus analysis under nonlinear discontinuous protocols with both fixed topology and switchingtopology, are carried out in Section 3. An application of the theoretical results to a general blink-ing network model with numerical simulations are given in Section 4. The paper is concluded inSection 5. In this section, we present some definitions and basic lemmas that will be used later. A weighted directed graph of order n is denoted by a triple {V , E , W } where V = { v , · · · , v n } isthe vertex set and E ⊆ V × V is the edge set, i.e., e ij = ( v i , v j ) ∈ E if there is an edge from v i to v j , and W = [ w ij ], i, j = 1 , · · · , n , is the weight matrix which is a nonnegative matrix such that4or i, j ∈ { , · · · , n } , w ij > i = j and e ji ∈ E . For a weighted directed graph G oforder n , the graph Laplacian L ( G ) = [ l ij ] ni,j =1 can be defined from the weight matrix W in thefollowing way: l ij = − w ij i = j n P j =1 ,j = i w ij j = i. And for a given Laplacian matrix L , the weighted directed graph corresponding to L is writtenas G ( L ).In this paper, we only consider simple graphes, i.e., there are no self links and multipleedges. A directed path of length r from v i to v j is an ordered sequence of r + 1 distinct vertices v k , · · · , v k r +1 with v k = v i and v k r +1 = v j such that ( v i s , v i s +1 ) ∈ E . A ( directed ) spanning tree is a directed graph such that there exists a vertex v r , called the root vertex, such that for anyother vertex v i ∈ V , there exists a directed path from v r to v i . We say a graph G has a spanningtree if a subgraph of G that has the same vertex set with G is a spanning tree. A graph G isstrongly connected if for any pair of vertices, say, v i , v j , there exist directed paths both from v i to v j and from v j to v i .If a graph has spanning trees, then the vertices of the graph can be divided into two disjointsets: S , S , where S contains the vertices that can be the root of some spanning tree, S contains all other vertices. We have the following lemma. Lemma 1.
If a graph G of n vertices has spanning trees, let S , S be defined as above, then(i) The subgraph of G induced by S is strongly connected.(ii) G is strongly connected if and only if S = n . Proof: (i): First, for any given vertices v , v ∈ S , since v can be the root of some spanningtree, then from definition, there is a directed path from v to v . On the other hand, v can alsobe the root of some spanning tree, so there also exists a directed path from v to v . Second,we prove that these two paths contain no vertices outside S . Otherwise, there exists a vertex v S such that v is on one of the paths. Suppose v is on the path from v to v , then thereis a directed path from v to v . Since v is a root, there exist directed paths from v to all othervertices. Thus there are directed paths from v to all other vertices, which implies v also canbe the root of some spanning tree. This contradicts the fact that v S .5ii): If S = n , then the subgraph induced by S is G itself. From (i), G is stronglyconnected. On the other hand, if G is strongly connected, from definition, each vertex can bethe root of some spanning tree. Thus, S = n . (cid:3) Remark 1.
It is known that in a leader-follower system, only the leader can influence the follower,but the follower can not influence the leader. So the final state of the system is determined onlyby the leader. In a strongly connected system, each agent can be seen as a leader. So the finalstate of the system is determined by all agents. Yet there are also many intermediate casesbetween these two extremes. In such cases, there are group of leaders, but the whole system isnot strongly connected. Lemma 1 unifies these three cases into a general framework.From the proof of Lemma 1, we can see that there exist no edges from vertices of S tovertices of S . then after a proper renumbering of its vertices, the graph Laplacian L of G canbe written in the following form: L = L ∗ L , (7)where the square submatrix L corresponds to the vertex set S . Since the subgraph induced by S is strongly connected, L is irreducible. By Perron-Frobenius theory, the left eigenvector of L corresponding to the eigenvalue 0 is positive. Thus we can define the following Definition 1. (Weighted root average) Let L = [ l ij ] ni,j =1 be the graph Laplacian of some weighteddirected graph G ( L ). Suppose that G has spanning trees and L is of the form (7). Let ξ =[ ξ , · · · , ξ n ] ⊤ be the positive left eigenvector of L corresponding to the eigenvalue 0 such that P n i =1 ξ i = 1, where n = S . Given some x = [ x , · · · , x n ] ⊤ ∈ R n , the weighted root average of x with respect to L is defined as: Wra( x, L ) = n X i =1 ξ i x i . Remark 2.
In a leader-follower system, the final state of the system is determined by the leaderonly. In the case that there are group of leaders, the final state of the system is determined bythe leader group. The weighted root average is also a generalization from the case of one leaderto the case of leader group.
Example 1.
Consider the graph in Fig. 1, it is obvious that S = { v , v } , S = { v , v } . If we6 Figure 1: Graph example 1take all the positive weight of the edges to be 1, then the graph Laplacian is L = − − − − − . Here, L = − − , and ξ = [1 / , / ⊤ . Thus, for any x = [ x , x , x , x ] ⊤ , Wra( x, L ) =( x + x ) / Metzler matrix is a matrix that has nonnegative off-diagonal entries. It is clear that − L is a Metzler matrix with zero row sum. Following Liu and Chen (2008), for a Metzler matrix M = [ m ij ], we define a function η ( M ) = max i,j {− ( m ij + m ji ) − X k = i,j min { m ik , m jk }} . and we say that M is scrambling if η ( M ) <
0. It is obvious that scramblingness is not influencedby the diagonal entries of a Metzler matrix, so L is scrambling if and only if W is scrambling.On the other hand, since there is a one to one correspondence between each weighted directedgraph and its weight matrix W (or Laplacian matix L ), we also say a graph is scrambling if W (or − L ) is scrambling. Remark 3.
It can be seen from the definition that if a graph is scrambling, then for each vertexpair ( v i , v j ), either there exists at least one directed edge between v i and v j , or there is another7ertex v k such that there are directed edges from v k to v i and v k to v j . From this, it can be seenthat the graph in Fig. 1 is scrambling. Since there exists directed edges between ( v , v ), ( v , v ),( v , v ), and ( v , v ). And there exist edges from v to v , v , and edges from v to v , v .If we incorporate a positive threshold δ on the graph G , then we get the concept of δ -graph (Moreau, 2004). The δ -graph of G is a graph that has the same vertex set and weight matrixwith G . Yet for each v i , v j , there is a directed edge from v j to v i if and only if w ij ≥ δ . We saya graph G is δ -scrambling if its δ -graph is scrambling. Remark 4.
It is obvious that if G is δ -scrambling, then η ( − L ( G )) ≥ δ . In this subsection, we will provide some concepts and lemmas concerning nonsmooth stabilityanalysis. First, we present some basic concepts and theorems from Filippov theory on differentialequations with discontinuous righthand sides. For more details, the readers are referred toFilippov (1988) directly.Consider the following differential equations:˙ x ( t ) = f ( x ( t )) (8)where x ∈ R n , and f : R n R n is a discontinuous map. Then the Filippov solution of (8) canbe defined as: Definition 2.
An absolutely continuous function ϕ : [ t , t + a ] R n is said to be a Filippovsolution to (8) on [ t , t + a ] if it is a solution of the differential inclusion:˙ x ( t ) ∈ \ δ> \ µ ( N )=0 K [ f ( B ( x, δ ) \ N )] , a.e.t ∈ [ t , t + a ] , (9)where K ( E ) is the closure of the convex hull of E , B ( x, δ ) is the open ball centered at x withradius δ >
0, and µ ( · ) denote the usual Lebesgue measure in R n .For the simplicity of notation, we denote K [ f ]( x ) = T δ> T µ ( N )=0 K [ f ( B ( x, δ ) \ N )], and (9)can be rewritten as: ˙ x ( t ) ∈ K [ f ]( x ( t )) , a.e.t ∈ [ t , t + a ] . (10)8 Filippov solution of (10) is a maximum solution if its domain of existence is maximum, i.e.,it can not be extended any further. A set S ⊆ R n is weakly invariant (resp. strongly invariant)with respect to (10) if for each x ∈ S , S contains a maximum solution (resp. all maximumsolutions) from x of (10).Let f : R n R , then the usual one-sided directional derivative of f at x in direction v isdefined as: f ′ ( x, v ) = lim t → + f ( x + tv ) − f ( x ) t . (11)The generalized directional derivative of f at x in direction v is defined as: f ◦ ( x, v ) = lim sup y → x,t → + f ( y + tv ) − f ( y ) t . (12) Definition 3. (Clarke,1983) Let f : R n R , f is said to be regular at x if for all v ∈ R n , theusual one-sided directional derivative f ′ ( x, v ) exists, and f ′ ( x, v ) = f ◦ ( x, v ).Following lemma can be used to derive regularity. Lemma 2. (Clarke,1983) Let f : R n R be Lipschitz near x , then1. If f is convex, then f is regular at x ;2. A finite linear combination (by nonnegative scalars) of functions regular at x is regular at x .From Rademacher’s Theorem (Clarke,1983), we know that locally Lipschitz functions aredifferentiable almost everywhere. Definition 4. (Clarke,1983) Let V : R n R be a locally Lipschitz continuous function. Let Ω V be the set of points where V fails to be differentiable, then the Clarke generalized gradient of V ( x ) at x is the set ∂V ( x ) , { lim i → + ∞ ∇ V ( x i ) : x i → x, x i Ω V ∪ S} (13)where S can be any set of zero measure. The set-valued Lie derivative of V with respect to(10) at x is: ˜ L f V ( x ) = { a ∈ R : ∃ v ∈ K [ f ]( x ) such that a = ζ · v, ∀ ζ ∈ ∂V ( x ) } . (14)9he following lemma shows that the evolution of the Filippov solutions can be measured bythe Lie derivative. Lemma 3.
Let x : [ t , t ] be a Filippov solution of (8). Let V : R n R be a locally Lipschitzand regular function. Then, t V ( x ( t )) is absolutely continuous, dV ( x ( t )) dt exists a.e. and dV ( x ( t )) dt ∈ ˜ L f V ( x ( t )) for a.e. t .In the following we first define a special class of discontinuous functions which will be usedthroughout this paper. Definition 5. (Function class A ) A function g : R R belongs to A , denoted by g ∈ A , if :1. g is continuous on R except for a set with zero measure, and on each finite interval, thenumber of discontinuous points of g is finite.2. On each interval where g is continuous, g is strictly increasing;3. If x is a discontinuous point of g , let g ( x +0 ) = lim x → x +0 g ( x ), g ( x − ) = lim x → x − g ( x ), then g ( x +0 ) > g ( x − ). Example 2.
Let g ( x ) = x + 1 , x > x, x < , (15)then g ∈ A with x = 0 being the only discontinuous point of g . The graph of g is shown in Fig.2. Definition 6. (shrinking condition) An absolutely continuous function x ( t ) = [ x ( t ) , · · · , x n ( t )] T : R + R n is shrinking if max i { x i ( t ) } is nonincreasing and min i { x i ( t ) } is nondecreasing withrespect to t . Furthermore, x ( t ) is completely shrinking if x ( t ) is shrinking andlim t → + ∞ max i { x i ( t ) } − min i { x i ( t ) } = 0 . Remark 5.
It is obvious that if x ( t ) is shrinking, then the limits of max i { x i ( t ) } and min i { x i ( t ) } exist as t → ∞ . 10 g ( x ) Figure 2: An example of function g ∈ A Definition 7. (Aubin & Frankowska, 1990) Let X , Y be metric spaces, A map F defined on E ⊆ X is called a set-valued map, if to each x ∈ E , there corresponds a set F ( x ) ⊆ Y . A set-valued map F is said to be upper semicontinuous at x ∈ E if for any opening set N containing F ( x ), there exists a neighborhood M of x such that F ( M ) ⊂ N . F is said to have closed(convex, compact) image, if for each x ∈ E , F ( x ) is closed (convex, compact, respectively). Definition 8. (Filippov, 1988) A set valued map F : R n R n is said to satisfy the basicconditions in a domain G ⊆ R n if for any x ∈ G , F ( x ) is non-empty, bounded, closed andconvex, and F is upper semicontinuous in x .As to the existence of Filippov solutions, we have the following Lemma 4. (Filippov,1988) If a set-valued map F ( x ) satisfies the basic conditions in the domain D ⊆ R n , then for any point x ∈ D , there exists a solution in D of the following differentialinclusion: ˙ x ( t ) ∈ F ( x ( t )) , x ( t ) = x (16)over an interval [ t , t ′ ) for some t ′ > t . Moreover, if F satisfies the basic conditions in a closedbounded domain D , then each solution of the differential inclusion (16) lying within D can becontinued either unboundedly as t increases (and decreases), i.e., as t → ∞ , or until it reachesthe boundary of the domain D . Lemma 5. (Filippov,1988) Let a set-valued map F ( x ) be upper semicontinuous on a compactum K and let for each x ∈ K the set F ( x ) be bounded, then F is bounded on K .11 emark 6. It is clear from lemma 5 that if F satisfies the basic conditions on some compactset K , then F is bounded on K . Lemma 6. (Filippov,1988) If M is a bounded closed set and if a function f is continuous,then the set f ( M ) = { f ( x ) : x ∈ M } is closed. If M is convex, f ( x ) = Ax + b , then the set f ( M ) = AM + b is convex. Remark 7.
It can be seen from lemma 6 that if a set-valued map F ( x ) satisfies the basiccondition, then for any n × n matrix T , the set-valued map T F ( x ) = { T y : y ∈ F ( x ) } alsosatisfies the basic condition.The following lemma is a generalization of LaSalle invariance principle for discontinuousdifferential equations. Lemma 7. (Cort´es, 2006) Let V : R n R be a locally Lipschitz and regular function, let x ∈ S ⊂ R n where S is compact and strongly invariant with respect to (8). Assume that eithermax ˜ L f V ( x ) ≤ L f V ( x ) = ∅ for all x ∈ S . Let Z f,V = { x ∈ R n | ∈ ˜ L f V ( x ) } . Then, anysolution x ( t ) starting from x converges to the largest invariant set M contained in Z f,V ∩ S . Let P denote the probability, and E be the mathematical expectation. The following are thesecond Borel-Cantelli Lemma concerning an independent sequence. Lemma 8. (Durrett, 2005) If the events { A n } are independent, then P P { A n } = ∞ implies P { A n i.o. } = 1, where i.o. means infinitely often. In this section, we will discuss consensus in a network under nonlinear discontinuous protocolswith both fixed topology and switching topologies.12 .1 Consensus in networks with fixed topology.
Consider the following consensus protocol in a network of multiagents with fixed graph topologies:˙ x i = − n X j =1 l ij g ( x j ) , (17)where g ∈ A and L = [ l ij ] is the graph Laplacian.Denote Φ( x ) = [Φ ( x ) , · · · , Φ n ( x )] ⊤ with Φ i ( x ) = − n P j =1 l ij g ( x j ), then we can define a set-valued map K [Φ i ]( x ) = − P nj =1 l ij γ j , with γ j ∈ K [ g ]( x j ), where K [ g ]( z ) = g ( z ) if g is continuousat z , and K [ g ]( z ) = [ g ( z − ) , g ( z + )] otherwise. Since for any x = [ x , x , · · · , x n ] ⊤ ∈ R n , the set { γ = [ γ , γ , · · · , γ n ] ⊤ : γ i ∈ K [ g ]( x i ) , i = 1 , , · · · , n. } is closed and convex, from Lemma 6, K [Φ]( x ) is a closed convex set. The Filippov solution x ( t ) to (17) is defined as the followingdifferential inclusion: ˙ x i ( t ) ∈ K [Φ i ]( x ( t )) , a.e. t. (18)First, we have the following lemma which says that all the Filippov solutions of (17) isshrinking. Lemma 9.
For any initial value x ∈ R n , the Filippov solution exists and is shrinking, thus, allthe solutions can be extended to [0 , + ∞ ). Proof:
It is clear that the set-valued map K [Φ]( x ) = P nj =1 l ij K [ g ]( x j ) satisfies the basic condi-tions on any bounded region of R n , which implies that for any initial value x ∈ R n , the Filippovsolution exists on the interval [0 , t ) for some t > V ∗ ( x ) = max i { x i } , V ∗ ( x ) = min i { x i } . It is easy to see that V ∗ ( x ) is locally Lipschitzand convex. In fact, for x = [ x , · · · , x n ] ⊤ , y = [ y , · · · , y n ] ⊤ , and λ ∈ [0 , | V ∗ ( x ) − V ∗ ( y ) | = | max i { x i } − max i { y i }| ≤ max i | x i − y i | and V ∗ ( λx + (1 − λ ) y ) = max i { λx i + (1 − λ ) y i }≤ λ max i { x i } + (1 − λ ) max i { y i } = λV ∗ ( x ) + (1 − λ ) V ∗ ( y ) , V ∗ is regular and dV ∗ ( x ( t )) dt ∈ ˜ L Φ V ∗ ( x ) , a.e. t. where ˜ L Φ V ∗ is the set-valued Lie derivative of V ∗ with respect to Φ.We will prove that V ∗ ( x ( t )) is nonincreasing and V ∗ ( x ( t )) is nondecreasing. Here, we onlyshow that V ∗ ( x ( t )) is nonincreasing, and a similar argument can apply to V ∗ ( x ( t )).Now, we will prove that for each x , either ˜ L Φ V ∗ ( x ) = ∅ or max { ˜ L Φ V ∗ ( x ) } ≤
0. Given x = [ x , · · · , x n ] ⊤ ∈ R n , let I x = { i ∈ { , · · · , n } : x i = max j { x j }} . We have ∂V ∗ ( x ) = co { e i :i ∈ I x } . If a ∈ ˜ L Φ V ∗ ( x ), then there exists some v = [ v , · · · , v n ] ⊤ ∈ K [Φ]( x ) such that a = v · ζ for each ζ ∈ ∂V ∗ ( x ). Therefore, v i = a for i ∈ I x .Noting v i = − n X j =1 l ij γ j = − n X j =1 ,j = i l ij ( γ j − γ i ) , for some γ j ∈ K [ g ]( x j ), if g is continuous at x i , then γ j = g ( x j ) = g ( x i ) = γ i for j ∈ I x , and γ j < γ i for j I x . So in this case we have v i ≤
0. Otherwise, g is discontinuous at x i . If a > i ∈ I x , v i = a >
0. Let i ∈ I x be one index satisfying γ i = max { γ i : i ∈ I x } . Thenwe obviously have v i ≤
0, which is a contradiction. So in this case we also have a ≤ dV ∗ ( x ( t )) dt ≤ , a.e. t. Thus V ∗ ( x ( t )) is nonincreasing. A similar argument can show that V ∗ ( x ( t )) is nondecreasing. So x ( t ) is shrinking. The second claim then directly follows from Lemma 4. (cid:3) Based on lemma 9, we can prove following theorem concerning the consensus of system (17).
Theorem 1.
The system (17) will achieve consensus for any initial value if and only if the graphof L has spanning trees. And the consensus value is Wra( x (0) , L ). Furthermore, if the graph of L is strongly connected, and g is discontinuous at Wra( x (0) , L ), then finite time convergence canbe achieved. Proof:
See Appendix A.It can be seen that Theorem 1 is quite similar to the result obtained in literature for continuousconsensus protocols. So the protocol (17) can be seen as natural extensions of the continuous14rotocols. Intuitively, if a networks has spanning trees, then the information from the roots canbe sent to all other nodes in the network. And the roots can exchange information with eachother. So the network can finally reach a consensus. If a network has no spanning trees, from theproof of Theorem 1, there are two possible cases. Case I: there exists an isolated subgraph thathas no connection with other parts of the network. In this case the isolated subgraph can notexchange information with other parts of the network, and consensus can not be reached. CaseII: there are no isolated subgraphs. In this case, the network has a subgraph that has spanningtrees. There are edges from nodes outside this subgraph to nodes of this subgraph which arenot roots. Fig. 5 provides an example. In this case, the roots in the subgraph can not exchangeinformation with nodes outside the subgraph, since they can neither send their information tothe nodes outside the subgraph, nor receive information from nodes outside the subgraph. Asa result, consensus also can not be reached. In the following, we will provide some examples toillustrate the theoretical results.
Example 3.
The graph shown in Fig. 3 may be called a “double-star” graph . It has spanningtrees, with { v , v } being the set of roots. Yet this graph is not strongly connected. If we takethe weight of each edge to be 1, then the graph Laplacian is L = L L I , with L = − − , L = − − − − − − − − − − ⊤ , and I being the 10 ×
10 identity matrix. For any x = [ x , x , · · · , x ] ⊤ ∈ R , Wra( x, L ) = ( x + x ) / g is given in Example 2, and the initial value x (0) is randomly chosen. The position of Wra( x (0) , L ) = ( x (0) + x (0)) / x (0) , L ). Example 4.
Fig. 5 provides an example of a graph that has no spanning trees. This graph hasno isolated subgraphs. The subgraph induced by { v , v , v , v } has spanning trees, with { v , v } being the root set. And there are edges from { v , v } to { v , v } . So this graph belongs to thesecond case discussed above. And it can not reach a consensus for arbitrary initial value. For15
12 11 1098765 4 3 2
Figure 3: A “double star” network x i ( t ) Figure 4: Consensus in a “double-star” networkeach edge, we take the weight as 1. Then the graph Laplacian is L = − − − − − −
10 0 0 0 1 −
10 0 0 0 − The simulation results are presented in Fig. 6, with g being given in Example 2. It can be seenthat no consensus is realized. 16
65 43 2
Figure 5: An example of a graph that has no spanning trees x i ( t ) Figure 6: No consensus can be reached in the graph of Fig. 5.17 .2 Consensus in networks with randomly switching topologies
In this section, we will investigate consensus in networks of multiagents under nonlinear protocolsover graphes with randomly switching topologies.Consider the following dynamical system:˙ x i ( t ) = − n X j =1 l kij g ( x j ) t ∈ [ t k , t k +1 ) , (19)where g ∈ A and L k = [ l kij ] is the graph Laplacian for the underlying graph on the time interval[ t k , t k +1 ). At each time point t k there is a switching of the network topology. We consider thecase that L k is a random sequence. Denote ∆ t k = t k +1 − t k , in this following, we make Assumption 2. { ∆ t j } is independent and identically distributed;2. the sequence { ∆ t i } and { L k } are independent;3. { L k } is uniformly bounded. Assumption 3.
There exists ε > α, β ∈ R with α = β and g is continuousat α , β , it satisfies that g ( α ) − g ( β ) α − β ≥ ε. Remark 8.
It is easy to verify that under Assumption 3, for any α, β ∈ R with α = β and v ∈ K [ g ]( α ), v ∈ K [ g ]( β ), it satisfies that v − v α − β ≥ ǫ. First, we will prove the following Theorem for almost sure consensus.
Theorem 2.
Under Assumption 2, 3, the system (19) will achieve consensus almost surely ifthere exists δ > P {G ( L k ) is δ -scrambling for infinitely k } = 1 . Proof:
See Appendix B.From Theorem 2, we can have the following corollary concerning switching sequence { L k } which is independent and identically distributed.18 orollary 1. Under Assumption 2,3, if { L k } is independent and identically distributed, thenthe system (19) will achieve consensus almost surely if E η ( − L k ) > Proof:
Denote δ = E η ( − L k ) >
0, and M = sup η ( − L k ) < ∞ . Then δ = E η ( − L k ) ≤ δ P { η ( − L k ) ≤ δ } + M P { η ( − L k ) > δ } ≤ δ M P { η ( − L k ) > δ } , which implies P { η ( − L k > δ/ } ≥ δ M .
From the second Borel-Cantelli Lemma (Lemma 8), we have P { η ( − L k ) > δ } = 1 . The conclusion follows from Theorem 2.
In this section, we will show how the theoretical results can be applied to analyze real worldnetwork models. For this purpose, we consider a generalized blinking network model.The original blinking model was proposed in Belykh, Belykh, & Hasler (2004). It is a kind ofsmall world networks that consists of a regular lattice of cells with constant 2 K nearest neighborcouplings and time dependent on-off couplings between any other pair of cells. In each timeinterval of duration τ each time dependent coupling is switched on with a probability p , and thecorresponding switching random variables are independent for different links and for differenttimes. It is a good model for many real-world dynamical networks such as computers networkedover the Internet interact by sending packets of information, and neurons in our brain interactby sending short pulses called spikes, etc.On the other hand, this model is still quite restrictive in several aspects. First, this modelis an undirected model. Second, the duration between any two successive switchings may notbe identical, nor may it be small sometimes. And it may even be not deterministic, but just arandom variable. Finally, the basic regular 2 K nearest neighbor coupling lattice may not exist,or we can say K = 0 in such case. 19
10 20 30 40 50−2.5−2−1.5−1−0.500.511.522.5 t x i ( t ) Figure 7: Consensus in a generalized blinking model.Based on the above analysis, we make the following generalizations on the original blinkingmodel. First, we assume the model to be a directed graph. For every two vertices v i , v j that haverandom switching links between them, the switching of the edge from v i to v j is independent ofthat from v j to v i . Second, we assume the duration between every two successive switchings isa random variable, and each duration is independent of others. Finally, we assume that K maybe zero in the basic 2 K nearest neighbor lattice. That is, no links exist with probability 1.It is obvious that in this generalized model, the sequence of the durations are independentand identically distributed. And the underlying graph sequence {G k } is also independent andidentically distributed. For each G k , since different links are switched on independently, it isobvious that there is a positive probability that G k is a complete graph. Since a complete graphis scrambling, if we set the weight of each link to be δ >
0, then G k is δ -scrambling for some δ > K = 0, p = 0 .
1, and the weight ofeach link to be 0 .
1. The duration between every two successive switching is a random variableuniformly distributed on (0 , g be as given in Example 2. The initial value is chosenrandomly. The simulation results are presented in Fig. 7. It can be seen that consensus can bereached almost surely. 20 Conclusion
In this paper, we investigate consensus in networks of multiagents under nonlinear discontin-uous protocols. First, we consider networks with fixed topology described by weighted directedgraphs. Compared to existing results concerning discontinuous consensus protocols, we do not re-quire the underlying graph to be strongly connected. Instead, we prove that a directed spanningtree is sufficient and necessary to realize consensus. And we can also locate the consensus value.This result can be seen as an extension of continuous protocols if we take continuous protocols asspecial case of discontinuous ones. Under this viewpoint, we establish a more generalized theo-retical framework for consensus analysis. Second, we consider networks with randomly switchingtopologies. We provide sufficient conditions for the network to achieve consensus almost surelybased on the scramblingness of the underlying graphs. Particularly, we consider the case whenthe switching sequence is independent and identically distributed. Compared to existing resultson discontinuous protocols, we do not require the network to be connected at each time point.Finally, as application of the theoretical results, we study a generalized blinking model and showthat consensus can be realized almost surely under the proposed discontinuous protocols.
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A Proof of Theorem 1
Sufficiency:
Let V = V ∗ − V ∗ , where V ∗ and V ∗ are defined as in lemma 9. Then V is locallyLipschitz and regular.Given any initial value x (0) ∈ R n , denote x = max i { x i (0) } , x = min i { x i (0) } , and S = { x = [ x , · · · , x n ] ⊤ ∈ R n : x ≤ x i ≤ x } . By lemma 9, S is strongly invariant. Let Z Φ ,V = { x ∈ R n : 0 ∈ ˜ L Φ V } and M be the largest weakly invariant set contained in Z Φ ,V ∩ S . By LasalleInvariance Principle (see Lemma 7), we haveΩ( x ( t )) ⊆ M, where Ω( x ( t )) is the positive limit set of x ( t ).Let C = { x = [ x · · · , x n ] ⊤ ∈ R n , x = · · · = x n } be the consensus manifold, we claimthat ( M ⊆ C ∩ S .) Z Φ ,V ⊂ C . Otherwise, there exists x = [ x , · · · , x n ] ⊤ ∈ M such thatmax i { x i } > min i { x i } and 0 = ˙ x i ∈ n X j =1 l ij K [ g ]( x j ) , which means that for some v = [ v , · · · , v n ] ⊤ with v i ∈ K [ g ]( x i ) and max i { v i } > min i { v i } , and X j ∈ N i l ij ( v j − v i ) = 0 , Let I v = { i : v i = max j { v j }} , and I v = { i : v i = min j { v j }} . First, from the monotonicity of g , we have that I v ⊆ I x and I v ⊆ I x . For i ∈ I v , we have0 = X j ∈ N i l ij ( v j − v i ) . N i ⊆ I v for all i ∈ I v . By induction arguments it can be seen that the root setof the spanning trees is contained in I v . A similar argument reveals that the root set of spanningtrees are contained in I v . But from the assumption that max i { x i } > min i { x i } , we can obviouslyhave that I v ∩ I v = ∅ , which is a contradiction.Based on previous derivation, we proved that Ω( x ( t )) ⊆ M ⊆ C . Next, we will showthat Ω( x ( t )) only contains one point. Otherwise, there exist u = [ a, · · · , a ] ⊤ ∈ Ω( x ( t )), v =[ b, · · · , b ] ⊤ ∈ Ω( x ( t )), a = b . Assume a > b . Then there exists a sequence t n → + ∞ as n → + ∞ such that lim n → + ∞ min i { x i ( t n ) } = a . By the fact that min i { x i ( t ) } is nondecreasing, we havelim t → + ∞ min i { x i ( t ) } = a , which implies b > a . A contradiction.Summing up, we have proved that lim t → + ∞ x ( t ) = x ∞ for some x ∞ ∈ C ∩ S . This completesthe proof of the sufficiency. Necessity:
Let G ( L ) = {V , E } be the graph of L , if G ( L ) doesn’t have a spanning tree, thenthere is a subgraph G s of G ( L ) that is a maximum spanning tree, i.e., if there exists a subgraph G s ′ of G ( L ) that is a spanning tree and contains G s , then G s = G s ′ . Let V s be the vertex set of G s ,and V c = V\V s . Then, V c = ∅ . Let V sr be the set of roots of G s , and let V s ′ = V s \V sr . Obviously,the following properties hold:1. There are no edges from V s to V c .2. There are no edges from V\V sr to V sr .Here, for two vertex sets, an edge from one to the other means an edge from some vertex in theformer to some vertex in the latter.Then there are two cases to be considered. For simplicity, we denote each vertex by index,and the vertex should be renumbered if necessary.1. V s ′ = ∅ .In this case, after proper renumbering, from the above mentioned two properties, thematrix L has the following form: L = L L , where L , L correspond to V sr , V c ,respectively. Let n be the dimension of L and x = [ a, · · · , a | {z } n , b, · · · , b | {z } n − n ] ⊤ with a = b , thenobviously, x ( t ) ≡ x is a solution which can not achieve any consensus.24. V s ′ = ∅ .In this case, after proper renumbering, from the above mentioned two properties, the matrix L has the following form: L = L ∗ L ∗ L , where L , L , L correspond to V sr , V s ′ , V c , respectively, and “*” can be anything. Let n i be the dimensions of L i for i = 1 , , x = [ a, · · · , a | {z } n , b , · · · , b n , c, · · · , c | {z } n ] T for some a = c and b i ∈ R , then we have ˙ x i ≡ i ∈ V sr ∪ V c . Therefore, for any solution x ( t ) starting from x , it holds that x i ( t ) ≡ a i ∈ V sr b i ∈ V c . no consensus will be achieved.At last, we prove the consensus value is Wra( x (0) , L ). Suppose that G ( L ) has spanning trees,and L is of the following form L = L ∗ L , (20)where L corresponds to the vertex set of all the roots of the spanning trees. In such case, wehave that L = L I r .Let ξ = [ ξ , · · · , ξ I r ] ⊤ be the eigenvector corresponding to the zero eigenvalue of L . Assume P I r i =1 ξ i = 1, and let x ( t ) = ξ ⊤ x I r ( t ), then for almost all t ,˙ x = I r X i =1 ξ i I r X j =1 l ij γ j ( t ) = I r X j =1 ( I r X i =1 ξ i l ij ) γ j ( t ) = 0 , where γ j ( t ) ∈ K [ g ]( x j ( t )). This implies that x ( t ) ≡ x (0). Since lim t → + ∞ x i ( t ) = x ∞ , we havelim t → + ∞ P I r i =1 ξ i x i ( t ) = x ∞ . thus x ∞ = x (0).At last, we prove finite time convergence when g is discontinuous at Wra( x (0) , L ). Denote¯ x = Wra( x (0) , L ), and let ¯ γ ∈ K [ g ](¯ x ). For x = [ x , · · · , x n ] ⊤ , define a function V L ( x ) = n X i =1 ξ i Z x i ¯ x [ g ( s ) − ¯ γ ] ds, where ξ = [ ξ , · · · , ξ n ] ⊤ is the positive left eigenvector corresponding to the zero eigenvalue of L such that P ni =1 ξ i = 1. Then it is obvious that V L ≥ V L ( x ) = 0 if and only if x i = ¯ x for each25 . Furthermore, since g ( x ) is strictly increasing, and ξ i > i = 1 , · · · , n , V L ( x ) is convex, thusregular. Also, V L ( x ) is locally Lipschitz. So from Lemma 3, dV L ( x ( t )) dt exists for a.e. t , and dV L ( x ( t )) dt = ˜ L Φ ( x ( t )) , a.e. t. Since from definition, ∂V L ( x ) = { [ γ , · · · , γ n ] ⊤ : γ i ∈ K [ g ]( x i ) , i = 1 , · · · , n } , if ˜ L Φ ( x ) = ∅ ,then either g is continuous at each x i , or there exists γ i ∈ K [ g ]( x i ), i = 1 , · · · , n such that P nj =1 l ij γ j = 0 for each i satisfying g is discontinuous at x i . Then let γ ( t ) = [ γ ( t ) , · · · , γ n ( t )] ⊤ be such that γ i ( t ) = K [ g ]( x i ( t )), i = 1 , · · · , n , and P nj =1 l ij γ j ( t ) = 0 for each i satisfying g isdiscontinuous at x i ( t ), we have dV L ( x ( t )) dt = − n X i =1 ξ i [ γ i ( t ) − ¯ γ ] n X j =1 l ij γ j ( t )= − n X i =1 ξ i [ γ i ( t ) − ¯ γ ] n X j =1 l ij [ γ j ( t ) − ¯ γ ]= − n X i =1 n X j =1 ξ i l ij [ γ i ( t ) − ¯ γ ][ γ j ( t ) − ¯ γ ]= 12 Γ( t ) ⊤ ( − Ξ L − L ⊤ Ξ)Γ( t ) ≤ λ t ) ⊤ ¯Γ( t ) , a.e. t, where Ξ = diag[ ξ , · · · , ξ n ], λ is the second largest eigenvalue of − Ξ L − L ⊤ Ξ, Γ( t ) = [ γ ( t ) − ¯ γ, · · · , γ n ( t ) − ¯ γ ] ⊤ , and ¯Γ( t ) = [ γ ( t ) − ˜ γ ( t ) , · · · , γ n ( t ) − ˜ γ ( t )] ⊤ with ˜ γ = P ni =1 γ i ( t ) /n . The lastinequality is due to the fact that largest eigenvalue of − Ξ L − L ⊤ Ξ is 0 with the corespondingeigenspace being k [1 , , · · · , ⊤ , k ∈ R . Since the graph of L is strongly connected, L is irre-ducible, so λ <
0. Let i be the index such that x i ( t ) = max i { x i ( t ) } , and i be the index such that x i ( t ) = min i { x i ( t ) } . In the case that x i ( t ) > x i ( t ), we have x i ( t ) < ¯ x < x i ( t ). Otherwise, either x i ( t ) ≤ ¯ x or x i ( t ) ≥ ¯ x . If x i ( t ) ≤ ¯ x , then Wra( x ( t ) , L ) < ¯ x . If x i ( t ) ≥ ¯ x , then Wra( x ( t ) , L ) > ¯ x .These all contradict the fact that Wra( x ( t ) , L ) is constant. Thus γ i ( t ) − γ i ( t ) > g (¯ x + ) − g (¯ x − ) > t ) ⊤ Γ( t ) = n X i =1 [ γ i ( t ) − ˜ γ ( t )] ≥ [ γ i ( t ) − ˜ γ ( t )] + [ γ i ( t ) − ˜ γ ( t )] ≥
12 [ γ i ( t ) − γ i ( t )] > [ g (¯ x + ) − g (¯ x − )] / . dV L ( x ( t )) dt < − λ g (¯ x + ) − g (¯ x − )] , a.e. t for V L >
0. This implies that V L will converge to zero in finite time upper bounded by4 V L ( x (0)) λ [ g (¯ x + ) − g (¯ x − )] . The proof is completed. (cid:3) B Proof of Theorem 2
Let V ∗ , V ∗ and V be defined as in the previous section. Given any initial value x (0) ∈ R n andany switching sequence of time points, denoted by 0 = t < t < t < · · · , we can construct thesolution in the following way. First, with initial value x (0), there exists a Filippov solution x ( t )on some interval [0 , δ ) ⊂ [0 , t ]. By similar arguments used in the proof of Lemma 9, we canprove that x ( t ) is shrinking and can be extended to the whole interval [0 , t ]. Repeating sucharguments, we can show that a solution of (19) can be defined as follows: x ( t ) = x k ( t ) , t ∈ [ t k , t k +1 ] , where x k ( t ) is a Filippov solution successively defined from x k − ( t ) on [ t k , t k +1 ] such that x k ( t k ) = x k − ( t k ). It is obvious that x ( t ) is shrinking and absolutely continuous. Let i ∗ , i ∗ be the indicessatisfying V ∗ ( x ) = x i ∗ , V ∗ ( x ) = x i ∗ , respectively. Similar to the arguments in previous section,on each interval [ t k , t k +1 ], we have dVdt = − n X j =1 l ki ∗ j γ j ( t ) + n X j =1 l ki ∗ j γ j ( t )= − n X j =1 ,j = i ∗ l ki ∗ j [ γ j ( t ) − γ i ∗ ( t )] + n X j =1 ,j = i ∗ l ki ∗ j [ γ j ( t ) − γ i ∗ ( t )]= ( l ki ∗ i ∗ + l ki ∗ i ∗ )[ γ i ∗ ( t ) − γ i ∗ ( t )] + X j =1 ,j = i ∗ ,i ∗ { l ki ∗ j [ γ i ∗ ( t ) − γ j ( t )] + l ki ∗ j [ γ j ( t ) − γ i ∗ ( t )] }≤ ( l ki ∗ i ∗ + l i ∗ i ∗ )[ γ i ∗ ( t ) − γ i ∗ ( t )] + X j =1 ,j = i ∗ ,i ∗ max { l ki ∗ j , l ki ∗ j } [ γ i ∗ ( t ) − γ i ∗ ( t )] ≤ − ǫ ( − l ki ∗ i ∗ − l ki ∗ i ∗ + X j =1 ,j = i ∗ ,i ∗ min {− l ki ∗ j , − l ki ∗ j } ) V ≤ − ǫη ( − L k ) V, (21)where γ j ( t ) ∈ K [ g ]( x j ( t )) for each j . Therefore, we have V ( x ( t k +1 )) ≤ e − ǫη ( − L k )∆ t k V ( x ( t k )) ≤ e − ǫ P ki =0 η ( − L i )∆ t i V ( x (0)) . (22)27hus if P + ∞ k =1 η ( − L k )∆ t k = ∞ , then lim k → + ∞ V ( x ( t k )) = 0 . (23)On the other hand, let S N denote the space of strictly increasing infinite sequence of thenatural numbers, we have P (cid:8) + ∞ X k =1 η ( − L k )∆ t k = ∞ (cid:9) ≥ P (cid:8) η ( − L n k ) ≥ δ, ∞ X k =1 ∆ t n k = ∞ , { n k } ∈ S N (cid:9) = X { n k }∈ S N P { η ( − L n k ) ≥ δ (cid:12)(cid:12) ∞ X k =1 ∆ t n k = ∞} P { ∞ X k =1 ∆ t n k = ∞} (24)= X { n k }∈ S N P { η ( − L n k ) ≥ δ } P { ∞ X k =1 ∆ t n k = ∞} (25)= X { n k }∈ S N P { η ( − L n k ) ≥ δ } (26)= P { η ( − L k ) ≥ δ infinitely often } = 1 . Due to the independence of { ∆ t i } and { L k } from Assumption 2, we can have the equality from(24) to (25). Since { ∆ t k } is independent and identically distributed, the subsequence { ∆ t n k } isalso independent and identically distributed for each { n k } ∈ S N . From the strong law of largenumbers, we have lim N →∞ N N X k =1 ∆ t n k = E ∆ t > P { ∞ X k =1 ∆ t n k = ∞} = 1 . Thus we get the equality from (25) to (26).This implies P { + ∞ X k =1 η ( − L k )∆ t k = ∞} = 1 , and P { lim k → + ∞ V ( x ( t k )) = 0 } = 1 . V ( x ( t )) is nonincreasing with respect to t , we conclude P { lim t → + ∞ V ( x ( t )) = 0 } = 1 , Theorem 2 is proved completely. (cid:3)(cid:3)