Multi-agent Systems Reaching Optimal Consensus with Time-varying Communication Graphs

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1 Preprints of the 8th IFAC Worl Congress Multi-agent Systems Reaching Optimal Consensus with Time-varying Communication Graphs Guoong Shi ACCESS Linnaeus Centre, School of Electrical Engineering, Royal Institute of Technology, Stockholm 0044, Sween ( Abstract: In this paper, an optimal consensus problem for continuous-time multi-agent systems is formulate an solve with time-varying interconnection topologies Base on a sharp connectivity assumption, the consiere multi-agent network with simple nonlinear istribute control rules achieves not only a consensus, but also an optimal one by agreeing within the global optimal solution set of a sum of objective functions corresponing to multiple agents Convergence analysis is presente, by establishing several key properties of a class of istance functions an invariant sets with the help of convex analysis an non-smooth analysis Keywors: Multi-agent systems, Optimal consensus, Set convergence, Distribute optimization INTRODUCTION In recent years, multi-agent ynamics has been intensively investigate in various areas incluing engineering, natural science, an social science Cooperative control of multi-agent systems is an active research topic, an rapi evelopments of istribute control protocols via interconnecte communication have been mae to achieve the collective tasks, eg, Tsitsiklis et al [986], Jababaie et al [003], Olfati-Saber an Murray [004], Moreau [005], Martinez et al [007], Ren an Bear [008], Shi an Hong [009] However, funamental challenges still lie in fining suitable tools to escribe an esign the ynamical behavior of these systems an thus proviing insights in their functioning principles Different from the classical control esign, the multi-agent stuies aim at fully exploiting, rather than avoiing, the interconnection between agents in analysis an synthesis in orer to eal with istribute esign an large-scale information processing Consensus is an important basic problem of multi-agent coorination, an it is usually require that all the agents achieve the same state, eg, a certain relative position an the same velocity In the stuy of whether collective behavior can be achieve, connectivity of the communication topologies plays a key role, an various connectivity conitions have been use to escribe the frequently switching topologies in ifferent cases The joint connection or similar concepts are important in the analysis of stability an convergence to guarantee a suitable convergence Uniform joint-connection, ie, the joint graph is connecte uring all time intervals longer than a constant, has been utilize for ifferent consensus problems (Tsitsiklis et al [986], Jababaie et al [003], Lin et al [007], Cheng et al [008]) On the other han, [t, )-joint connecteness, ie, the joint graph is connecte in time intervals [t, ), is the This work has been supporte in part by the Knut an Alice Wallenberg Founation an the Sweish Research Council most general form to secure the global coorination, which is also prove to be necessary in many situations (Moreau [005], Shi an Hong [009]) Moreover, multi-agent networks optimizing a sum of convex objective functions via sub-graient methos have attracte much attention in recent years, e g, Johansson et al [007, 008], Neić et al [009, 00] In these stuies, agents have to take both consensus an optimization into consieration, an it is usually har for the network to reach both of them unless the weights rule of the links, the step size an the connecteness of the communication graph are properly selecte Moreover, most of the literature on optimization an consensus algorithms are on iscrete-time systems Therefore, an interesting problem is on whether similar optimization ieas can be use for continuous-time multi-agent systems, especially on whether consensus an optimization can be achieve in the mean time with limite information exchange, relaxe weights rule an time-varying communications The goal of this paper is to stuy istribute optimization of continuous-time multi-agent systems by proviing an optimal consensus protocol for switching communication topologies We assume that the optimization objective is a sum of functions with convex optimal solution sets Different from the existing results, we obtain a global consensus an convergence to the optimal solution set of the couple objective function Although each agent only knows the information from the optimal solution set of its own objective function an the system topology is irecte an time-varying, we show that an optimal consensus can be achieve by a simple nonlinear control law base on jointly connecte connectivity assumptions The contributions of the paper inclue: We provie a simple, istribute control protocol using very little information exchange for continuoustime multi-agent networks to reach both consen- Copyright by the International Feeration of Automatic Control (IFAC) 5693

2 Preprints of the 8th IFAC Worl Congress sus an optimum with time-varying communication graphs The connectivity conition for the inter-agent connection uner consieration is base on [t, )-joint connections, which is less restrictive than existing results By a simple example, we show that the propose connectivity assumption is sharp in the sense that optimal consensus cannot be guarantee by weaker connecteness The multi-agent system is in the form of continuoustime nonlinear ynamics, which covers many practical situations with uncertainty an isturbance A new metho base on invariant sets is propose to eal with the optimization problem The paper is organize as follows Section introuces some preliminary knowlege Then in Section 3, we formulate the consiere optimal consensus problem an propose the main result base on several key assumptions In Section 4, we stuy the property of the istance functions, base on which we prove the optimal solution set convergence Next, consensus analysis is carrie out via a metho stuying a class of invariant sets in Section 5 We show a simulation result in Section 6, an finally in Section 7 concluing remarks are given Graph Theory PRELIMINARIES A irecte graph (igraph) G = (V, E) consists of a finite set V of noes an an arc set E (see Gosil et al [00] for etails) Noe j is sai to be a neighbor of i if there is an arc (i, j) E A igraph G is calle to be biirectional when for any two noes i an j, i is a neighbor of j if an only if j is a neighbor of i Ignoring the irection of the arcs, the connecteness of a biirectional igraph will be transforme to that of the corresponing unirecte graph A time-varying graph is efine as G σ(t) = (V, E σ(t) ) with σ : [0, + ) Q as a piecewise constant function, where Q is a finite set inicating all possible graphs Moreover, the joint graph of G σ(t) in time interval [t, t ) with t < t + is enote as G([t, t )) = t [t,t )G(t) = (V, t [t,t )E σ(t) ) Convex Analysis A set K R m is sai to be convex if ( λ)x + λy K whenever x K, y K an 0 λ (more etails can be foun in Rockafellar [97]) For any set S R m, the intersection of all convex sets containing S is calle the convex hull of S, enote by co(s) The next lemma can be foun in Aubin an Cellina [984] Lemma Let K be a subset of R m The convex hull co(k) of K is the set of elements of the form x = λ i x i, where λ i 0, i =,, m + with λ i = an x i K Let K be a close convex subset in R m an enote x K = infy K x y as the istance between x R m an K, where enotes the Eucliean norm There is a unique element P K (x) K satisfying x P K (x) = x K associate to any x R m The map P K is calle the projector onto K We also have P K (x) x, P K (x) y 0, y K () Moreover, P K has the following non-expansiveness property: P K (x) P K (y) x y, x, y R m () Clearly, x K is continuously ifferentiable at point x, an (see Aubin an Cellina [984]) x K = (x P K (x)) (3) The following lemma was obtaine in Shi an Hong [009], which is useful in what follows Lemma Suppose K R m is a convex set an x a, x b R m Then x a P K (x a ), x b x a x a K x a K x b K (4) Particularly, if x a K > x b K, then x a P K (x a ), x b x a x a K ( x a K x b K ) (5) 3 Dini Derivative The upper Dini erivative of a function h : (a, b) R ( a < b ) at t is efine as D + h(t + s) h(t) h(t) = lim sup s 0 s + When h is continuous on (a, b), h is non-increasing on (a, b) if an only if D + h(t) 0 for any t (a, b) (more etails can be foun in Rouche et al [977]) The next result is given for the calculation of Dini erivative (see Danskin et al [966], Lin et al [007]) Lemma 3 Let V i (t, x) : R R m R (i =,, n) be C an V (t, x) = max,,n V i (t, x) If I(t) = {i {,,, n} : V (t, x(t)) = V i (t, x(t))} is the set of inices where the maximum is reache at t, then D + V (t, x(t)) = max i I(t) Vi (t, x(t)) 3 PROBLEM FORMULATION AND MAIN RESULT In this section, we first propose the consiere multi-agent optimization problem, an then the main result is shown 3 Multi-agent Moel Consier a multi-agent system with continuous-time integrator agent ynamics: ẋ i = u i, i =,, N (6) where x i R m represents the state of agent i, an u i is the control input Denote x = (x,, x N ) T The initial time is t 0 = 0, an the initial conition is x 0 = (x T (0),, x T N (0))T R mn Let V = {,,, N} be the noe set The communications over the network are moele as a time-varying igraph G σ(t) = (V, E σ(t) ) We assume that there is a lower boun τ D > 0 (well time) between two consecutive switching instances of σ(t) Let N i (σ(t)) represent i s neighbor set 5694

3 Preprints of the 8th IFAC Worl Congress The objective for this group of autonomous agents is to reach a consensus, an meanwhile to cooperatively solve the following optimization problem min z R m N F (z) = f i (z) where f i : R m R represents the cost function of agent i, observe by agent i only, an z is a ecision vector 3 Distribute Control Suppose each noe can observe the optimal solution set of f i, enote by X i = {v fi (v) = min z R m f i (z)}, i =,, N We assume that X i, i =,, N are convex sets in R m Let the continuous function a ij (x, t) > 0 be the weight of arc (j, i), for i, j V Then we present the following istribute control rule: u i = a ij (x, t)(x j x i ) + P Xi (x i ) x i (7) j N i(σ(t)) Consier (6) with protocol (7) for initial conition x(0) = x 0 Let X 0 be the optimal solution set of F (z) The consiere optimal consensus is efine as follows (see Fig ) Fig The goal of the agents is to achieve a consensus in X 0 Definition 4 For system (6) with protocol (7), (i) a global optimal set convergence is achieve if for any initial conition x 0 R mn, lim t + x i(t) X0 = 0, i =,, N; (ii) a global consensus is achieve if for any initial conition x 0 R mn, lim t + x i(t) x j (t) = 0, i, j =,, N; (iii) a global optimal consensus is achieve if both (i) an (ii) hol 33 Main Result We nee the following staning assumptions to get the main result A) (Biirectional Communications) G σ(t) is biirectional for all t 0 A) (Nonempty Intersection) compact N X i is nonempty an A3) (Weights Rule) There are a > 0 an a > 0 such that a a ij (x, t) a, t R +, x R mn Throughout the rest of the paper, we assume that A), A) an A3) hol The main result of this paper is state as follows Theorem 5 System (6) with protocol (7) achieves a global optimal consensus if G([t, + )) is connecte for all t 0 Remark [t, + )-joint connecteness for all t 0 is equivalent to that there exists an unboune time sequence 0 t < < t k < t k+ < such that G([t k, t k+ )) is connecte for all k =,, Note that it oes not require an upper boun for t k+ t k in the efinition In this sense this connectivity assumption is quite weak Remark Biirectional communication assumption A oes not require symmetric weight functions, ie, a ij (x, t) a ji (x, t), i, j =,, N Essentially, the communication graph is still irecte Remark 3 If the weight functions a ij, are only stateepenent, then Theorem 5 will still stan even with A3) being remove Example Suppose X = = X N = [0, ] Then G([t, + )) being connecte for all t 0 is also necessary to guarantee a global optimal consensus (Moreau [005], Shi an Hong [009]) In fact, initial conitions with x i (0) [0, ], i =,, N can always be foun such that the network fails to reach a consensus Example shows that connectivity conition iscusse in Theorem 5 is also partially necessary Therefore, as a matter of fact, Theorem 5 gives sharp connectivity conition for the system to achieve a global optimal consensus We will arrive at a complete proof of Theorem 5 by investigating the optimal set convergence an consensus, respectively in Propositions 9 an 4 OPTIMAL SET CONVERGENCE In this section, we investigate the optimal solution set convergence by establishing a metho which analyzes the asymptotic properties of the istance functions between the agents an the solution sets 4 Distance Functions Let g(t) = max x i(t) X i V 0 be the maximum istance among all the agents away from the optimal solution set Note that, g(t) is not continuously ifferentiable We have to stuy the Dini erivative of g(t) We present the following result inicating that g(t) is nonincreasing 5695

4 Preprints of the 8th IFAC Worl Congress Lemma 6 D + g(t) 0 for all t 0 Proof Accoring to (3), one has t x i(t) X 0 = x i P X0 (x i ), ẋ i = x i P X0 (x i ), j N i(σ(t)) a ij (x, t)(x j x i ) +P Xi (x i ) x i (8) In light of (), we obtain P Xi (x i ) P X0 (x i ), P Xi (x i ) x i 0 since we always have P X0 (x i ) X i for all i =,, N Therefore, for any i V, we obtain x i P X0 (x i ), P Xi (x i ) x i x i P Xi (x i ), P Xi (x i ) x i = x i X i (9) Moreover, let I(t) enote the set containing all the agents that reach the maximum in the efinition of g(t) at time t For any i I(t), accoring to (5) of Lemma, one has x i P X0 (x i ), x j x i 0 (0) since it always hols that x j (t) X0 x i (t) X0 for any j L i (σ(t)) accoring to the efinition of I(t) Therefore, with (8), (9), (0) an base on Lemma 3, we obtain D + g(t) = max i I(t) t x i(t) X 0 max ( x i X i I(t) i ) 0, which completes the proof With Lemma 6, there exists a constant g 0 such that lim g(t) = g There also exist constants 0 θ i η i g, i =,, N such that lim inf x i(t) X 0 = θ i, lim sup x i (t) X 0 = η i Then we propose another two lemmas Their proofs can be foun in the Appenix Lemma 7 Suppose i V Then lim t + x i (t) Xi = 0 if θ i = η i = g Lemma 8 Suppose that G([t, + )) is connecte for all t 0 Then θ i = η i = g for all i =,, N if g > 0 Remark 4 Lemma 6 is the key lemma of the whole iscussion, which points out that the trajectories of each agent always lie in an area with boune istance away from X 0 However, the limit sets of this multi-agent system may still be har to analyze because of the time-varying communications Then with properly selecte metric, Lemmas 7 an 8 provie characterizations of these limit sets viewing their specific structures 4 Set Convergence Now we are reay to show the optimal set convergence Proposition 9 System (6) with protocol (7) achieves the optimal solution set convergence if G([t, + )) is connecte for all t 0 Proof Suppose g > 0 Accoring to Lemmas 7 an 8, we have that for all i =,, N, lim x i(t) X0 = g ; lim x i (t) Xi = 0 () This implies, for any ε > 0, we have that x i (t) B 0 (ε) B i (ε) for sufficiently large t, where B 0 (ε) = {y g + ε y X0 g + ε} an B i (ε) = {y y Xi ε}, i =,, N Then we see from (8) that the erivative of x i (t) X 0 is globally Lipschitz Therefore, base on Barbalat s lemma, we know lim t x i(t) X 0 = 0 () Define E = {(i, j) (i, j) Eσ(t) for infinitely long time} Then G = (V, E ) is connecte since G([t, + )) is connecte for all t 0 Let Ni be the neighbor set of noe i in graph G With Lemma, () an () yiel that for any i =,, N an j Ni, lim x i(t) P X0 (x i (t)), x j (t) x i (t) = 0 (3) Taking i 0 V, we efine two hyperplane: H (t) = {v x i0 (t) P X0 (x i0 (t)), v x i0 (t) = 0}; H (t) = {v x i0 (t) P X0 (x i0 (t)), v P X0 (x i0 (t)) = 0} Then j N i 0, (3) implies that lim x j(t) H (t) = 0; lim x j (t) H (t) = g, which leas to lim P X 0 (x j (t)) P H (t)(x j (t)) = 0 (4) Because G is connecte, we can repeat the analysis over the network, then arrive that (4) hols for all j =,, N Let C i0 (t) = co{p Xi0 (x i0 (t)), P X0 (x (t)),, P X0 (x N (t))} Then C i0 (t) X i0, t 0 Therefore, with () an (4) an accoring to the structure of H (t) an H (t), there will be a point z N i 0 = C i 0 (t) X 0 for sufficiently large t such that x i0 (t) P X0 (x i0 (t)), z P X0 (x i0 (t)) > 0, which contraicts () Therefore, g > 0 oes not hol, an then the optimal set convergence follows 5 CONSENSUS ANALYSIS This section focuses on the consensus analysis By constructing an stuying a class of invariant sets for the consiere system, we show that optimal set convergence finally results in a global consensus 5 Invariant Set We efine a multi-projection function: N : R m P ik i k i X i 5696

5 Preprints of the 8th IFAC Worl Congress with i,, i k {,, N}, k =,, in the following way P ik i k i (x) = P Xik P Xik P Xi (x) Particularly, P is enote by P (x) = x as the case for k = 0 Let Γ = {P ik i k i i,, i k {,, N}, k = 0,,, } be the set which contains all the multi-projection functions we efine Furthermore, let K be a convex set in R m, an efine K by = co{p (y) y K, P Γ} K Therefore, enoting ĝ(t) = max x i(t),,n K, base on a similar analysis as the proof of Lemma 6, we see that D + ĝ(t) 0, t 0 This implies, ĝ(t) 0 for all t t 0 once we have ĝ(t 0 ) = 0, which leas to the following conclusion immeiately (see Fig 5) Proposition 0 N K = K K is positively invariant for system (6) with protocol (7) Fig Constructing an invariant set from K = co{y, y } We next establish an important property of the constructe invariant set N K Lemma y K max z K z X0, y K Proof With Lemma, any y K has the following form y = λ i P i (z i ), where λ i = with λ i 0, P i Γ an z i K, i =,, Then, by the non-expansiveness property (), we have that for any z R m an P ik i k i Γ, P X0 (z) P ik i k i (z) = P ik i k i (P X0 (z)) P ik i k i (z) P X0 (z) z = z X0 This leas to λ i P i (z i ) λ i z i λ i z i P X0 (z i ) + max z K z X 0, λ i P X0 (z i ) P i (z i ) which implies the conclusion because λ iz i K 5 Global Consensus We are now in a place to propose the consensus analysis Proposition System (6) with protocol (7) achieves a global consensus if G([t, + )) is connecte for all t 0 Proof Accoring to Proposition 9, lim x i (t) X0 = 0, i =,, N Therefore, ε > 0, ˆT (ε) > 0 such that, t ˆT x i (t) X0 ε, i =,, N Moreover, Proposition 0 inicates that N co{x (t),,x N (t)} is an invariant set, an therefore, Lemma will lea to x i (s) co{x (t),,x N (t)} ε (5) for all ˆT t s an i =,, N The consensus analysis focuses on each coorinate respectively Denote x l i (t) as the l-th coorinate of x i(t), an let ϕ(t) = min i N {xl i(t)}, φ(t) = max i N {xl i(t)} be the minimum an the maximum within all the agents Base on (5), one has that for all T < t s, ϕ(s) ϕ(t) ε; φ(s) φ(t) + ε (6) We ivie the following proof into 3 steps Step : Take i 0 as a noe with x l i 0 (t ) = ϕ(t ) with t = ˆT If there is no link connecting i 0 uring t (t, s), we have x l i 0 (t) x l i 0 (t ) ε, t [t, s) Denote the first moment when i 0 has at least one neighbor uring t t as t, an let the neighbor set of i 0 for t [ t, t + τ D ) be V Then, one has t xl i 0 (t) (N )a (φ(t ) + ε x l i 0 (t)) + ε (7) for all t [ t, t + τ D ) As a result, we obtain x l i 0 (t) m 0 ϕ(t ) + ( m 0 )φ(t ) + (N )a + (N )a ε = µ for all t [ t, t + τ D ], where m 0 = e (N )a τ D On the other han, for any j V, we have that when t [ t, t + τ D ), t xl j(t) a (µ x l j(t)) + (N )a (φ(t ) + ε x l j(t)) + ε which yiels x l j( t + τ D ) m 0 w 0 ϕ(t ) + ( m 0 w 0 )φ(t ) + L 0 ε, where w 0 = a ( e [(N )a +a ]τ D ) (N )a +a an L 0 = ( + (N )a ) Step : Applying Proposition 0 on the subsystem forme by noes in {i 0 } V, we have that if there is no other noe connecte to the noe set {i 0 } V uring t ( t + τ D, s), then for any j {i 0 } V, x l j(t) x l j( t + τ D ) ε, t [ t + τ D, s) 5697

6 Preprints of the 8th IFAC Worl Congress Therefore, since G([t, + )) is connecte for all t 0, we can always procee the upper process until V = {i 0 } V V j0, an then obtain x l k( t j0 + τ D ) (m 0 w 0 ) j 0 ϕ(t ) + ( (m 0 w 0 ) j 0 )φ(t ) +j 0 L 0 ε for all k V Thus, letting t = t j0 + τ D, we obtain φ(t ) (m 0 w 0 ) j0 ϕ(t ) + ( (m 0 w 0 ) j0 )φ(t ) + j 0 L 0 ε Denote H(t) = φ(t) ϕ(t) Then H(t ) ( (m 0 w 0 ) j 0 )H(t ) + (j 0 L 0 + )ε (8) Step 3: Noting the fact that j 0 N, we enote ρ = (m 0 w 0 ) N an repeat the estimate by viewing t k as t for k =, 3,, an then we can get a time sequence T < t < t < with t k t k + τ D such that H(t k ) ( ρ )H(t k ) + ((N )L 0 + )ε, for all k =,,, which leas to H(t k+ ) ( ρ ) k H(t ) + (N )L 0 + ρ ε (9) Therefore, letting k ten to infinity in (9) an by (6), we obtain lim sup H(t) ( (N )L )ε ρ Then lim H(t) = 0 since ε can be arbitrarily small This completes the proof 6 SIMULATIONS Example Consier a multi-agent network with three noes V = {,, 3} The communication links are timevarying, efine by { (, ) if n = 0 mo 3 E σ(t) = (, 3) if n = mo 3 (, 3) if n = mo 3 when t [ n(n ), n(n+) ], n =,, Take a ij (x, t) = + x i x j an the optimal solution sets corresponing to the noes are isks with raius an centers (0, 0), (, ) an (, 0) respectively The trajectories for (6) with protocol (7) are shown in Fig 6 It can be seen that as time goes on, an optimal consensus is achieve asymptotically t = 0 to t = 000 Circle Circle Circle 3 x x x Fig 3 Agreeing within the optimal solution set 7 CONCLUSIONS This paper aresse an optimal consensus problem for multi-agent systems With jointly connecte graphs, the consiere multi-agent system achieve not only consensus, but also optimum by agreeing within the global solution set of a sum of objective functions Moreover, control laws applie to the agents were nonlinear an istribute The work showe that a global optimization problem can be solve over a multi-agent network with a simple protocol an limite interactions REFERENCES Boy, S an Vanenberghe, L (004) Convex Optimization New York, NY: Cambrige University Press Aubin, J an Cellina, A (984) Differential Inclusions Berlin: Speringer-Verlag Rockafellar, R T (97) Convex Analysis New Jersey: Princeton University Press Gosil, C an Royle, G (00) Algebraic Graph Theory New York: Springer-Verlag Rouche, N, Habets, P an Laloy, M (977) Stability Theory by Liapunov s Direct Metho New York: Springer- Verlag Danskin, J (966) The theory of max-min, with applications SIAM J Appl Math, vol 4, Clarke, F, Leyaev, YuS, Stern, R an Wolenski, P (998) Nonsmooth Analysis an Control Theory Speringer-Verlag Martinez, S, Cortés J an Bullo, F (007) Motion coorination with istribute information IEEE Control Systems Magazine, vol 7, no 4, Ren, W an Bear, R (008) Distribute Consensus in Multi-vehicle Cooperative Control Springer-Verlag, Lonon Tsitsiklis, J, Bertsekas, D an Athans, M (986) Distribute asynchronous eterministic an stochastic graient optimization algorithms IEEE Trans Automatic Control, 3, Olfati-Saber, R an Murray, R (004) Consensus problems in the networks of agents with switching topology an time ealys IEEE Trans Automatic Control, vol 49, no 9, Jababaie, A, Lin, J an Morse, A S (003) Coorination of groups of mobile agents using nearest neighbor rules IEEE Trans Automatic Control, vol 48, no 6, Tanner, H G, Jababaie, A, an Pappas G J (007) Flocking in fixe an switching networks IEEE Trans Automatic Control, 5(5): Cheng, D, Wang, J an Hu, X (008) An extension of LaSalle s invariance principle an its applciation to multi-agents consensus IEEE Trans Automatic Control, 53, Shi, G an Hong, Y (009) Global target aggregation an state agreement of nonlinear multi-agent systems with switching topologies Automatica, vol 45, Moreau, L (005) Stability of multiagent systems with time-epenent communication links IEEE Trans Automatic Control, 50,

7 Preprints of the 8th IFAC Worl Congress Lin, Z, Francis, B an Maggiore, M (007) State agreement for continuous-time couple nonlinear systems SIAM J Control Optim, vol 46, no, Neić, A an Ozaglar, A (009) Distribute subgraient methos for multi-agent optimization IEEE Transactions on Automatic Control, vol 54, no, 48-6 Neić, A, Ozaglar, A an Parrilo, P A(00) Constraine Consensus an Optimization in Multi-Agent Networks IEEE Transactions on Automatic Control, vol 55, no 4, Johansson, B, Rabi, M an Johansson, M (007) A simple peer-to-peer algorithm for istribute optimization in sensor networks in Proc IEEE Conference on Decision an Control, New Orleans, LA, Johansson, B, Keviczky, T, Johansson, M an Johansson, K H (008) Subgraient methos an consensus algorithms for solving convex optimization problems Proc IEEE Conference on Decision an Control, Cancun, Mexico, A Proof of Lemma 7 APPENDIX Suppose θ i0 = η i0 = g Then the efinition of g implies that, ε > 0, T (ε) > 0 such that, t T x i0 (t) X 0 [g ε, g + ε], an x i (t) X 0 [0, g + ε], i =,, N (0) When g = 0, then the conclusion hols straightforwarly because x i (t) Xi x i (t) X0, t 0 We just assume g > 0 in the following Accoring to (8) an (9), we have t x i 0 (t) X 0 x i0 P X0 (x i0 ), x i0 (t) X i j N i0 (σ(t)) a ij (x j x i0 ) Furthermore, base on (0) an Lemmas an 6, one has that for any t T an j N i0 (σ(t)), x i0 P X0 (x i0 ), x j x i0 g(0)ε If the conclusion oes not hol, there will be a constant M 0 > 0 an a time serial 0 < t < < t k < with lim k t k = such that x i0 (t k ) Xi0 = M 0 () Since X 0 is compact, there is a constant L > 0 such that a b L, a, b {y y X 0 g(0)} Thus, t x i 0 (t) X i0 g(0) + (N )a g(0)l () Denoting τ 0 = M 0 () an (), we obtain g(0)+(n )a g(0)l an accoring to t x i 0 (t) X 0 M 0 + g(0)ε 4 M 0, for any t [t k, t k + τ 0 ] with t k > T an ε M 0 8 g(0) As a result, x i0 (t k + τ 0 ) X 0 g + ε M 0 τ 0 (3) 4 However, (3) contraicts (0) when ε < M 0 τ0 8 The proof is complete A Proof of Lemma 8 Suppose there exists a noe i 0 V such that 0 θ i0 < η i0 Then ε > 0, T (ε) > 0 such that, t > T (ε) x i (t) X 0 (0, g + ε), i =,, N (4) Take ζ 0 = (θ i 0 + η i0 ) Then there exists a time serial 0 < ˆt < < ˆt k < with lim ˆt k = such that x i0 (ˆt k ) X0 = ζ 0 for all k =,, Take ˆt k0 > T with x i0 (ˆt k0 ) X0 = ζ 0 If there is no other noe connecte to i 0 uring t (ˆt k0, s) for ˆt k0 < s, applying Lemma 6 on noe i 0 will lea to x i0 (t) X0 x i0 (ˆt k0 ) X0, t (ˆt k0, s) (5) Next, enote the first moment when i 0 has at least one neighbor when t ˆt k0 as t, an let V be the neighbor set of i 0 uring t [ t, t + τ D ) Accoring to (4) an Lemma, we have that for all t > ˆt k0, t x i 0 (t) X 0 (N )a x i0 (t) X0 ( g + ε x i0 (t) X0 ), which will lea to D + x i0 (t) X0 (N )a x i0 (t) X0 + (N )a g + ε As a result, for all t [ t, t + τ D ), we have x i0 (t) X0 m 0 ζ 0 + ( m 0 ) g + ε = ξ We procee to estimate the noes in V When t [ t, t + τ D ), again by Lemma, we obtain that for all i V, which yiels D + x i (t) X0 [(N )a + a ] x i (t) X0 +(N )a g + ε + a ξ, x i ( t + τ D ) X0 ( w 0 ) g + ε + w 0 ξ = m 0 w 0 ζ 0 + ( m 0 w 0 ) g + ε Similarly, since G([t, + )) is connecte for all t 0, we can always procee the upper process until V = {i 0 } V V j0, an then we obtain x j ( t j0 + τ D ) X0 (m 0 w 0 ) j0 ζ 0 + ( (m 0 w 0 ) j0 ) g + ε for all j V Therefore, for sufficiently small ε, we obtain x j ( t j0 + τ D ) X0 < g, j =,, N This contraicts the efinition of g Then the proof is complete 5699