constrained consensus in continuous time multiagent systems under weighted graph this paper studies constrained consensus for continuous time multiagent systems where agents are subject to different state constraints a projection based consensus protocol is presented where a projection operator is used on top of the traditional consensus protocol in order to keep the state staying inside the constraint set this operator can ensure the smoothness of the state trajectory under projection and it will only be used when the state is close to the boundary of the constraint set and about to move outside under this protocol each individual agent only needs to know its own state and constraint set as well as the relative states to its neighboring agents with the assumption that all constraint sets are convex and have nonempty intersection it is proven that global consensus can be achieved without violating any state constraints during the transience

asynchronous distributed algorithms for solving linear algebraic equations two asynchronous distributed algorithms are presented for solving a linear equation of the form ax b with at least one solution the equation is simultaneously and asynchronously solved by m agents assuming that each agent knows only a subset of the rows of the partitioned matrix a b the estimates of the equation s solution generated by its neighbors and nothing more neighbor relationships are characterized by a time dependent directed graph whose vertices correspond to agents and whose arcs depict neighbor relationships each agent recursively updates its estimate of a solution at its own event times by utilizing estimates generated by its neighbors which are transmitted with delays the event time sequences of different agents are not assumed to be synchronized it is shown that for any matrix vector pair a b for which the equation has a solution and any repeatedly jointly strongly connected sequence of neighbor graphs defined on the merged sequence of all agents event times the algorithms cause all agents estimates to converge exponentially fast to the same solution to ax b the first algorithm requires a specific initialization step at each agent and the second algorithm works for arbitrary initializations explicit expressions for convergence rates are provided and a relation between local initializations and limiting consensus solutions is established which is used to solve the least 2 norm solution

exponential convergence of a distributed algorithm for solving linear algebraic equations abstract in a recent paper a distributed algorithm was proposed for solving linear algebraic equations of the form a x b assuming that the equation has at least one solution the equation is presumed to be solved by m agents assuming that each agent knows a subset of the rows of the matrix a b the current estimates of the equation s solution generated by each of its neighbors and nothing more neighbor relationships are represented by a time dependent directed graph n t whose vertices correspond to agents and whose arcs characterize neighbor relationships sufficient conditions on n t were derived under which the algorithm can cause all agents estimates to converge exponentially fast to the same solution to a x b these conditions were also shown to be necessary for exponential convergence provided the data about a b available to the agents is non redundant the aim of this paper is to relax this non redundant assumption this is accomplished by establishing exponential convergence under conditions which are the weakest possible for the problem at hand the conditions are based on a new notion of graph connectivity an improved bound on the convergence rate is also derived

state consensus for multi agent systems with switching topologies and time varying delays in this paper we investigate state consensus problems for discrete time multi agent systems with changing communications topologies and bounded time varying communication delays the analysis in this paper is based on the properties of non negative matrices we first extend the model of networks of dynamic agents to the case with multiple time delays and prove that if the communication topology time delays and weighting factors are time invariant then the necessary and sufficient condition that the multi agent system solves a consensus problem is that the communication topology represented by a directed graph has spanning trees then we allow for dynamically changing communication topologies and bounded time varying communication delays and present some sufficient conditions for state consensus of system finally as a special case of our model the problem of asynchronous information exchange is also discussed

orthogonal polynomial solutions of linear ordinary differential equations abstract this paper surveys the latest known results concerning the classification of differential equations of the form k 1 n a k x y k x y x having a sequence of polynomial eigenfunctions p n x n 0 that are orthogonal with respect to some real bilinear form since the publishing of the erice report in 1990 several new significant results and generalizations have been discovered

distributed consensus for multiagent systems with communication delays and limited data rate this paper considers the average consensus problem for multiagent networks with communication delays and limited data rate on one hand communication delays often exist in information acquisition and transmission on the other hand only limited state information of agents can be transmitted to their neighbors at each time step due to bandwidth constraints the average consensus problem becomes much more complicated when both delays and data rate constraints are to be considered in this paper a distributed consensus protocol is proposed based on dynamic encoding and decoding it is shown that for a connected network as long as the time delays are bounded the average consensus can be achieved with a finite communication data rate in particular it is shown that merely a one bit information exchange between each pair of adjacent agents at each time step suffices to guarantee the average consensus

the complexity of counting stable marriages in an instance of size n of the stable marriage problem each of n men and n women ranks the members of the opposite sex in order of preference a stable matching is a complete matching of men and women such that no man and woman who are not partners both prefer each other to their actual partners under the matching it is well known that at least one stable matching exists for every stable marriage instance so that the decision version of the problem always has a yes answer furthermore efficient algorithms are known for the determination of such a stable matching so that the search version of the problem is polynomially solvable however by exploring the structure of the set of stable matchings for any particular instance of the problem and exploiting its relationship with the set of antichains of an associated partially ordered set we prove that the enumeration version of the problem determining the number of stable matchings is p complete and therefore cannot be solved in polynomial time if

robust consensus for continuous time multiagent dynamics this paper investigates consensus problems for continuous time multiagent systems with time varying communication graphs subject to input noise based on input to state stability and integral input to state stability robust consensus and integral robust consensus are defined with respect to l and l1 norms of the noise function respectively sufficient and or necessary connectivity conditions are obtained for the system to reach robust consensus or integral robust consensus under mild assumptions the results answer the question on how much interaction is required for a multiagent network to converge despite a certain amount of input disturbance the convergence time is obtained for the consensus computation on directed and k bidirectional graphs

consensus conditions of continuous time multi agent systems with time delays and measurement noises this work is concerned with stochastic consensus conditions of multi agent systems with both time delays and measurement noises for the case of additive noises we develop some necessary conditions and sufficient conditions for stochastic weak consensus by estimating the differential resolvent function for delay equations by the martingale convergence theorem we obtain necessary conditions and sufficient conditions for stochastic strong consensus for the case of multiplicative noises we consider two kinds of time delays appeared in the measurement term and the noise term respectively we first show that stochastic weak consensus with the exponential convergence rate implies stochastic strong consensus then by constructing degenerate lyapunov functional we find the sufficient consensus conditions and show that stochastic consensus can be achieved by carefully choosing the control gain according to the noise intensities and the time delay in the measurement term

the symmetric form of the koekoeks laguerre type differential equation abstract koekoek and koekoek 1991 proved that the laguerre type polynomials ln n x n 0 which are orthogonal on 0 with respect to the measure d n defined on 0 by d n x x e x 1 n x dx satisfy a differential equation l2 4 n y x ny x of order 2 4 when is a nonnegative integer in this paper we announce that the differential expression l2 4 n is symmetrizable on 0 with symmetry factor x e x and explicitly give this formally symmetric expression x e xl2 4 n y

distributed cooperative adaptive identification and control for a group of continuous time systems with a cooperative pe condition via consensus in this paper we first address the uniformly exponential stability ues problem of a group of distributed cooperative adaptive systems in a general framework inspired by consensus theory distributed cooperative adaptive laws are proposed to estimate the unknown parameters of these systems it is shown that not only is the entire closed loop system stable but also both the identification tracking error and the parameter estimation error converge to zero uniformly exponentially under a cooperative persistent excitation pe condition of a regressor matrix in each system which is weaker than the traditionally defined pe condition the effects of network topology on ues of the closed loop system are also explored if the topology is time invariant it needs to be undirected and connected however when the topology is time varying it is just required that the integration of the topology over an interval with fixed length is undirected and connected the established results are then employed to identify and control several classes of linearly parameterized systems simulation examples are also provided to demonstrate the effectiveness and applications of the proposed distributed cooperative adaptive laws

an extension of lasalle s invariance principle and its application to multi agent consensus in the paper an extension of lasalle s invariance principle to a class of switched linear systems is studied one of the motivations is the consensus problem in multi agent systems unlike most existing results in which each switching mode in the system needs to be asymptotically stable this paper allows that the switching modes are only lyapunov stable under certain ergodicity assumptions an extension of lasalle s invariance principle for global asymptotic stability is obtained then it is used to solve the consensus reaching problem of certain multi agent systems in which each agent is modeled by a double integrator and the associated interaction graph is switching and is assumed to be only jointly connected

formation stability analysis of unmanned multi vehicles under interconnection topologies in this paper the overall formation stability of an unmanned multi vehicle is mathematically presented under interconnection topologies a novel definition of formation error is first given and followed by the proposed formation stability hypothesis based on this hypothesis a unique extension decomposition aggregation scheme is then employed to support the stability analysis for the overall multi vehicle formation under a mesh topology it is proved that the overall formation control system consisting of n number of nonlinear vehicles is not only asymptotically stable but also exponentially stable in the sense of lyapunov within a neighbourhood of the desired formation this technique is shown to be applicable for a mesh topology but is equally applicable for other topologies a simulation study of the formation manoeuvre of multiple aerosonde uavs unmanned aerial vehicles in 3 d space is finally carried out verifying the achieved formation stability result

connectivity and set tracking of multi agent systems guided by multiple moving leaders in this paper we investigate distributed multi agent tracking of a convex set specified by multiple moving leaders with unmeasurable velocities various jointly connected interaction topologies of the follower agents with uncertainties are considered in the study of set tracking based on the connectivity of the time varying multi agent system necessary and sufficient conditions are obtained for set input to state stability and set integral input to state stability for a nonlinear neighbor based coordination rule with switching directed topologies conditions for asymptotic set tracking are also proposed with respect to the polytope spanned by the leaders

constrained consensus and optimization in multi agent networks we present distributed algorithms that can be used by multiple agents to align their estimates with a particular value over a network with time varying connectivity our framework is general in that this value can represent a consensus value among multiple agents or an optimal solution of an optimization problem where the global objective function is a combination of local agent objective functions our main focus is on constrained problems where the estimates of each agent are restricted to lie in different convex sets to highlight the effects of constraints we first consider a constrained consensus problem and present a distributed projected consensus algorithm in which agents combine their local averaging operation with projection on their individual constraint sets this algorithm can be viewed as a version of an alternating projection method with weights that are varying over time and across agents we establish convergence and convergence rate results for the projected consensus algorithm we next study a constrained optimization problem for optimizing the sum of local objective functions of the agents subject to the intersection of their local constraint sets we present a distributed projected subgradient algorithm which involves each agent performing a local averaging operation taking a subgradient step to minimize its own objective function and projecting on its constraint set we show that with an appropriately selected stepsize rule the agent estimates generated by this algorithm converge to the same optimal solution for the cases when the weights are constant and equal and when the weights are time varying but all agents have the same constraint set

distributed attitude synchronization control of multi agent systems with switching topologies this paper addresses the attitude synchronization problem in multi agent systems with directed and switching interconnection topologies two cases for the synchronization problem are discussed under different assumptions about the measurable information in the first case the agents can measure their rotations relative to a global reference coordinate frame whilst in the second case they can only measure the relative rotations between each other two intuitive distributed control laws based on the axis angle representations of the rotations are proposed for the two cases respectively the invariance of convex balls in so 3 is guaranteed moreover attitude synchronization is ensured under the well known mild switching assumptions the joint strong connection for the first case and joint quasi strong connection for the second case to show the effectiveness of the proposed control schemes illustrative examples are provided

what matchings can be stable the testable implications of matching theory this paper studies the falsifiability of two sided matching theory when agents preferences are unknown a collection of matchings is rationalizable if there are preferences for the agents involved so that the matchings are stable we show that there are nonrationalizable collections of matchings hence the theory is falsifiable we also characterize the rationalizable collections of matchings which leads to a test of matching theory in the spirit of revealed preference tests of individual optimizing behavior

constrained consensus in unbalanced networks with communication delays in this note a constrained consensus problem is studied for multi agent systems in unbalanced networks in the presence of communication delays here each agent needs to lie in a closed convex constraint set while reaching a consensus the communication graphs are directed dynamically changing and not necessarily balanced and only the union of the graphs is assumed to be strongly connected among each time interval of a certain bounded length the analysis is performed based on an undelayed equivalent system that is composed of a linear main body and an error auxiliary to tackle the loss of symmetry caused by unbalanced graphs and communication delays a novel approach is proposed the idea is to estimate the distance from each agent to the intersection set of all agents constraint sets based on the properties of the projection on convex sets so as to show consensus convergence by contradiction it is shown that the error auxiliary vanishes as time evolves and the linear main body converges to a vector with an exponential rate as a separate system it is also shown that the communication delays do not affect the consensus stability and constrained consensus is reached even if the communication delays are arbitrarily bounded finally a numerical example is included to illustrate the obtained theoretical results

differential equations having orthogonal polynomial solutions abstract necessary and sufficient conditions for an orthogonal polynomial system ops to satisfy a differential equation with polynomial coefficients of the form l n y i 1 n l i x y i x n y x were found by h l krall here we find new necessary conditions for the equation to have an ops of solutions as well as some other interesting applications in particular we obtain necessary and sufficient conditions for a distribution w x to be an orthogonalizing weight for such an ops and investigate the structure of w x we also show that if the equation has an ops of solutions which is orthogonal relative to a distribution w x then the differential operator ln in must be symmetrizable under certain conditions on w x

constrained consensus of asynchronous discrete time multi agent systems with time varying topology the union graph is assumed to be strongly connected over each finite interval an approach is proposed to transform the original network to a synchronous one we show that the linear part converges and the projection error vanishes over time in this paper the constrained consensus problem is studied for asynchronous discrete time multi agent system where each agent needs to lie in a closed convex constraint set while reaching consensus the communication graphs are assumed to be directed unbalanced dynamically changing in addition their union graph is assumed to be strongly connected among certain intervals of finite length to deal with the asynchronous communications among agents the original asynchronous system is equivalently transformed to the synchronous one by adding new agents by employing the properties of the projection on the convex sets the distance from the states of the agents in the newly constructed system to the intersection set of all agents constraint sets is estimated based on this estimation the original system is proven to reach consensus by showing that the linear parts of the newly constructed systems converge and the nonlinear parts vanish over time finally two numerical examples are provided to show the effectiveness of the theoretical results

distributed nonlinear control of mobile autonomous multi agents abstract this paper studies the distributed nonlinear control of mobile autonomous agents with variable and directed topology a new distributed nonlinear design scheme is presented for multi agent systems modeled by double integrators with the new design the outputs of the controlled agents asymptotically converge to each other as long as a mild connectivity condition is satisfied moreover the velocity derivative of the output of each agent can be restricted to be within any specified neighborhood of the origin which is of practical interest for systems under such physical constraint the new design is still valid if one of the agents is a leader and the control objective is to achieve leader following as an illustration of the generality and effectiveness of the presented methodology the formation control of a group of unicycle mobile robots with nonholonomic constraints is revisited instead of assuming the point robot model the unicycle model is transformed into two double integrators by dynamic feedback linearization and the proposed distributed nonlinear design method is used to overcome the singularity problem caused by the nonholonomic constraint by properly restricting the velocities simulation results are included to illustrate the theoretical results

the revealed preference theory of stable and extremal stable matchings we investigate the testable implications of the theory of stable matchings we provide a characterization of the matchings that are rationalizable as stable matchings when agents preferences are unobserved the characterization is a simple nonparametric test for stability in the tradition of revealed preference tests we also characterize the observed stable matchings when monetary transfers are allowed and the stable matchings that are best for one side of the market extremal stable matchings we find that the theory of extremal stable matchings is observationally equivalent to requiring that there be a unique stable matching or that the matching be consistent with unrestricted monetary transfers

synchronizing nonlinear complex networks via switching disconnected topology the current theoretical study on the synchronization mechanism of complex networks mainly focuses on fixed connected topology or switching balanced topology which is heavily based on constructing common smooth lyapunov functions however for complex networks with nonlinear node dynamics it is still unknown whether such kind of networks can synchronize under the condition of switching directed and disconnected topology by employing the ideas of sequential connectivity and joint connectivity this paper finds that complex networks with one sided lipschitz node dynamics can realize synchronization even if the network topology is not connected at any time instant by iteratively estimating the maximal distance between different nodes this paper gives several sufficient conditions on synchronization of nonlinear complex networks under switching disconnected topology finally simulation examples validate the main results

consensus seeking in multiagent systems under dynamically changing interaction topologies this note considers the problem of information consensus among multiple agents in the presence of limited and unreliable information exchange with dynamically changing interaction topologies both discrete and continuous update schemes are proposed for information consensus this note shows that information consensus under dynamically changing interaction topologies can be achieved asymptotically if the union of the directed interaction graphs have a spanning tree frequently enough as the system evolves

jacobi stirling numbers jacobi polynomials and the left definite analysis of the classical jacobi differential expression we develop the left definite analysis associated with the self adjoint jacobi operator generated from the classical second order jacobi differential expression in the hilbert space where w t 1 t 1 t that has the jacobi polynomials as eigenfunctions here 1 and k is a fixed non negative constant more specifically for each we explicitly determine the unique left definite hilbert sobolev space and the corresponding unique left definite self adjoint operator in associated with the pair the jacobi polynomials form a complete orthogonal set in each left definite space and are the eigenfunctions of each moreover in this paper we explicitly determine the domain of each as well as each integral power of the key to determining these spaces and operators is in finding the explicit lagrangian symmetric form of the integral composite powers of l k in turn the key to determining these powers is a double sequence of numbers which we introduce in this paper as the jacobi stirling numbers some properties of these numbers which in some ways behave like the classical stirling numbers of the second kind are established including a remarkable and yet somewhat mysterious identity involving these numbers and the eigenvalues of

distributed velocity constrained consensus of discrete time multi agent systems with nonconvex constraints switching topologies and delays in this technical note a distributed velocity constrained consensus problem is studied for discrete time multi agent systems where each agent s velocity is constrained to lie in a nonconvex set a distributed constrained control algorithm is proposed to enable all agents to converge to a common point using only local information the gains of the algorithm for all agents need not to be the same or predesigned and can be adjusted by each agent itself based on its own and neighbors information it is shown that the algorithm is robust to arbitrarily bounded communication delays and arbitrarily switching communication graphs provided that the union of the graphs has directed spanning trees among each certain time interval the analysis approach is based on multiple novel model transformations proper control parameter selections boundedness analysis of state dependent stochastic matrices1 exploitation of the convexity of stochastic matrices and the joint connectivity of the communication graphs numerical examples are included to illustrate the theoretical results

distributed h constrained consensus problem abstract this paper is concerned with the distributed h constrained consensus problem for multi agent systems the objective is to use local information to make all agents reach a consensus in a given convex set while satisfying a desired h performance index in the presence of external disturbances a nonlinear distributed algorithm is adopted to solve this constrained consensus problem to measure the effects of the disturbance on the consensus of all agents and the convergence of all agents to the convex set a nonlinear function is constructed as the h output function by analyzing the convexity of the convex set a sufficient condition is given to solve the constrained consensus problem with the desired h performance finally a numerical simulation example is provided to show the effectiveness of our theoretical results

containment control of multi agent systems in a noisy communication environment the containment problem of first order and second order integral multi agent systems with communication noises is investigated in this paper motivated by the previous work a time varying gain a t is introduced to attenuate the noise effect it is proved that the proposed protocol can solve the mean square containment problem if the following conditions hold 1 the communication topology graph has a spanning forest whose roots are leaders of the multi agent system 2 0 a t dt 3 lim t a t 0 moreover if a t is uniformly continuous these conditions are necessary for ensuring the mean square containment in addition the proposed protocol is also extended to allow the case where each agent has its own gain a i t it is proved that the mean square containment problem can still be solved if all agent dependent gains are infinitesimals of the same order as time goes to infinity finally two simulation examples are provided to demonstrate the effectiveness of the proposed protocols

left definite theory with applications to orthogonal polynomials in the past several years there has been considerable progress made on a general left definite theory associated with a self adjoint operator a that is bounded below in a hilbert space h the term left definite has its origins in differential equations but littlejohn and wellman l l littlejohn r wellman a general left definite theory for certain self adjoint operators with applications to differential equations j differential equations 181 2 2002 280 339 generalized the main ideas to a general abstract setting in particular it is known that such an operator a generates a continuum h r r 0 of hilbert spaces and a continuum of a r r 0 of self adjoint operators in this paper we review the main theoretical results in l l littlejohn r wellman a general left definite theory for certain self adjoint operators with applications to differential equations j differential equations 181 2 2002 280 339 moreover we apply these results to several specific examples including the classical orthogonal polynomials of laguerre hermite and jacobi

distributed consensus of second order multiagent systems with nonconvex velocity and control input constraints this paper is devoted to a distributed consensus problem for second order multiagent systems with nonconvex velocity and control input constraints a distributed control algorithm is introduced using local information it is shown that consensus can be achieved as long as the union of the communication graphs has directed spanning trees among each time interval of certain bounded length in the presence of arbitrarily bounded communication delays finally a numerical example is included to illustrate the theoretical results

lyapunov based approach to multiagent systems with switching jointly connected interconnection this note addresses a coordination problem of a multiagent system with jointly connected interconnection topologies neighbor based rules are adopted to realize local control strategies for these continuous time autonomous agents described by double integrators although the interagent connection structures vary over time and related graphs may not be connected a sufficient condition to make all the agents converge to a common value is given for the problem by a proposed lyapunov based approach and related space decomposition technique

reaching an optimal consensus dynamical systems that compute intersections of convex sets in this paper multi agent systems minimizing a sum of objective functions where each component is only known to a particular node is considered for continuous time dynamics with time varying interconnection topologies assuming that each node can observe a convex solution set of its optimization component and the intersection of all such sets is nonempty the considered optimization problem is converted to an intersection computation problem by a simple distributed control rule the considered multi agent system with continuous time dynamics achieves not only a consensus but also an optimal agreement within the optimal solution set of the overall optimization objective directed and bidirectional communications are studied respectively and connectivity conditions are given to ensure a global optimal consensus in this way the corresponding intersection computation problem is solved by the proposed decentralized continuous time algorithm we establish several important properties of the distance functions with respect to the global optimal solution set and a class of invariant sets with the help of convex and non smooth analysis

sobolev orthogonal polynomials and spectral differential equations we find necessary and sufficient conditions for a spectral differential equation n ln y x 52ti x yw x kny x 1 to have sobolev orthogonal polynomials of solutions which are orthogonal rel ative to the sobolev pseudo inner product c t p q pqdn i p q du

extensions of weak invariance principle for nonlinear switched systems with time invariant and time varying subsystems abstract this paper addresses the stability problem for nonlinear switched systems with both time invariant and time varying subsystems given that lyapunov like functions are difficult to construct for nonlinear switched systems generalized invariance principles are established based on observer functions for nonlinear switched systems where the subsystems share lyapunov like functions the generalized invariance principles can be specialized to lyapunov based invariance principles where accurate convergent region can be obtained in addition to the above efforts the definitions of p limit system and limit system set are presented under which the proposed invariance principles are extended to nonlinear switched systems with time varying subsystems illustrative examples show the effectiveness of the theoretical results

symmetrizability of differential equations having orthogonal polynomial solutions abstract we show that if a linear differential equation of spectral type with polynomial coefficients l n y x i 0 n l i x n y x has an orthogonal polynomial system of solutions then the differential operator ln must be symmetrizable we also give a few applications of this result

stochastic strongly convex optimization via distributed epoch stochastic gradient algorithm this article considers the problem of stochastic strongly convex optimization over a network of multiple interacting nodes the optimization is under a global inequality constraint and the restriction that nodes have only access to the stochastic gradients of their objective functions we propose an efficient distributed non primal dual algorithm by incorporating the inequality constraint into the objective via a smoothing technique we show that the proposed algorithm achieves an optimal o 1 t t is the total number of iterations convergence rate in the mean square distance from the optimal solution in particular we establish a high probability bound for the proposed algorithm by showing that with a probability at least 1 the proposed algorithm converges at a rate of o ln ln t t finally we provide numerical experiments to demonstrate the efficacy of the proposed algorithm

dunkl shift operators and bannai ito polynomials abstract we consider the most general dunkl shift operator l with the following properties i l is of first order in the shift operator and involves reflections ii l preserves the space of polynomials of a given degree iii l is potentially self adjoint we show that under these conditions the operator l has eigenfunctions which coincide with the bannai ito polynomials we construct a polynomial basis which is lower triangular and two diagonal with respect to the action of the operator l this allows to express the bi polynomials explicitly we also present an anti commutator aw 3 algebra corresponding to this operator from the representations of this algebra we derive the structure and recurrence relations of the bi polynomials we introduce new orthogonal polynomials referred to as the complementary bi polynomials as an alternative q 1 limit of the askey wilson polynomials these complementary bi polynomials lead to a new explicit expression for the bi polynomials in terms of the ordinary wilson polynomials

stability of multiagent systems with time dependent communication links we study a simple but compelling model of network of agents interacting via time dependent communication links the model finds application in a variety of fields including synchronization swarming and distributed decision making in the model each agent updates his current state based upon the current information received from neighboring agents necessary and or sufficient conditions for the convergence of the individual agents states to a common value are presented thereby extending recent results reported in the literature the stability analysis is based upon a blend of graph theoretic and system theoretic tools with the notion of convexity playing a central role the analysis is integrated within a formal framework of set valued lyapunov theory which may be of independent interest among others it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence even for the simple models discussed in the present paper