distributed consensus for multi agent systems with delays and noises in transmission channels this paper studies the distributed consensus problem for linear discrete time multi agent systems with delays and noises in transmission channels due to the presence of noises and delays existing techniques such as the lifting technique and the stochastic lyapunov theory are no longer applicable to the analysis of consensus in this paper a novel technique is introduced to overcome the difficulties induced by the delays and noises a consensus protocol with decaying gains satisfying persistence condition is adopted necessary and sufficient conditions for strong consensus and mean square consensus are respectively given for non leader follower and leader follower cases under a fixed topology under dynamically switching topologies and randomly switching topologies sufficient conditions for strong consensus and mean square consensus are also obtained numerical examples are given to demonstrate the effectiveness of the proposed protocols

a nonlinear parallel algorithm with application to the stefan problem the algebraic problem b gamma e a beta e eta which may evolve from elliptic or parabolic partial differential equations is studied conditions are given so that a parallel algorithm of a nonlinear gauss seidel type converges to the unique solution an application is given to the solidification of a channel in which the continuum problem is discretized by the finite element method these computations show a significant decrease in computing times as compared to serial algorithms

consensus of high order multi agent systems with large input and communication delays we study in this paper the consensus problem for multi agent systems with agents characterized by high order linear systems with time delays in both the communication network and inputs provided that the open loop dynamics of the agents is not exponentially unstable but may be polynomially unstable and the communication topology contains a directed spanning tree a truncated predictor feedback approach is established to solve the consensus problem it is shown that if the delays are constant and exactly known the consensus problems can be solved by both full state feedback and observer based output feedback protocols for arbitrarily large yet bounded delays if it is further assumed that the open loop dynamics of the agents only contains zero eigenvalues the delays are allowed to be time varying and unknown numerical examples are worked out to illustrate the effectiveness of the proposed approaches

generalizations of the nonstationary multisplitting iterative method for symmetric positive definite linear systems in this paper we generalize the nonstationary parallel multisplitting iterative method for solving the symmetric positive definite linear systems with several choices of variable weighting matrices the convergence properties of these generalized methods can be improved finally the numerical comparison of several nonstationary parallel multisplitting methods are shown

on the convergence of parallel multisplitting asynchronous gaor method for h matrix in this paper we establish several algorithms of parallel chaotic generalized aor iterative methods for solving large nonsingular systems based on some given models under some different assumptions of coefficient matrix a and its multisplittings we obtain corresponding sufficient conditions of convergence for some relaxed parameters

igaor and multisplitting igaor methods for linear complementarity problems in this paper we propose an interval version of the generalized accelerated overrelaxation methods which we refer to as igaor for solving the linear complementarity problems lcp m q and develop a class of multisplitting igaor methods which can be easily implemented in parallel in addition in regards to the h matrix with positive diagonal elements we prove the convergence of these algorithms and illustrate their efficiency through our numerical results

convergence of parallel multisplitting iterative methods for m matrices abstract despite the major recent activity in parallel processing few effective new algorithms designed exclusively for multiprocessors have been put forth one such is the multisplitting iterative algorithm suggested by o leary and white although they have given some sufficient conditions for convergence a general convergence theory has not been developed even for the classical situation where the coefficient matrix a is an m matrix or is symmetric positive definite in this paper we study the m matrix case in detail the multisplitting process for a rn n is recast as an ordinary iterative process for a certain block matrix a r kn kn where k is the number of processors and standard convergence results are used to develop a convergence theory for multisplitting iterative methods where a is an m matrix comparison results between multisplitting methods are established in terms of monotonic norms and for the case where a is irreducible in terms of the asymptotic convergence rate a key observation here is that in certain cases the rate of global convergence of these parallel iterative methods is inherent in the splitting of a and is independent of the manner in which the work is distributed among the processors thus in general one can distribute the work for load balancing purposes without affecting the convergence rate

consensus seeking over markovian switching networks with time varying delays and uncertain topologies stochastic consensus problems for linear time invariant multi agent systems over markovian switching networks with time varying delays and topology uncertainties are dealt with by using the linear matrix inequality method and the stability theory of markovian jump linear system we show that consensus can be achieved for appropriate time delays and topology uncertainties which are not caused by the markov process provided the union of topologies associated with the positive recurrent states of the markov process admits a spanning tree and the agent dynamics is stabilizable feasible linear matrix inequalities are established to determine the maximal allowable upper bound of time varying delays numerical examples are given to show the feasibility of theoretical results

technical communique average consensus in networks with nonlinearly delayed couplings and switching topology the paper addresses consensus under nonlinear couplings and bounded delays for multi agent systems where the agents have the single integrator dynamics the network topology is undirected and may alter as time progresses the couplings are uncertain and satisfy a conventional sector condition with known sector slopes the delays are uncertain time varying and obey known upper bounds the network satisfies a symmetry condition that resembles the newton s third law explicit analytical conditions for the robust consensus are offered that employ only the known upper bounds for the delays and the sector slopes

distributed robust adaptive equilibrium computation for generalized convex games this paper considers a class of generalized convex games where each player is associated with a convex objective function a convex inequality constraint and a convex constraint set the players aim to compute a nash equilibrium through communicating with neighboring players the particular challenge we consider is that the component functions are unknown a priori to associated players we study two distributed computation algorithms and analyze their convergence properties in the presence of data transmission delays and dynamic changes of network topologies the algorithm performance is verified through demand response on the ieee 30 bus test system our technical tools integrate convex analysis variational inequalities and simultaneous perturbation stochastic approximation

the hausdorff dimension of some fractals and attractors of overlapping construction we introduce a method to show that the hausdorff dimension of certain fractals of overlapping construction is almost always what would be observed if no overlapping occurred the method is also used to examine the dimension of attractors in some noninjective piecewise linear baker s type transformations

robust synchronization of uncertain linear multi agent systems this paper deals with robust synchronization of uncertain multi agent networks given a network with for each of the agents identical nominal linear dynamics we allow uncertainty in the form of additive perturbations of the transfer matrices of the nominal dynamics the perturbations are assumed to be stable and bounded in h norm by some a priori given desired tolerance we derive state space formulas for observer based dynamic protocols that achieve synchronization for all perturbations bounded by this desired tolerance it is shown that a protocol achieves robust synchronization if and only if each controller from a related finite set of feedback controllers robustly stabilizes a given single linear system our protocols are expressed in terms of real symmetric solutions of certain algebraic riccati equations and inequalities and also involve weighting factors that depend on the eigenvalues of the graph laplacian for undirected network graphs we show that within the class of such dynamic protocols a guaranteed achievable tolerance can be obtained that is proportional to the quotient of the second smallest and the largest eigenvalue of the laplacian we also extend our results to additive nonlinear perturbations with l2 gain bounded by a given tolerance

a unifying framework for robust synchronization of heterogeneous networks via integral quadratic constraints a general framework for analysing robust synchronization in large scale heterogenous networks is proposed based on the theory of integral quadratic constraints iqcs dynamic agents are represented as linear time invariant single input single output systems the agents exchange information according to a sparse dynamical interconnection operator in order to achieve synchronization where their outputs are steered to the same possibly time varying signal the main technical hindrance to applying iqcs in this context lies with the presence of the marginally stable dynamics which define the trajectory to which the agents outputs synchronize it is shown that by working with conditions defined on modified signal spaces of interest and exploiting the graph structure underlying the connections between the dynamic systems iqc methods can be applied directly to synchronization analysis without recourse to loop transformations which may obscure the inherent structural properties of the multi agent networked systems decentralized and scalable conditions for synchronization are proposed within this setting the iqc framework is demonstrated to unify and generalize some of the existing results in the literature including certain nyquist type consensus certificates for time delay systems moreover it allows the role of feedback in robustness against uncertainty to be better manifested within the context of synchronization

consensus reaching in multi agent packet switched networks with non linear coupling an important task for multi agent systems mas is to reach a consensus e g to align their velocity vectors recent results propose appropriate consensus protocols to achieve such tasks but most of them do not consider the effect of communication constraints such as the presence of time delays in the exchange of information between the agents in this article we provide conditions for a non linear locally passive mas of arbitrary size to reach a consensus when the agents communicate over a packet switched network that is characterised by a given topology both the cases of constant and switching topologies are considered the nature of the communication channel imposes constraints that are modelled using stochastic delays of arbitrary distribution we first embed this model in another distributed but deterministic delay model and provide conditions for the error introduced by this simplification in our main result we provide conditions for the locally passive mas with distributed delays to reach

chaotic attractors of a locally conservative hyperbolic map with overlap abstract we introduce a class of hyperbolic maps which are locally conservative but globally dissipative the dissipation stems not from local contraction of phase space but from the overlap that arises from the noninvertibility of the map the dynamics of these maps are chaotic and the attractors are strange although not necessarily fractal though several examples of this class are provided our analysis concentrates on a single example which is obtained as a modification of the baker s transform we discuss the morphology of the attractors as the map parameter is varied we find that some attractors are thin fractals and others are fat sets of positive measure we are able to characterize the parameter values that lead to these two kinds of attractors and for certain parameter values we can derive analytically exact values for the fractal dimension or for the fat fractal scaling exponent taken together these results lead us to conjecture that the generic attractor of this map has zero measure and full dimension

nonstationary parallel relaxed multisplitting methods abstract relaxed nonstationary multisplitting methods are studied for the parallel solution of nonsingular linear systems convergence results of the synchronous and asynchronous versions for systems with h matrices are presented computational results of these methods on a shared memory multiprocessor vector computer are reported these results show that nonstationary methods synchronous and asynchronous are better than the standard ones especially when the matrix of the linear system has a relatively small bandwidth moreover asynchronous versions always behave better than the synchronous ones

mas consensus and delay limits under delayed output feedback in this technical note we study the consensus problem for discrete time multi agent systems over an undirected fixed network communication graph focusing on the robustness of consensus with respect to communication delay we assume that the agents input is subject to a constant albeit possibly unknown time delay and is generated by a distributed dynamic output feedback control protocol drawing upon concepts and techniques from robust control notably those concerning gain margin optimization and analytic interpolation we derive explicit closed form conditions for general linear agents to achieve consensus our results display an explicit dependence of the consensus condition on the delay value as well as on the agent s unstable poles and non minimum phase zeros showing that delayed communication between agents will generally hinder consensus and impose restrictions on the network topology we also show that a lower bound on the maximal delay allowable for consensus can be computed by a simple line search method

input delay margin for consensusability of multi agent systems in this paper we are concerned with the consensus of multi agent systems with input delay among all standard static protocols that achieve the consensus for the multi agent system under no input delay we aim to find the maximum input delay such that the system remains consensusable under the same protocols in the case of continuous time systems in view of the continuity of stability with respect to the time delay the maximum delay margin for consensusability is given for scalar systems and vector systems with a single unstable open loop pole for scalar discrete time systems we show that the maximum delay margin for consensusability is strictly greater than zero if and only if the open loop pole of the system is located in a specified interval

asynchronous multisplitting two stage iterations for systems of weakly nonlinear equations for the large sparse systems of weakly nonlinear equations arising in the discretizations of many classical differential and integral equations this paper presents a class of asynchronous parallel multisplitting two stage iteration methods for getting their solutions by the high speed multiprocessor systems under suitable assumptions we study the global convergence properties of these asynchronous multisplitting two stage iteration methods moreover for this class of new methods we establish their local convergence theories and precisely estimate their asymptotic convergence factors under some reasonable assumptions when the involved nonlinear mapping is only assumed to be directionally differentiable numerical computations show that our new methods are feasible and efficient for parallely solving the system of weakly nonlinear equations

deviation bounds in multi agent systems described by undirected graphs the theory of port hamiltonian systems is used to derive upper bounds for the state deviations in multi agent systems described by undirected graphs pinned to a reference signal the upper bounds for the deviations in networks of first or second order agents respectively depend on the minimal eigenvalue of the extended laplacian of the system in networks of first order agents the deviations decay exponentially with a rate depending on the same minimal eigenvalue in case networks of second order systems meet specific design properties it can be shown that the deviations also decay exponentially with half the rate compared to first order systems

convergence of block iterative methods for linear systems arising in the numerical solution of euler equations we discuss block matrices of the forma a ij wherea ij is ak k symmetric matrix a ij is positive definite anda ij is negative semidefinite these matrices are natural block generalizations of z matrices and m matrices matrices of this type arise in the numerical solution of euler equations in fluid flow computations we discuss properties of these matrices in particular we prove convergence of block iterative methods for linear systems with such system matrices

synchronization of networks of heterogeneous agents with common nominal behavior this paper deals with the problem of synchronization in networks of heterogeneous agents with common nominal behavior the agents are modeled as possibly nonlinear perturbed versions of a common siso nominal lti operator and they are interconnected via a sparse memoryless interconnection operator coherent with the communication graph underlying the network the network is said to synchronize if the outputs of the agents tend to align along given directions the most important case being consensus or agreement the paper provides a general result based on iqcs that ensures synchronization of the network with robustness w r t uncertainties in the interconnection and in each agent s dynamics scalability issues are discussed in the scenario where the interconnection operator is a constant normal matrix yielding the generalization of the very popular linear consensus algorithm the wide range of applicability of the proposed criterion is shown by providing synchronization conditions in some important examples whenever possible graphical criteria are proposed for checking the required conditions

on structure oriented hybrid two stage iteration methods for the large and sparse blocked system of linear equations in this paper we first present a class of structure oriented hybrid two stage iteration methods for solving the large and sparse blocked system of linear equations as well as the saddle point problem as a special case and the new methods converge to the solution under suitable restrictions for instance when the coefficient matrix is positive stable matrix generally numerical experiments for a model generalized saddle point problem are given and the results show that our new methods are feasible and efficient and converge faster than the classical uzawa method

a class of parallel hybrid two stage iteration methods for block bordered linear systems a class of parallel hybrid two stage iteration method and its accelerated overrelaxation variant are established for solving the large sparse block bordered system of linear equations and their convergence are proved when the coefficient matrix of this linear system is an m matrix and an h matrix respectively

models of parallel chaotic iteration methods abstract we consider two models of parallel multisplitting chaotic iterations for solving large nonsingular systems of equations ax b in the first model each processor can carry out an arbitrary number of local iterations before the next global approximation to the solution is formed in the second model any processor can update the global approximation which resides in the central processor at any time this model is a generalization of a sequential iterative scheme due to ostrowski called the free steering group jacobi iterative scheme and a chaotic relaxation point iterative scheme due to chazan and miranker we show that when a is a monotone matrix and all the splittings are weak regular both models lead to convergent schemes

brief paper oscillation analysis of linearly coupled piecewise affine systems application to spatio temporal neuron dynamics this paper discusses oscillation analysis of a large number of linearly coupled piecewise affine pwa systems motivated by various kinds of reaction diffusion systems including cell signaling dynamics and neural dynamics we derive a sufficient condition under which the system shows an oscillatory behavior called y oscillation it is known that the analysis of pwa systems is difficult due to their switching nature an important feature of the result obtained is that under the assumption that every subsystem has a specific property in common the criteria can be rewritten in terms of coupling topology in an easily checkable way so it is applicable to large scale systems the results obtained are applied to theoretical investigation of the cardiac action potential generation propagation represented by spatio temporal fitzhugh nagumo equations

generalized multisplitting asynchronous iteration abstract we give a generalized asynchronous iteration based on the multisplitting of linear or nonlinear operators to solve systems of equations the condition for convergence can be imposed only on the components which actually enter the computation the results contain synchronous multisplitting iterations as a special case

consensus of networked mechanical systems with communication delays a unified framework this technical note addresses the consensus problem of networked uncertain mechanical systems which interact on directed graphs containing a spanning tree and are subjected to nonuniform communication delays the challenge lies in the unclear input output property of a linear networked system containing communication delays and the unclear convergent property of this system under an external input we establish a new input output property of this linear networked system and moreover its convergent property under an external input upon which we establish a unified framework to resolve the consensus problem of multiple mechanical systems the proposed consensus framework unifies extends the existing results and in addition yields a fully cascaded closed loop system with lyapunov like analysis and frequency domain input output analysis we show that the proposed unified consensus control scheme ensures consensus without the integral action of the sliding vector and scaled weighted average consensus with the integral action of the sliding vector simulation results are provided to demonstrate the performance of the proposed consensus schemes

iterated function systems and the global construction of fractals iterated function systems i f ss are introduced as a unified way of generating a broad class of fractals these fractals are often attractors for i f ss and occur as the supports of probability measures associated with functional equations the existence of certain p balanced measures for i f ss is established and these measures are uniquely characterized for hyperbolic i f ss the hausdorff besicovitch dimension for some attrac tors of hyperbolic i f ss is estimated with the aid of p balanced measures what appears to be the broadest framework for the exactly computable moment theory of p balanced measures that of linear i f ss and of probabilistic mixtures of iterated riemann surfaces is presented this extensively generalizes earlier work on orthogonal polynomials on julia sets an example is given of fractal reconstruction with the use of linear i f ss and moment theory

on a class of matrices which arise in the numerical solution of euler equations we study block matricesa aij where every blocka ij k k is hermitian anda ii is positive definite we call such a matrix a generalized h matrix if its block comparison matrix is a generalized m matrix these matrices arise in the numerical solution of euler equations in fluid flow computations and in the study of invariant tori of dynamical systems we discuss properties of these matrices and we give some equivalent conditions for a matrix to be a generalized h matrix

distributed adaptive consensus for multiple mechanical systems with switching topologies and time varying delay abstract this paper studies the adaptive consensus problem of networked mechanical systems with time varying delay and jointly connected topologies two different consensus protocols are proposed first we present an adaptive consensus protocol for the connected switching topologies based on graph theory lyapunov stability theory and switching control theory the stability of the proposed algorithm is demonstrated then we investigate the problem under the more general jointly connected topologies and with concurrent time varying communication delay the proposed consensus protocol consists of two parts one is for the connected agents which contains the current states disagreement among them and the other is designed for the isolated agents which contains the states difference between the current and past a distinctive feature of this work is to address the consensus control problem of mechanical systems with unknown parameters time varying delay and switching topologies in a unified theoretical framework numerical simulation is provided to demonstrate the effectiveness of the obtained results

consensus in multi agent systems with coupling delays and switching topology the robustness of consensus in single integrator multi agent systems mas to coupling delays and switching topologies is investigated it is shown that consensus is reached for arbitrarily large constant time varying or distributed delays if consensus is reached without delays this delay robustness holds under the weakest possible connectivity assumptions on the underlying graph i e as long as the graph is uniformly quasi strongly connected and switches with a dwell time the proof is based on a contraction argument and allows to remove technical assumptions that were used in previous publications moreover the result also applies to non scalar single integrators an important extension toward consensus and rendezvous in higher dimensions

average consensus in multi agent systems with uncertain topologies and multiple time varying delays abstract in this paper we investigate average consensus problem in continuous time multi agent systems with uncertain topologies as well as multiple time varying communication delays the network of dynamic agents is directed and both fixed and switching topologies are considered by using the linear matrix inequality method we prove that all the nodes in the network can achieve average consensus asymptotically for appropriate delays and uncertainties if the network topology is weakly connected and balanced feasible linear matrix inequalities are established to determine the maximal allowable upper bounds of multiple delays numerical examples are presented to illustrate the availability of the theoretical results

h consensus of second order multi agent systems with asymmetric delays abstract this paper is concerned with the problem of consensus in the h sense for second order continuous time multi agent systems with multiple asymmetric time varying delays by using a model transformation approach and matrix theory we establish several conditions in terms of linear matrix inequalities such that consensus of multi agent systems can be achieved in the h sense the feasibility of the consensus conditions is also analyzed as an application we consider the case of intermittent measurement between agents numerical examples are presented to illustrate the theoretical results which can be applied to the case of negative information weights

convergence theorems for parallel alternating iterative methods the parallel multisplitting nonstationary iterative model a was introduced by bru elsner and neumann linear algebra appl 103 1988 175 192 for solving nonsingular linear system ax b using a weak nonnegative multisplitting of the first type in this paper new results using a weak nonnegative multisplitting of the second type are introduced when a is a monotone matrix and using p regular multisplitting when a is a symmetric positive definite matrix combining model a and alternating iterative methods two new models of parallel multisplitting nonstationary iterations are introduced it is shown that when matrix a is monotone and the multisplittings are weak nonnegative of the first or second type both models lead to convergent schemes when matrix a is symmetric positive definite and the multisplittings are p regular the schemes are also convergent

spectral radii of certain iteration matrices and cycle means of digraphs abstract motivated by questions arising in the study of asynchronous iterative methods for solving linear systems we consider the spectral radius of products of certain one cycle matrices the spectral radius of a matrix in our class is a monotonic increasing function of the length of the cycle of the matrix but this is known to be false for products of such matrices the thrust of our investigation is to determine sufficient conditions under which the spectral radius of the product increases decreases when the lengths of the cycles of the factors increase decrease we also find sufficient conditions for the spectral radius of the product to be independent of the order of the factors our chief tool is an auxiliary directed weighted graph whose cycle means determine the eigenvalues of the matrix product and our main results are stated in terms of the maximal cycle mean of this graph

monotonic sequences and rates of convergence of asynchronized iterative methods abstract in a recent paper b vemmer and the authors investigated the effect of varying the number of processors on the rate of convergence of the asynchronized parallel block jacobi method associated with monotone matrices it was found that under certain simplifying assumptions increasing the number of processors in relation to the number of blocks or what comes to the same in more general settings the number of iteration operators slows down the convergence one interpretation for these results given in that paper was that increasing the number of processors means that when the current global approximation is updated by a local approximation from one of the processors that local approximation was computed from a much earlier global approximation received from the host node hence the slowdown in the rate of convergence the principal purpose of this paper is to remove some of the simplifying assumptions that were made in the above mentioned paper and to prove that many of the results there hold under much more general conditions our present assumptions do not yield a fixed iteration matrix which models the process as was the case previously this means that different tools have to be developed to establish results comparing the rate of convergence of two assynchronized processes

dynamical consensus seeking of second order multi agent systems based on delayed state compensation abstract by introducing delayed state compensation into the normal asynchronously coupled consensus algorithm a new consensus algorithm is constructed to solve the dynamical consensus problem of second order multi agent systems subjected to communication delay based on frequency domain analysis a sufficient and necessary condition which depends on the communication delay and the control parameters is obtained for two coupled second order dynamic agents converging to the dynamical consensus and it is proved that the proposed algorithm can tolerate higher communication delay than the synchronously coupled consensus algorithm according to linear fractional transformation and the small gain theorem sufficient conditions are gained for second order multi agent systems with communication delay under a general digraph simulation illustrates the correctness of the results

continuous time average consensus under dynamically changing topologies and multiple time varying delays abstract the average consensus in continuous time multi agent systems with dynamically changing topologies and multiple time varying delays is studied in this paper the network topology is captured by weighted digraphs which are weakly connected and balanced some feasible linear matrix inequalities are established to determine the allowable upper bounds of multiple delays that guarantee the average consensus of the system numerical examples are provided to show the usefulness of the theoretical results

on modified block ssor iteration methods for linear systems from steady incompressible viscous flow problems abstract in order to solve the large sparse systems of linear equations arising from numerical solutions of two dimensional steady incompressible viscous flow problems in primitive variable formulation we present block ssor and modified block ssor iteration methods based on the special structures of the coefficient matrices in each step of the block ssor iteration we employ the block lu factorization to solve the sub systems of linear equations we show that the block lu factorization is existent and stable when the coefficient matrices are block diagonally dominant of type ii by columns under suitable conditions we establish convergence theorems for both block ssor and modified block ssor iteration methods in addition the block ssor iteration and af adi method are considered as preconditioners for the nonsymmetric systems of linear equations numerical experiments show that both block ssor and modified block ssor iterations are feasible iterative solvers and they are also effective for preconditioning krylov subspace methods such as gmres and bicgstab when used to solve this class of systems of linear equations

a class of asynchronous multisplitting two stage iterations for large sparse block systems of weakly nonlinear equations abstract for the block system of weakly nonlinear equations ax g x where a r n n is a large sparse block matrix and g r n r n is a block nonlinear mapping having certain smoothness properties we present a class of asynchronous parallel multisplitting block two stage iteration methods in this paper these methods are actually the block variants and generalizations of the asynchronous multisplitting two stage iteration methods studied by bai and huang journal of computational and applied mathematics 93 1 1998 13 33 and they can achieve high parallel efficiency of the multiprocessor system especially when there is load imbalance under quite general conditions that a r n n is a block h matrix of different types and g r n r n is a block p bounded mapping we establish convergence theories of these asynchronous multisplitting block two stage iteration methods numerical computations show that these new methods are very efficient for solving the block system of weakly nonlinear equations in the asynchronous parallel computing environment

delay robustness in non identical multi agent systems we investigate the robustness of consensus protocols on sensor networks to different types of feedback delays we provide set valued conditions for consensus of linear multi agent systems mas with nonidentical agent dynamics and heterogeneous delays these conditions consider different feedback delay configurations and are robust and scalable to unknown arbitrary large topologies and unknown but bounded self delays and communication delays moreover we provide consensus conditions for relative degree two mas with different feedback delays in order to illustrate the applicability of these set valued conditions

relating the various scaling exponents used to characterize fat fractals in nonlinear dynamical systems abstract fat fractals sets with fractal structure but nonzero measure are becoming increasingly important in nonlinear dynamical systems these sets have allowed quantitative analyses of sensitivity to parameters final state sensitivity and quantum chaos and have played important roles in discussions of nonlinear twist maps one dimensional maps with quadratic maxima arnol d tongues in circle maps and practal basin boundaries unfortunately since fat fractals have nonzero measure they must have the same integer dimension as the underlying space therefore their dimension is insensitive to the fractal structure and does not provide an appropriate means of characterization hence several scaling exponents have been suggested for this purpose and there has been some debate over which is best also there has been considerable confusion over how these exponents differ and whether they are simply related we shed light on this issue by examining these exponents and their properties and deriving relationships between them

asynchronous two stage iterative methods parallel block two stage iterative methods for the solution of linear systems of algebraic equations are studied convergence is shown for monotone matrices and for h matrices two different asynchronous versions of these methods are considered and their convergence investigated

new comparison results for parallel multisplitting iterative methods abstract the parallel multisplitting nonstationary iterative model a and models 1 2 were introduced by bru et al bru elsner neumann models of parallel chaotic iteration methods linear algebra appl 103 1988 175 192 and climent et al climent perea tortosa zamora convergence theorems for parallel alternating iterative methods appl math comput 148 2004 497 517 respectively using nonnegative splitting theory we present some new comparison theorems between two general parallel multisplittings of model a and models 1 2 when the coefficient matrix is a monotone matrix particularly we establish some sufficient conditions for strict inequalities in comparisons of spectral radii of different iteration matrices

on the convergence of parallel chaotic nonlinear multisplitting newton type methods abstract a class of parallel chaotic nonlinear multisplitting newton type methods for solving the nonlinear system of equations f x 0 f d r n r n is established and its local convergence theory is presented

design of a recursive shape controlling mesh generator a recursive shape controlling triangulation method is described the method is designed to produce a labelling which implies reduced fill in the solution of finite element equations assembled from such a triangulation and allows simple implementation of a nested disection algorithm for irregular domains this approach saves a substantial amount of time usually spent on discovering a suitable relabelling of the triangulation in addition the matrix of the resulting system is then endowed with a recursive doubly bordered block diagonal form this allows us to develop a recursive parallel bisection method for the solution of the system of equations

on the convergence of iterative methods for solving singular linear systems in this paper we first show that for the stationary iterative methods for solving consistent singular linear systems the convergence and the quotient convergence are equivalent then we use this assertion to analyze the convergence of the multisplitting algorithms for the solution of linear systems when the coefficient matrices of the linear systems are singular hermitian positive semidefinite and the linear systems are consistent

reaching a consensus in a dynamically changing environment a graphical approach this paper presents new graph theoretic results appropriate for the analysis of a variety of consensus problems cast in dynamically changing environments the concepts of rooted strongly rooted and neighbor shared are defined and conditions are derived for compositions of sequences of directed graphs to be of these types the graph of a stochastic matrix is defined and it is shown that under certain conditions the graph of a sarymsakov matrix and a rooted graph are one and the same as an illustration of the use of the concepts developed in this paper graph theoretic conditions are obtained which address the convergence question for the leaderless version of the widely studied vicsek consensus problem

on the convergence of asynchronous paracontractions with application to tomographic reconstruction from incomplete data abstract convergence of iterative processes in c k of the form x i r i j i x 1 r i 1 1 p j i x i where j i 1 2 n i 1 2 is analyzed it is shown that if the matrices p 1 p n are paracontracting in the same smooth strictly convex norm and if the sequence j i i 1 has certain regularity properties then the above iterates converge this result implies the convergence of a parallel asynchronous implementation of the algebraic reconstruction technique art algorithm often used in tomographic reconstruction from incomplete data

a unified framework for the construction of various matrix multisplitting iterative methods for large sparse system of linear equations abstract a unified framework for the construction of various synchronous and asynchronous parallel matrix multisplitting iterative methods suitable to the simd and mimd multiprocessor systems respectively is presented and its convergence theory is established under rather weak conditions these afford general method models and systematical convergence criterions for studying the parallel iterations in the sense of matrix multisplitting in addition how the known parallel matrix multisplitting iterative methods can be classified into this new framework and what novel ones can be generated by it are shown in detail

exact stability analysis of second order leaderless and leader follower consensus protocols with rationally independent multiple time delays abstract an investigation of double integrator agents with directed asymmetric consensus protocols and multiple rationally independent time delays is presented in this paper from two novel perspectives first we complement the group consensus literature on crucial stability analysis using a recent technique called the cluster treatment of characteristic roots ctcr for the first time on this class of time delayed systems the ctcr paradigm is pursued after a block diagonalization mode decoupling transformation on the system this treatment produces some unique stability tables for the dynamics in the space of the delays which are non conservative and exhaustive second a novel concept of spectral delay space is presented as an overture to the ctcr for the determination of the complete set of stability crossing switching hypersurfaces in the delay space examples are provided to display the strengths and efficiency of this new stability analysis mechanism

on the convergence of nonstationary multisplitting two stage iteration methods for hermitian positive definite linear systems convergence properties of the nonstationary multisplitting two stage iteration methods for solving large sparse system of linear equations are further studied when the coefficient matrices are hermitian positive definite matrices

stability of asynchronous two dimensional fornasini marchesini dynamical systems abstract conditions for stability of two dimensional 2d fornasini marchesini fm models with variable shifts are derived using nonnegative matrix theory the conditions are shift independent the stability is proved under a general boundary condition bc

on stein rosenberg type theorems for nonnegative and perron frobenius splittings abstract the stein rosenberg theorem is extended and generalized to the class of nonnegative splittings a m 1 n 1 m 2 n 2 as well as to the most generalized class of perron frobenius splittings two types of the stein rosenberg theorem are stated and proved for both classes these theorems allow us to obtain comparison results for the rate of convergence of the associated iterative methods specific assumptions are given under which the inequalities of the spectral radii become equalities or strict inequalities the theoretical results are confirmed by numerical examples

consensus problems in networks of agents with double integrator dynamics and time varying delays in this article we consider consensus problems in networks of agents with double integrator dynamics and non uniform time varying delays the agent dynamics is adopted as a typical point mass model based on newton s law without assuming that the weighting factors in the information update schemes are non negative we propose two protocols such that both the state and the velocity of agents achieve consensus an equivalent reduced order system is introduced to analyse the convergence of the protocols some necessary and or sufficient conditions for consensus are established in terms of linear matrix inequalities simulation results are provided that demonstrate the effectiveness of our theoretical results

on the convergence of the parallel multisplitting aor algorithm abstract we present a class of relaxed parallel multisplitting algorithms called the parallel multisplitting aor algorithm for solving large nonsingular systems of equations ax b this new algorithm is a generalization and improvement of the relaxed parallel multisplitting method linear algebra appl 119 141 152 1989 based on the new algorithm model we establish another algorithm called the relaxed parallel multisplitting aor algorithm the convergence of these algorithms is discussed under the condition that a is a monotone matrix we obtain corresponding convergence results these convergence conditions are convenient to verify

convergence and comparison theorems for a generalized alternating iterative method using as a principal tool the convergence results of standard iterative process for the solution of linear systems alternating iterative methods are studied we extend the convergence theorem for the stationary alternating iterative method of benzi and szyld numererische mathematik 76 1997 309 for weak nonnegative splittings of the first type of a monotone matrix to weak nonnegative splitting of the second type on the other hand we introduce a more general method the nonstationary alternating iterative method establishing convergence results for weak nonnegative splittings of a monotone matrix and for p regular splittings of a symmetric positive definite matrix

delay robustness in consensus problems we investigate the robustness of consensus schemes for linear multi agent systems mas to feedback delays to achieve this we develop a unified framework that considers linear mas models with different feedback delays e g affecting only the neighbor s output or affecting both the agent s own and its neighbors output this framework has the advantage of providing scalable simple and accurate set valued conditions for consensus using these set valued conditions previous results on consensus in mas with delays can be recovered and generalized moreover we use them to derive conditions for the convergence rate of single integrator mas with feedback delays finally building on this framework we propose a scalable delay dependent design algorithm for consensus controllers for a large class of linear mas

asynchronous parallel methods for enclosing solutions of nonlinear equations we consider interval arithmetic based parallel methods for enclosing solutions of nonlinear systems of equations where processors are allowed to proceed asynchronously we present a general study on the convergence of asynchronous iterations in interval spaces and apply them to a variety of methods for the nonlinear equations case the convergence results turn out to be very similar to those known for synchronous methods several practical examples on shared memory architectures are included the asynchronous methods sometimes perform substantially better than their synchronous counterparts