on the spaces lp x ω and wm p x ω in this paper we present some basic results on the generalized lebesgue spaces l p x omega and generalized lebesgue sobolev spaces w m w p x omega these results provide the necessary framework for the study of variational problems and elliptic equations with non standard p x growth conditions c 2001 academic press

existence of solutions for a p x kirchhoff type equation this paper is concerned with the existence and multiplicity of solutions to a class of p x kirchhoff type problem with dirichlet boundary data by means of a direct variational approach and the theory of the variable exponent sobolev spaces we establish conditions ensuring the existence and multiplicity of solutions for the problem

infinitely many solutions for a hemivariational inequality involving the p x laplacian abstract this paper deals with a hemivariational inequality involving the p x laplacian by using nonsmooth variational methods and the theory of the variable exponent sobolev spaces we establish the existence of infinitely many distinct non negative solutions of the problem in both cases one at the origin and the other at infinity under suitable hypotheses

distributed average consensus with dithered quantization in this paper we develop algorithms for distributed computation of averages of the node data over networks with bandwidth power constraints or large volumes of data distributed averaging algorithms fail to achieve consensus when deterministic uniform quantization is adopted we propose a distributed algorithm in which the nodes utilize probabilistically quantized information i e dithered quantization to communicate with each other the algorithm we develop is a dynamical system that generates sequences achieving a consensus at one of the quantization values almost surely in addition we show that the expected value of the consensus is equal to the average of the original sensor data we derive an upper bound on the mean square error performance of the probabilistically quantized distributed averaging pqda moreover we show that the convergence of the pqda is monotonic by studying the evolution of the minimum length interval containing the node values we reveal that the length of this interval is a monotonically nonincreasing function with limit zero we also demonstrate that all the node values in the worst case converge to the final two quantization bins at the same rate as standard unquantized consensus finally we report the results of simulations conducted to evaluate the behavior and the effectiveness of the proposed algorithm in various scenarios

infinitely many solutions for a differential inclusion problem in r n abstract in this paper we consider the differential inclusion problem di p u u p 2 u x f u x x r n u w 1 p r n where 2 n p l 1 r n l r n is radially symmetric and f stands for the generalized gradient of a locally lipschitz function f r r under suitable oscillatory assumptions on the potential f at zero or at infinity we show the existence of infinitely many radially symmetric solutions of di no symmetry requirement on f is needed our approach is based on a non smooth ricceri type variational principle developed by marano and motreanu j differential equations 182 2002 108 120

convergence of infinite products of stochastic matrices a graphical decomposition criterion this technical note presents a convergence criterion for infinite products of stochastic matrices which is based on graphical decomposition of the associated graphs we show that if the associated graphs of a set of stochastic matrices share a common graphical decomposition and the corresponding reduced graphs are rooted then any infinite products of the given set of stochastic matrices is convergent specifically we propose a numerical algorithm for finding the common graphical decomposition of the associated graphs which has been proved to be polynomial time fast the proposed criterion can be applied directly to a series of classical results in distributed coordination algorithm

about the number of c words of form let p i n denote the number of the c words of form email protected xw with length 2n i and gap i where i is the length of word x in this paper we prove the following result p i n 6 for all n 1 and i 2 3 4 moreover we provide a complete solution of email protected email protected c b w for x 4

eigenvalues of p x laplacian dirichlet problem abstract this paper studies the eigenvalues of the p x laplacian dirichlet problem div u p x 2 u u p x 2 u in u 0 on where is a bounded domain in rn and p x is a continuous function on such that p x 1 we show that the set of eigenvalues is a nonempty infinite set such that sup we present some sufficient conditions for inf 0 and for inf 0 respectively

finite time semistability and consensus for nonlinear dynamical networks this paper focuses on semistability and finite time stability analysis and synthesis of systems having a continuum of equilibria semistability is the property whereby the solutions of a dynamical system converge to lyapunov stable equilibrium points determined by the system initial conditions in this paper we merge the theories of semistability and finite time stability to develop a rigorous framework for finite time semistability in particular finite time semistability for a continuum of equilibria of continuous autonomous systems is established continuity of the settling time function as well as lyapunov and converse lyapunov theorems for semistability are also developed in addition necessary and sufficient conditions for finite time semistability of homogeneous systems are addressed by exploiting the fact that a homogeneous system is finite time semistable if and only if it is semistable and has a negative degree of homogeneity unlike previous work on homogeneous systems our results involve homogeneity with respect to semistable dynamics and require us to adopt a geometric description of homogeneity finally we use these results to develop a general framework for designing semistable protocols in dynamical networks for achieving coordination tasks in finite time

combinatorial properties of smooth infinite words we describe some combinatorial properties of an intriguing class of infinite words called smooth connected with the kolakoski one k defined as the fixed point of the run length encoding it is based on a bijection on the free monoid over 1 2 that shows some surprising mixing properties all words contain the same finite number of square factors and consequently they are cube free this suggests that they have the same complexity as confirmed by extensive computations we further investigate the occurrences of palindromic subwords finally we show that there exist smooth words obtained as fixed points of substitutions realized by transducers as in the case of k

infinitely many solutions for a neumann type differential inclusion problem involving the p x laplacian abstract in this paper we consider a differential inclusion problem involving the p x laplacian of the type div u p x 2 u x u p x 2 u f x u g x u in u 0 on we prove the existence of infinitely many solutions of this problem under suitable hypotheses by applying a non smooth ricceri type variational principle and the theory of the variable exponent sobolev spaces

how to decide consensus a combinatorial necessary and sufficient condition and a proof that consensus is decidable but np hard a set of stochastic matrices cal p is a consensus set if for every sequence of matrices p 1 p 2 ldots whose elements belong to cal p and every initial state x 0 the sequence of states defined by x t p t p t 1 cdots p 1 x 0 converges to a vector whose entries are all identical in this paper we introduce an avoiding set condition for compact sets of matrices and prove in our main theorem that this explicit combinatorial condition is both necessary and sufficient for consensus we show that several of the conditions for consensus proposed in the literature can be directly derived from the avoiding set condition the avoiding set condition is easy to check with an elementary algorithm and so our result also establishes that consensus is algorithmically decidable direct verification of the avoiding set condition may require more than a polynomial time number of operations this is however likely to be the case for any consensus checking algorithm since we also prove in th

consensus on complete riemannian manifolds in finite time abstract in view of the fact that prototypes of riemannian manifolds range from euclidean surfaces to lie groups operator spaces etc the paper explores common consensus schemes for connected complete riemannian manifolds after studying the restriction exponential map we introduce a local consensus protocol with switching communication topologies for the manifolds via the isometric mapping based on one of the euclidean consensus protocols equipped with the compression decompression functions we extend the local consensus protocol and put forth a global consensus algorithm for any such manifolds these consensus schemes can provide common solutions to the consensus problems arising from various nonlinear spaces or abstract spaces belonging to connected complete riemannian manifolds in particular they are simple and concise and can converge in finite time simulation on the unit sphere is provided to verify our consensus schemes and to demonstrate the usage of our proposed techniques

a note on the complexity of c words let c n be the number of c words of length n say that a c word w is left doubly extendable lde if both 1w and 2w are c we show that for any positive real number f and positive integer n such that the proportion of 2 s is greater than email protected in each lde word of length exceeding n there are positive constants c 1 and c 2 such that c 1n l o g 3 l o g 3 2 f 2 n c n

smooth words on 2 letter alphabets having same parity in this paper we consider smooth words over 2 letter alphabets a b where a b are integers having same parity with 0

existence and multiplicity of solutions for p x laplacian equations in rn abstract this paper investigates the existence and multiplicity of solutions for p x laplacian equations in r n

finite time consensus using stochastic matrices with positive diagonals we discuss the possibility of reaching consensus in finite time using only linear iterations with the additional restrictions that the update matrices must be stochastic with positive diagonals and consistent with a given graph structure we show that finite time average consensus can always be achieved for connected undirected graphs for directed graphs we show some necessary conditions for finite time consensus including strong connectivity and the presence of a simple cycle of even length

feedback stabilization and lyapunov functions given a locally defined nondifferentiable but lipschitz lyapunov function we employ it in order to construct a discontinuous feedback law which stabilizes the underlying system to any given tolerance a converse result shows that suitable lyapunov functions of this type exist under mild assumptions we also establish that the feedback in question possesses a robustness property relative to measurement error despite the fact that it may not be continuous

finite time weighted average consensus with respect to a monotonic function and its application abstract the consensus state is an important and fundamental quantity for consensus problems of multi agent systems which indicates where all the dynamical agents reach in this paper weighted average consensus with respect to a monotonic function which means that the trajectories of the monotonic function along the state of each agent reach the weighted average of their initial values is studied for a group of kinematic agents with time varying topology by constructing a continuous nonlinear distributed protocol such a consensus problem can be solved in finite time even though the time varying topology involves unconnected graphs then the distributed protocol is employed to compute the maximum likelihood estimation of unknown parameters over sensor networks compared with the existing results the estimate scheme proposed here may reduce the costs of data communication storage memory book keeping and computational overheads

step size sequence design for finite time average consensus in secure wireless sensor networks this paper concerns the study of average consensus in wireless sensor networks with aim of providing a way to reach consensus in a finite number of steps in particular we investigate the design of consensus protocols when for security reasons for instance the underlying graph is constrained to be strongly regular or distance regular the proposed design method is based on parameters of the intersection array characterizing the underlying graph with this protocol at execution time average consensus is achieved in a number of steps equal to the diameter of the graph i e the smallest possible number of steps to achieve consensus

finite time information consensus for multi agent systems with fixed and switching topologies in this paper we study the consensus problems for a group of interacting agents first we analytically establish the explicit expression of the consensus state for the entire group second we prove that the agents of the group under a particular type of nonlinear interaction can reach the consensus state in finite time in the scenarios with fixed and switching undirected topologies the results are also extended to the case where the topology of the group is directed and satisfies a detailed balance condition on coupling weights third some numerical examples are provided to analyze the influencing factors of the convergence time that is the parameter of the particular interaction function and the algebraic connectivity of graphs finally an application of the theoretical results in sensor networks is given namely computing the maximum likelihood estimate of unknown parameters

coordination and geometric optimization via distributed dynamical systems this paper discusses dynamical systems for disk covering and sphere packing problems we present facility location functions from geometric optimization and characterize their differentiable properties we design and analyze a collection of distributed control laws that are related to nonsmooth gradient systems the resulting dynamical systems promise to be of use in coordination problems for networked robots in this setting the distributed control laws correspond to local interactions between the robots the technical approach relies on concepts from computational geometry nonsmooth analysis and the dynamical system approach to algorithms

smooth words over arbitrary alphabets smooth infinite words over 1 2 are connected to the kolakoski word k 221121 defined as the fixpoint of the function that counts the length of the runs of 1 s and 2 s in this paper we extend the notion of smooth words to arbitrary alphabets and study some of their combinatorial properties using the run length encoding every word is represented by a word obtained from the iterations of in particular we provide a new representation of the infinite fibonacci word f as an eventually periodic word on the other hand the thue morse word is represented by a finite one

a note on differentiable palindromes abstract we give a characterization of the palindromes in a class of infinite words over 1 2 related to the kolakoski word k this characterization based on the left palindromic closure of all prefixes of k is obtained by using a bijection between the class of right infinite words over and a class of words over the same alphabet and reveals the first link between the existence of some palindromes and the recurrence of k indeed the existence of arbitrarily long palindromes implies the recurrence of k and a stronger assumption implies the closure of the set of its factors by permutation of the letters in

on primitivity of sets of matrices a nonnegative matrix a is called primitive if a k is positive for some integer k 0 a generalization by protasov and voynov 2012 of this concept to finite sets of matrices is as follows a set of matrices m a 1 a 2 a m is primitive if a i 1 a i 2 a i k is positive for some indices i 1 i 2 i k the concept of primitive sets of matrices comes up in a number of problems within the study of discrete time switched systems in this paper we analyze the computational complexity of deciding if a given set of matrices is primitive and we derive bounds on the length of the shortest positive product we show that while primitivity is algorithmically decidable unless p n p it is not possible to decide primitivity of a matrix set in polynomial time moreover we show that the length of the shortest positive sequence can be superpolynomial in the dimension of the matrices on the other hand defining p to be the set of matrices with no zero rows or columns we give a combinatorial proof of the protasov voynov characterization 2012 of primitivity for matrices in p which can be tested in polynomial time this latter observation is related to the well known 1964 conjecture of ern on synchronizing automata in fact any bound on the minimal length of a synchronizing word for synchronizing automata immediately translates into a bound on the length of the shortest positive product of a primitive set of matrices in p in particular any primitive set of n n matrices in p has a positive product of length o n 3

the complexity of c bw words of the form w w let p n denote the number of c b w words of the form w xw with gap n and p n k denote the number of c words of the form w xw with length 2k n and gap n where n is the length of the word x s brlek a ladouceur a note on differentiable palindromes theoret comput sci 302 2003 167 178 proved that c palindromes are characterized by the left palindromic closure of the prefixes of the well known kolakoski sequences and revealed an interesting perspective for understanding some of the conjectures in fact they found all infinite c palindromes and established p 0 k p 1 k 2 for all k n where n is the set of positive integers y b huang about the number of c words of form w xw theoret comput sci 393 2008 280 286 obtained p n k 6 for all k n and n 2 3 4 and gave all c b w words of the form w xw with gap less than 5 which imply p n 2 for n 0 1 and p n 6 for n 2 3 4 in this paper we prove the following intriguing results 1 if w xw c b w and x 7 then the first and last letters of the word x are the same 2 p n 14 for n 5 3 for every positive integer n there exists a positive integer h n such that for all k n if k h n then p n k p 5 k if k is odd and p n k p 6 k if k is even which would help us understand better the complexity of finite c words of the form w xw moreover we provide all twenty eight c b w words of the form w xw

distributed average tracking for reference signals with bounded accelerations in this technical note the distributed average tracking dat problem for reference signals with bounded accelerations is considered a sliding mode surface is designed such that trajectories on the surface will achieve dat a discontinuous control algorithm is then proposed to guarantee that the surface can be reached in finite time it is shown that if the switching network topology is connected at all time then dat can be solved by properly choosing a gain parameter of the algorithm

finite time consensus with disturbance rejection by discontinuous local interactions in directed graphs in this technical note we propose a decentralized discontinuous interaction rule which allows to achieve consensus in a network of agents modeled by continuous time first order integrator dynamics affected by bounded disturbances the topology of the network is described by a directed graph the proposed discontinuous interaction rule is capable of rejecting the effects of the disturbances and achieving consensus after a finite transient time an upper bound to the convergence time is explicitly derived in the technical note simulation results referring to a network of coupled kuramoto like oscillators are illustrated to corroborate the theoretical analysis

variational methods for non differentiable functionals and their applications to partial differential equations the interest in extending the theory of nonlinear partial differential equations pde to pde with discontinuous nonlinearities dnde has been increasing daily the reasons are twofold 1 many free boundary problems may be reduced to boundary value problems of dnde cf chang 16 71 chang and jiang 9 and fleishman and mahar 121 2 sometimes in dealing with a pde problem it is beneficial to put the original pde into a large category for instance dnde cf chang 7 simply speaking a dnde is as follows

distributed continuous time optimization nonuniform gradient gains finite time convergence and convex constraint set in this paper a distributed optimization problem with general differentiable convex objective functions is studied for continuous time multi agent systems with single integrator dynamics the objective is for multiple agents to cooperatively optimize a team objective function formed by a sum of local objective functions with only local interaction and information while explicitly taking into account nonuniform gradient gains finite time convergence and a common convex constraint set first a distributed nonsmooth algorithm is introduced for a special class of convex objective functions that have a quadratic like form it is shown that all agents reach a consensus in finite time while minimizing the team objective function asymptotically second a distributed algorithm is presented for general differentiable convex objective functions in which the interaction gains of each agent can be self adjusted based on local states a corresponding condition is then given to guarantee that all agents reach a consensus in finite time while minimizing the team objective function asymptotically third a distributed optimization algorithm with state dependent gradient gains is given for general differentiable convex objective functions it is shown that the distributed continuous time optimization problem can be solved even though the gradient gains are not identical fourth a distributed tracking algorithm combined with a distributed estimation algorithm is given for general differentiable convex objective functions it is shown that all agents reach a consensus while minimizing the team objective function in finite time fifth as an extension of the previous results a distributed constrained optimization algorithm with nonuniform gradient gains and a distributed constrained finite time optimization algorithm are given it is shown that both algorithms can be used to solve a distributed continuous time optimization problem with a common convex constraint set numerical examples are included to illustrate the obtained theoretical results

sobolev embedding theorems for spaces wk p x ω this paper gives a sobolev type embedding theorem for the generalized lebesgue sobolev space wk p x where is an open domain in rn n 2 with cone property and p x is a lipschitz continuous function defined on satisfying 1 p p nk the main result can be stated as follows for any measurable function q x x formula with p x q x p x np x n kp x there exists a continuous embedding from wk p x to lq x

kolakoski 2m 2n are limit periodic model sets we consider generalized kolakoski sequences on an alphabet with two even numbers they can be related to a primitive substitution rule of constant length l using this connection we prove that they have pure point dynamical and pure point diffractive spectrum where we make use of the strong interplay between these two concepts since these sequences can then be described as model sets with l adic internal space we add an approach to visualize such internal spaces

arbitrarily many solutions for a perturbed p x laplacian equation involving oscillatory terms this paper deals with a perturbed p x laplacian equation involving oscillatory terms div u p x email protected u u p x 2u q x f u email protected u email protected r n u 0 u x 0as x by using variational methods and the non smooth version symmetric criticality principle we establish a the unperturbed problem p 0 has infinitely many distinct solutions b the number of distinct solutions for p e becomes greater and greater whenever e is smaller and smaller our results are a generalization of the case of laplacian from kristaly to the case of p x laplacian

min max time consensus tracking with communication guarantee in this paper a collection of double integrator agents with bounded inputs is considered communication is possible between any two agents only if the inter agent distance is less than a fixed threshold a special node referred to as the leader generates an unknown reference trajectory to which all the other agents are required to converge in the shortest possible time assuming the initial communication graph to be connected a directed spanning tree rooted at the leader is identified using a local algorithm the dynamics of any two agents connected by an edge in the selected tree are modeled as a time optimal pursuit evasion game while maintaining the inter agent communication link using the corresponding feedback saddle point strategies local min max time control laws for each pair of agents are formulated for the selected tree the proposed collection of local min max time control strategies is shown to be the communication preserving global min max time strategy

on the sub supersolution method for p x laplacian equations abstract this paper deals with the sub supersolution method for the p x laplacian equations a sub supersolution principle for the dirichlet problems involving the p x laplacian is established it is proved that the local minimizers in the c 1 topology are also local minimizers in the w 1 p x topology for given energy functionals a strong comparison theorem for the p x laplacian equations is presented some applications of the abstract theorems obtained in this paper to the eigenvalue problems for the p x laplacian equations are given

existence of solutions for p x laplacian dirichlet problem this paper presents several sufficient conditions for the existence of solutions for the dirichlet problem of p x laplacian div u p x 2 u f x u x u 0 x especially an existence criterion for infinite many pairs of solutions for the problem is obtained the discussion is based on the theory of the spaces lp x and w01 p x

decentralised minimum time consensus we consider the discrete time dynamics of a network of agents that exchange information according to a nearest neighbour protocol under which all agents are guaranteed to reach consensus asymptotically we present a fully decentralised algorithm that allows any agent to compute the final consensus value of the whole network in finite time using the minimum number of successive values of its own state history we show that the minimum number of steps is related to a jordan block decomposition of the network dynamics and present an algorithm to compute the final consensus value in the minimum number of steps by checking a rank condition of a hankel matrix of local observations furthermore we prove that the minimum number of steps is related to graph theoretical notions that can be directly computed from the laplacian matrix of the graph and from the minimum external equitable partition

infinitely many solutions for a differential inclusion problem in rn involving the p x laplacian abstract in this paper we consider the differential inclusion problem in r n involving the p x laplacian of the type div u p x 2 u u p x 2 u f x u in r n u w 1 p x r n we prove the existence of infinitely many radially symmetric solutions of the problem under suitable hypotheses by applying a nonsmooth variational principle due to marano and motreanu and the theory of the variable exponent sobolev spaces

graph diameter eigenvalues and minimum time consensus we consider the problem of achieving average consensus in the minimum number of linear iterations on a fixed undirected graph we are motivated by the task of deriving lower bounds for consensus protocols and by the so called definitive consensus conjecture which states that for an undirected connected graph g with diameter d there exist d matrices whose nonzero pattern complies with the edges in g and whose product equals the all ones matrix our first result is a counterexample to the definitive consensus conjecture which is the first improvement of the diameter lower bound for linear consensus protocols we then provide some algebraic conditions under which this conjecture holds which we use to establish that all distance regular graphs satisfy the definitive consensus conjecture

generalized sarymsakov matrices within the set of stochastic indecomposable aperiodic sia matrices the class of sarymsakov matrices is the largest known subset that is closed under matrix multiplication and more critically whose compact subsets are all consensus sets this paper shows that a larger subset with these two properties can be obtained by generalizing the standard definition for sarymsakov matrices the generalization is achieved by introducing the notion of the sia index of a stochastic matrix whose value is 1 for sarymsakov matrices and then exploring those stochastic matrices with larger sia indices in addition to constructing the larger set this paper introduces another class of generalized sarymsakov matrices which contains matrices that are not sia and studies their products sufficient conditions are provided for an infinite product of matrices from this class converging to a rank one matrix finally as an application of the results just described and to confirm their usefulness a necessary and sufficient combinatorial condition the avoiding set condition for deciding whether or not a compact set of stochastic matrices is a consensus set is revisited in addition a necessary and sufficient combinatorial condition is established for deciding whether or not a compact set of doubly stochastic matrices is a consensus set

distributed model predictive control of linear discrete time systems with local and global constraints abstract this paper proposes a distributed model predictive control dmpc approach for a family of discrete time linear systems with local uncoupled and global coupled constraints the proposed approach is based on the dual problem of a mpc optimization problem involving all systems this dual problem is then distributedly solved based on the alternating direction multiplier method admm with several known simplifications when the network of systems is large or sparsely connected the computation of the optimal control using admm can be expensive the proposed approach mitigates this problem by allowing early termination of the admm process this is made possible via a finite time consensus algorithm that determines the satisfaction of the termination condition and by appropriate tightening of the coupled constraints under reasonable assumptions the approach is guaranteed to converge to a small neighborhood of the optimal so long as the network is connected recursive feasibility and exponential stability of the closed loop system are shown the performance of the proposed approach is demonstrated by a numerical example

tight bound for deciding convergence of consensus systems abstract we analyze the asymptotic convergence of all infinite products of matrices taken in a given finite set by looking only at finite or periodic products it is known that when the matrices of the set have a common nonincreasing polyhedral norm all infinite products converge to zero if and only if all infinite periodic products with period smaller than a certain value converge to zero moreover bounds on that value are available lagarias and wang 1995 we provide a stronger bound that holds for both polyhedral norms and polyhedral seminorms in the latter case the matrix products do not necessarily converge to zero but all trajectories of the associated system converge to a common invariant subspace we prove that our bound is tight for all seminorms our work is motivated by problems in consensus systems where the matrices are stochastic nonnegative with rows summing to one and hence always share a same common nonincreasing polyhedral seminorm in that case we also improve existing results

global consensus for discrete time multi agent systems with input saturation constraints in this paper we consider the global consensus problem for discrete time multi agent systems with input saturation constraints under fixed undirected topologies we first give necessary conditions for achieving global consensus via a distributed protocol based on relative state measurements of the agent itself and its neighboring agents we then focus on two special cases where the agent model is either neutrally stable or a double integrator for the neutrally stable case any linear protocol of a particular form which solves the consensus problem for the case without input saturation constraints also solves the global consensus problem for the case with input saturation constraints for the double integrator case we show that a subset of linear protocols which solve the consensus problem for the case without saturation constraints also solve the global consensus problem for the case with input saturation constraints the results are illustrated by numerical simulations

on the stabilization in finite time of locally controllable systems by means of continuous time varying feedback law it is proven that if for any positive time t there exists an open loop control u a t depending continuously on the initial data a vanishing for a 0 and steering a small neighborhood of 0 into 0 in time t then the control system can be locally stabilized in small time by means of continuous time varying feedback law provided that the dimension of the state space is at least 4 and the strong accessibility rank condition holds

infinitely many positive solutions for a p x kirchhoff type equation this paper is concerned with the existence of infinitely many positive solutions to a class of p x kirchhoff type problem by means of a direct variational approach and the theory of the variable exponent sobolev spaces we establish the existence of infinitely many distinct positive solutions whose w1 p x norms and l norms tend to zero under suitable hypotheses about nonlinearity

convergence speed in distributed consensus and averaging we study the convergence speed of distributed iterative algorithms for the consensus and averaging problems with emphasis on the latter we first consider the case of a fixed communication topology we show that a simple adaptation of a consensus algorithm leads to an averaging algorithm we prove lower bounds on the worst case convergence time for various classes of linear time invariant distributed consensus methods and provide an algorithm that essentially matches those lower bounds we then consider the case of a time varying topology and provide a polynomial time averaging algorithm

stability analysis of swarms in this note we specify an individual based continuous time model for swarm aggregation in n dimensional space and study its stability properties we show that the individuals autonomous agents or biological creatures will form a cohesive swarm in a finite time moreover we obtain an explicit bound on the swarm size which depends only on the parameters of the swarm model

finite time distributed consensus via binary control protocols this paper investigates the finite time distributed consensus problem for multi agent systems using a binary consensus protocol and the pinning control scheme compared with other consensus algorithms which need the complete state or output information of neighbors the proposed algorithm only requires sign information of the relative state measurements that is the differences between a node s state and that of its neighbors this corresponds to only requiring a single bit quantization error relative to each neighbor this signum protocol is realistic in terms of observed behavior in animal groups where relative motion is determined not by full time signal measurements but by coarse estimates of relative heading differences between neighbors the signum protocol does not require explicit measurement of time signals from neighbors and hence has the potential to significantly reduce the requirements for both computation and sensing analysis of discontinuous dynamical systems is used including the filippov solutions and set valued lie derivative based on the second order information on the evolution of lyapunov functions the conditions that guarantee the finite time consensus for the systems are identified numerical examples are given to illustrate the theoretical results

infinitely many non negative solutions for a p x kirchhoff type problem with dirichlet boundary condition abstract in this paper we consider the dirichlet problem involving the p x kirchhoff type a b 1 p x u p x d x div u p x 2 u f x u in u 0 on we prove the existence of infinitely many non negative solutions of the problem by applying a general variational principle due to b ricceri and the theory of the variable exponent sobolev spaces

optimal strategies in the average consensus problem let a set of communicating agents compute the average of their initial positions where every agent is restricted to communicate to a given small number of other agents average consensus problem we prove that the optimal topology of communication is given by a de bruijn graph consensus is then reached in finitely many steps this solution is valid when the number of agents is an exact power of the out degree of the communication graph we introduce an algebraic tool the shifted kronecker product and a more general family of strategies also based on a de bruijn communication graph those strategies are compared to cayley strategies in terms of the speed of convergence we also show that quantized communication between the agents still allows finite convergence to a consensus which is not in general the average of the initial positions

a new generalized consensus problem and its crt based solution this technical note introduces a new generalized consensus problem where nodes in a network are guaranteed to reach the final consensus on a common vector whose elements are exactly the initial values arbitrarily chosen by nodes we propose a fully decentralized algorithm able to solve the above problem and derive conditions to guarantee that consensus is reached in a finite number of steps in particular we prove that in a network composed by n nodes the proposed consensus problem is solvable in at most 2n steps moreover we introduce a finite field solution based on the chinese remainder theorem crt able to reduce the complexity of the proposed approach in the case of capacity constraints and we discuss an illustrative case study

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

infinitely many critical points of non differentiable functions and applications to a neumann type problem involving the p laplacian abstract for a family of functionals in a banach space which are possibly non smooth and depend also on a positive real parameter the existence of a sequence of critical points according to motreanu and panagiotopoulos minimax theorems and qualitative properties of the solutions of hemivariational inequalities nonconvex optimization applications vol 29 kluwer dordrecht 1998 chap 3 is established by mainly adapting a new technique due to ricceri 2000 j comput appl math 113 401 410 two applications are then presented both of them treat the neumann problem for an elliptic variational hemivariational inequality with p laplacian

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

the complexity of smooth words on 2 letter alphabets let a b n be the number of smooth words of length n over the alphabet a b with a b say that a smooth word w is left fully extendable lfe if both aw and bw are smooth in this paper we prove that for any positive number and positive integer n0 such that the proportion of b s is larger than for each lfe word of length exceeding n0 there are two constants c1andc2 such that for each positive integer n one has c1 nlog 2b 1 log 1 a b 2 1 a b n c2 nlog 2b 1 log 1 a b 2 moreover if both a and b are even then there are two suitable constants c1 c2 such that c1 nlog 2b 1 log a b 2 a b n c2 nlog 2b 1 log a b 2 for each positive integer n

three solutions for a differential inclusion problem involving the p x laplacian abstract in this paper we consider differential inclusion problem involving the p x laplacian of the type div u p x 2 u f x u in u 0 on applying a version of the non smooth three critical points theorem we obtain the existence of three solutions of the problem in w 0 1 p x

finite time convergent gradient flows with applications to network consensus this paper introduces the normalized and signed gradient dynamical systems associated with a differentiable function extending recent results on nonsmooth stability analysis we characterize their asymptotic convergence properties and identify conditions that guarantee finite time convergence we discuss the application of the results to consensus problems in multi agent systems and show how the proposed nonsmooth gradient flows achieve consensus in finite time

overview of differential equations with non standard growth abstract differential equations with non standard growth have been a very active field of investigation in recent years in this survey we present an overview of the field as well as several of the most important results we consider both existence and regularity questions finally we provide a comprehensive list of papers published to date

fixed time consensus tracking for multiagent systems with high order integrator dynamics this paper addresses the fixed time leader follower consensus problem for high order integrator multiagent systems subject to matched external disturbances a new cascade control structure based on a fixed time distributed observer is developed to achieve the fixed time consensus tracking control a simulation example is included to show the efficacy and the performance of the proposed control structure with respect to different initial conditions

fixed time stable gradient flows applications to continuous time optimization this paper proposes novel gradient flow schemes that yield convergence to the optimal point of a convex optimization problem within a fixed time from any given initial condition for unconstrained optimization constrained optimization and min max problems it is shown that the solution of the modified gradient flow dynamics exists and is unique under certain regularity conditions on the objective function while fixed time convergence to the optimal point is shown via lyapunov based analysis the application of the modified gradient flow to unconstrained optimization problems is studied under the assumption of gradient dominance a relaxation of strong convexity then a modified newton s method is presented that exhibits fixed time convergence under some mild conditions on the objective function building upon this method a novel technique for solving convex optimization problems with linear equality constraints that yields convergence to the optimal point in fixed time is developed finally the general min max problem is considered and a modified scheme to obtain the optimal solution of saddle point dynamics in fixed time is developed

adding a single state memory optimally accelerates symmetric linear maps previous papers have proposed to add memory registers to the dynamics of discrete time linear systems in order to accelerate their convergence in particular it has been proved that adding one memory slot per agent allows faster convergence towards average consensus we here prove that this situation cannot be improved by adding more memory slots when the knowledge about the self adjoint linear map to be accelerated reduces to bounds on its extreme eigenvalues

implicit lyapunov krasovski functionals for stability analysis and control design of time delay systems the method of implicit lyapunov krasovski functional ilkf for stability analysis of time delay systems is introduced theorems on lyapunov asymptotic hyper exponential finite time and fixed time stability analysis using ilkf are presented the hyper exponential stabilization algorithm for a time delay system is presented the theoretical result is supported by numerical simulation

lyapunov stability theory of nonsmooth systems this paper develops nonsmooth lyapunov stability theory and lasalle s invariance principle for a class of nonsmooth lipschitz continuous lyapunov functions and absolutely continuous state trajectories computable tests based on filipov s differential inclusion and clarke s generalized gradient are derived the primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right hand side such as in nonsmooth dynamic systems or variable structure control

from global finite time linear computations to local edge based interaction rules a network of locally interacting agents can be thought of as performing a distributed computation but not all computations can be faithfully distributed this technical note investigates which global linear transformations can be computed in finite time using local rules with time varying weights i e rules which rely solely on information from adjacent nodes in a network the main result states that a linear transformation is computable in finite time using local rules if and only if the transformation has positive determinant an optimal control problem is solved for finding the local interaction rules and simulations are performed to elucidate how optimal solutions can be obtained