master symmetries for differential difference equations of the volterra type abstract it is demonstrated that in the case of integrable differential difference equations similar to the well known volterra equation unlike the korteweg de vries and nonlinear schrodinger equations there are many instances in which local master symmetries can be found those master symmetries are new interesting examples of local evolution chains explicitly depending on the time and discrete variable and integrable in a special sense the examples are constructed by a direct and elementary approach which enables one to get new integrable equations using miura type transformations

on two not so new integrable partial difference equations we examine two integrable discrete lattice equations obtained by levi and yamilov we show that the first one is a form of the lattice kdv equation already obtained by hirota and tsujimoto while the second one is a discrete form of mkdv we present the miura transformations between the various equations involved including the more familiar potential form of the lattice mkdv

consensus in multi agent systems with random delays governed by a markov chain abstract this paper investigates the consensus problem in a multi agent system with random delays governed by a markov chain the communication topology is assumed to be directed and fixed with first order dynamics under the sampled data setting we first convert the original system into a reduced order one featuring the error dynamics accordingly the consensus problem is transformed into the stabilization of the error dynamic system thereafter based on the theory in stochastic stability for time delay systems a sufficient condition is established in terms of a set of linear matrix inequalities lmis the mean square stability of the error dynamics is shown to guarantee consensus of the multi agent system by explicitly incorporating the transition probability of the random delay into consideration the conservativeness in control design is reduced a delay dependent switching control scheme is developed by redesigning the adjacency matrix finally simulation results are provided to verify the effectiveness of the proposed approach

symmetries as integrability criteria for differential difference equations in this paper we review the results obtained by the generalized symmetry method in the case of differential difference equations during the last 20 years together with general theory of the method classification results are discussed for classes of equations which include the volterra toda and relativistic toda lattice equations

superposition principles associated with the moutard transformation an integrable discretization of a 2 1 dimensional sine gordon system superposition principles both linear and nonlinear associated with the moutard transformation are found on suitable reinterpretation these constitute an integrable discrete nonlinear system and its associated linear system further it is shown that in a particular form this system is an integrable discretization of a 2 1 dimensional sine gordon system solutions of the discrete nonlinear system are constructed by means of a discrete analogue of the moutard transformation included in these solutions are discrete analogues of the kink solutions of the continuous system

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

peculiar symmetry structure of some known discrete nonautonomous equations we study the generalized symmetry structure of three known discrete nonautonomous equations one of them is the semidiscrete dressing chain of shabat two others are completely discrete equations defined on the square lattice the first one is a discrete analogue of the dressing chain introduced by levi and yamilov the second one is a nonautonomous generalization of the potential discrete kdv equation or in other words the h1 equation of the well known adler bobenko suris list we demonstrate that these equations have generalized symmetries in both directions if and only if their coefficients depending on the discrete variables are periodic the order of the simplest generalized symmetry in at least one direction depends on the period and may be arbitrarily high we substantiate this picture by some theorems in the case of small periods in case of an arbitrarily large period we show that it is possible to construct two hierarchies of generalized symmetries and conservation laws the same picture should take place in case of any nonautonomous equation of the adler bobenko suris list

generalized symmetry integrability test for discrete equations on the square lattice we present an integrability test for discrete equations on the square lattice which is based on the existence of a generalized symmetry we apply this test to a number of equations obtained in different recent papers as a result we prove the integrability of seven equations which differ essentially from the qv equation introduced by viallet and thus from the adler bobenko suris list of equations contained therein

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

output feedback stabilization of networked control systems with random delays modeled by markov chains this note investigates the output feedback stabilization of networked control systems ncss the sensor to controller s c and controller to actuator c a random network induced delays are modeled as markov chains the focus is on the design of a two mode dependent controller that depends on not only the current s c delay but also the most recent available c a delay at the controller node the resulting closed loop system is transformed to a special discrete time jump linear system then the sufficient and necessary conditions for the stochastic stability are established further the output feedback controller is designed via the iterative linear matrix inequality lmi approach simulation examples illustrate the effectiveness of the proposed method

generalized symmetry classification of discrete equations of a class depending on twelve parameters we carry out the generalized symmetry classification of polylinear autonomous discrete equations defined on the square which belong to a twelve parametric class the direct result of this classification is a list of equations containing no new examples however as an indirect result of this work we find a number of integrable examples pretending to be new one of them has a nonstandard symmetry structure the others are analogues of the liouville equation in the sense that they are darboux integrable we also enumerate all equations of the class which are linearizable via a two point first integral and specify the nature of integrability of some known equations

the generalized symmetry method for discrete equations the generalized symmetry method is applied to a class of completely discrete equations including the adler bobenko suris list assuming the existence of a generalized symmetry we derive a few integrability conditions suitable for testing and classifying the equations of this class those conditions are used at the end to test for integrability discretizations of some well known hyperbolic equations

distributed leader follower flocking control for multi agent dynamical systems with time varying velocities abstract using tools from algebraic graph theory and nonsmooth analysis in combination with ideas of collective potential functions velocity consensus and navigation feedback a distributed leader follower flocking algorithm for multi agent dynamical systems with time varying velocities is developed where each agent is governed by second order dynamics the distributed leader follower algorithm considers the case in which the group has one virtual leader with time varying velocity for each agent i this algorithm consists of four terms the first term is the self nonlinear dynamics which determines the final time varying velocity the second term is determined by the gradient of the collective potential between agent i and all of its neighbors the third term is the velocity consensus term and the fourth term is the navigation feedback from the leader to avoid an impractical assumption that the informed agents sense all the states of the leader the new designed distributed algorithm is developed by making use of observer based pinning navigation feedback in this case each informed agent only has partial information about the leader yet the velocity of the whole group can still converge to that of the leader and the centroid of those informed agents having the leader s position information follows the trajectory of the leader asymptotically finally simulation results are presented to demonstrate the validity and effectiveness of the theoretical analysis surprisingly it is found that the local minimum of the potential function may not form a commonly believed lattice

classification of integrable equations on quad graphs the consistency approach a classification of discrete integrable systems on quad graphs i e on surface cell decompositions with quadrilateral faces is given the notion of integrability laid in the basis of the classification is the three dimensional consistency this property yields among other features the existence of the discrete zero curvature representation with a spectral parameter for all integrable systems of the obtained exhaustive list the so called three leg forms are found this establishes lagrangian and symplectic structures for these systems and the connection to discrete systems of the toda type on arbitrary graphs generalizations of these ideas to the three dimensional integrable systems and to the quantum context are also discussed

symmetries and integrability of discrete equations defined on a black white lattice we study the deformations of the h equations presented recently by adler bobenko and suris which are naturally defined on a black white lattice for each one of these equations two different three leg forms are constructed leading to two different discrete toda type equations their multidimensional consistency leads to backlund transformations relating different members of this class as well as to lax pairs their symmetry analysis is presented yielding infinite hierarchies of generalized symmetries

three point partial difference equations linearizable by local and nonlocal transformations we consider a class of nonlinear partial difference equations defined on three points of a plane lattice we construct conditions for this class of partial difference equations to be linearizable through a point or a cole hopf transformation using these conditions we are able to classify all multilinear linearizable equations belonging to this class

conditions for the existence of higher symmetries of evolutionary equations on the lattice in this paper we construct a set of five conditions necessary for the existence of generalized symmetries for a class of differential difference equations depending only on nearest neighboring interaction these conditions are applied to prove the existence of new integrable equations belonging to this class

on the integrability of a new lattice equation found by multiple scale analysis in this paper we discuss the integrability properties of a nonlinear partial difference equation on the square obtained by the multiple scale integrability test from a class of multilinear dispersive equations defined on a four point lattice

affine linear and d4 symmetric lattice equations symmetry analysis and reductions we consider lattice equations on which are autonomous affine linear and possess the symmetries of the square some basic properties of equations of this type are derived as well as a sufficient linearization condition and a conservation law a systematic analysis of the lie point and the generalized three and five point symmetries is presented it leads to the generic form of the symmetry generators of all the equations in this class which satisfy a certain non degeneracy condition finally symmetry reductions of certain lattice equations to discrete analogs of the painleve equations are considered

consensus conditions of multi agent systems with time varying topologies and stochastic communication noises this paper investigates the average consensus problem of first order discrete time multi agent networks in uncertain communication environments each agent can only use its own and neighbors information to design its control input to attenuate the communication noises a distributed stochastic approximation type protocol is used by using probability limit theory and algebraic graph theory consensus conditions for this kind of protocols are obtained a for the case of fixed topologies a necessary and sufficient condition for mean square average consensus is given which is also sufficient for almost sure consensus b for the case of time varying topologies sufficient conditions for mean square average consensus and almost sure consensus are given respectively especially if the network switches between jointly containing spanning tree instantaneously balanced graphs then the designed protocol can guarantee that each individual state converges both almost surely and in mean square to a common random variable whose expectation is right the average of the initial states of the whole system and whose variance describes the static maximum mean square error between each individual state and the average of the initial states of the whole system

strict aperiodic property of polynomials with perturbed coefficients let a family of polynomials be p s t sub 0 s sup n t sub 1 s sup n or sup 1 t sub n where 0

symmetries and conservation laws of the abs equations and corresponding differential difference equations of volterra type a sequence of canonical conservation laws for all the adler bobenko suris abs equations is derived and is employed in the construction of a hierarchy of master symmetries for equations h1 h3 q1 q3 for the discrete potential and schwarzian kdv equations it is shown that their local generalized symmetries and nonlocal master symmetries in each lattice direction form centerless virasoro type algebras in particular for the discrete potential kdv the structure of its symmetry algebra is explicitly given interpreting the hierarchies of symmetries of equations h1 h3 q1 q3 as differential difference equations of yamilov s discretization of krichever novikov equation corresponding hierarchies of master symmetries along with isospectral and nonisospectral zero curvature representations are derived for all of them

global and robust formation shape stabilization of relative sensing networks this paper proposes a simple distributed algorithm that achieves global stabilization of formations for relative sensing networks in arbitrary dimensions with fixed topology assuming the network runs an initialization procedure to equally orient all agent reference frames convergence to the desired formation shape is guaranteed even in partially asynchronous settings we characterize the algorithm robustness against several sources of errors link failures measurement errors and frame initialization errors the technical approach combines algebraic graph theory multidimensional scaling and distributed linear iterations

on the largest polytope of polynomials with a precise number of distinct real zeros abstract this paper presents a procedure to construct the largest polytope of polynomials with every polynomial having a precise number of distinct real zeros in a specific real line segment this polytope can be specified from the zeros of a finite number of polynomials

asymptotic diagonalization of the discrete lax pair around singularities and conservation laws for dynamical systems a method of the formal diagonalization of the discrete linear operator with a parameter is studied in the case when the operator provides a lax operator for a nonlinear quad system the formal diagonalization method allows one to describe effectively conservation laws and generalized symmetries for this system asymptotic representation of the lax operators eigenfunctions are constructed and infinite series of conservation laws are described for the quad system connected with affine lie algebra for the modified discrete boussinesq system and for the discrete tzitzeica equation for a newly found multiquadratic discrete model conservation laws and several generalized symmetries are presented

exponential stability analysis for delayed semi markovian recurrent neural networks a homogeneous polynomial approach this paper investigates the exponential stability analysis issue for a class of delayed recurrent neural networks rnns with semi markovian parameters by constructing a stochastic lyapunov functional and using some zoom techniques to estimate its weak infinitesimal operator the exponential mean square stability criteria have been proposed for the markovian neural networks with certain transition probabilities we then generalize the homogeneous polynomial approach for the delayed markovian rnns with uncertain transition probabilities during the stability analysis theoretical results have obtained by introducing an appropriate technique for dealing with a large number of complex homogeneous polynomial matrix inequalities finally numerical examples are provided to demonstrate the effectiveness of the proposed technique

infinitely many symmetries and conservation laws for quad graph equations via the gardner method the application of the gardner method for the generation of conservation laws to all the abs equations is considered it is shown that all the necessary information for the application of the gardner method namely b cklund transformations and initial conservation laws follows from the multidimensional consistency of abs equations we also apply the gardner method to an asymmetric equation which is not included in the abs classification an analog of the gardner method for the generation of symmetries is developed and applied to the discrete korteweg de vries equation it can also be applied to all the other abs equations

stochastic stability and stabilization conditions of semi markovian jump systems with mode transition dependent sojourn time distributions this paper is concerned with the stochastic stability and stabilization problem for a class of continuous time semi markovian jump systems s mjss with mode transition dependent sojourn time distributions in contrast to other approaches this paper offers the possibility to explore more refined probability models on the sojourn time by considering the mode transition dependent sojourn time distributions instead of mode dependent ones here to construct a finite set of linear matrix inequalities lmis from sojourn time dependent matrix inequalities a less conservative relaxation method is proposed in such a way that all available constraints on time varying transition rates can be exploited simulation examples report the effectiveness of the proposed method

soliton solutions for abs lattice equations i cauchy matrix approach in recent years there have been new insights into the integrability of quadrilateral lattice equations i e partial difference equations which are the natural discrete analogues of integrable partial differential equations in 1 1 dimensions in the scalar i e single field case there now exist classification results by adler bobenko and suris abs leading to some new examples in addition to the lattice equations of kdv type that were known since the late 1970s and early 1980s in this paper we review the construction of soliton solutions for the kdv type lattice equations and use those results to construct n soliton solutions for all lattice equations in the abs list except for the elliptic case of q4 which is left to a separate treatment

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

a survey of extreme point results for robustness of control systems abstract this paper surveys a subset of the body of research which was sparked by kharitonov s theorem the focal point is extreme point results for robust stability and robust performance that is we give conditions under which satisfaction of a performance specification is ascertained for a family of systems by only checking a finite subset of the extreme members of this family the results which are surveyed apply mainly to systems with structured real parametric uncertainty in addition a number of counterexamples are given to illustrate cases for which an extreme point result does not hold for such cases a solution via the so called edge theorem is often possible

invariance of the aperiodic property for polynomials with perturbed coefficients a procedure for obtaining the maximal intervals of the coefficients for a perturbed polynomial to be still strictly aperiodic is presented lemmas theorems and proofs are given an illustrative example is presented

cooperative adaptive fuzzy tracking control for networked unknown nonlinear multiagent systems with time varying actuator faults in this paper the cooperative adaptive fault tolerant fuzzy tracking control caftftc problem of networked high order multiagent with time varying actuator faults is studied and a novel caftftc scheme is proposed to guarantee that all follower nodes asymptotically synchronize a leader node with tracking errors converging to a small adjustable neighborhood of the origin in spite of actuator faults the leader node is modeled as a higher order nonautonomous nonlinear system it acts as a command generator giving commands only to a small portion of the networked group each follower is assumed to have nonidentical unknown nonlinear dynamics and the communication network is also assumed to be a weighted directed graph with a fixed topology a distributed robust adaptive fuzzy controller is designed for each follower node such that the tracking errors are cooperative uniform ultimate boundedness cuub moreover these controllers are distributed in the sense that the controller designed for each follower node only requires relative state information between itself and its neighbors the adaptive compensation term of the optimal approximation errors and external disturbances is adopted to reduce the effects of the errors and disturbances which removes the assumption that the upper bounds of unknown function approximation errors and disturbances should be known analysis of stability and parameter convergence of the proposed algorithm are conducted that are based on algebraic graph theory and lyapunov theory comparing with results in the literature the caftftc scheme can minimize the time delay between fault occurrence and accommodation and reduce its adverse effect on system performance in addition the ftc scheme requires no additional fault isolation model which is necessary in the traditional active ftc scheme finally an example is provided to validate the theoretical results

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

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 discrete 2d integrable equations of higher order we study two dimensional discrete integrable equations of order 1 with respect to one independent variable and m with respect to another one a generalization of the multidimensional consistency property is proposed for this type of equations the examples are related to the backlund darboux transformations for the lattice equations of bogoyavlensky type

brief paper some necessary and sufficient conditions for second order consensus in multi agent dynamical systems this paper studies some necessary and sufficient conditions for second order consensus in multi agent dynamical systems first basic theoretical analysis is carried out for the case where for each agent the second order dynamics are governed by the position and velocity terms and the asymptotic velocity is constant a necessary and sufficient condition is given to ensure second order consensus and it is found that both the real and imaginary parts of the eigenvalues of the laplacian matrix of the corresponding network play key roles in reaching consensus based on this result a second order consensus algorithm is derived for the multi agent system facing communication delays a necessary and sufficient condition is provided which shows that consensus can be achieved in a multi agent system whose network topology contains a directed spanning tree if and only if the time delay is less than a critical value finally simulation examples are given to verify the theoretical analysis

searching for integrable lattice maps using factorization we analyse the factorization process for lattice maps searching for integrable cases the maps were assumed to be at most quadratic in the dependent variables and we required minimal factorization one linear factor after two steps of iteration the results were then classified using the algebraic entropy some new models with polynomial growth strongly associated with integrability were found one of them is a nonsymmetric generalization of the homogeneous quadratic maps associated with kdv modified and schwarzian for this new model we have also verified the consistency around a cube

quantized consensus of second order continuous time multi agent systems with a directed topology via sampled data abstract this paper addresses the quantized consensus problem of continuous time second order multi agent systems via sampled data where the interconnection topology has a directed spanning tree a necessary and sufficient condition on the sampling period and the design parameters is obtained to guarantee that the consensus errors converge to a neighborhood of the origin which disappears as the quantized errors tend to zero this paper also gives the final consensus values which is different from the existing works on the quantized consensus of second order multi agent systems the method is further extended to the case when the velocities of agents are unavailable two simulation examples are provided to illustrate the effectiveness of the proposed consensus protocols

stabilization of markovian systems via probability rate synthesis and output feedback this technical note is concerned with the stabilization problem of markovian jump linear systems via designing switching probability rate matrices and static output feedback gains a novel necessary and sufficient condition is established to characterize the switching probability rate matrices that guarantee the mean square stability of markovian jump linear systems based on this a necessary and sufficient condition is provided for the existence of desired controller gains and probability rate matrices extensions to the polytopic uncertain case are also provided all the conditions are formulated in terms of linear matrix inequalities with some equality constraints which can be solved by two modified cone complementarity linearization algorithms examples are given to show the effectiveness of the proposed method

construction scheme for discrete miura transformations a direct and elementary scheme for the construction of miura type transformations and discrete differential equations related to them scalar and vector is presented the scheme is illustrated using as examples the volterra and toda models a discrete differential analogue of the calogero degasperis equation is discussed in detail this example is used to show how to construct conservation laws higher symmetries and solutions for an equation obtained with the help of the scheme

linearization through symmetries for discrete equations we show that one can devise through the symmetry approach a procedure to check the linearizability of a difference equation via a point or a discrete cole hopf transformation if the equation is linearizable then the symmetry provides the linearizing transformation at the end we present a few examples of applications for equations defined on four lattice points

emergent behavior in flocks we provide a model for both continuous and discrete time describing the evolution of a flock our model is parameterized by a constant beta capturing the rate of decay which in our model is polynomial of the influence between birds in the flock as they separate in space our main result shows that when beta 1 2 convergence of the flock to a common velocity is guaranteed while for betages1 2 convergence is guaranteed under some condition on the initial positions and velocities of the birds only

non invertible transformations of differential difference equations we discuss aspects of the theory of non invertible transformations of differential difference equations and in particular the notion of miura type transformation we introduce the concept of non miura type linearizable transformation and we present techniques that allow one to construct simple linearizable transformations and might help one to solve classification problems this theory is illustrated by the example of a new integrable differential difference equation depending on five lattice points interesting from the viewpoint of the non invertible transformation which relate it to an itoh narita bogoyavlensky equation

method for searching higher symmetries for quad graph equations a generalized symmetry integrability test for discrete equations on the square lattice is studied integrability conditions are discussed a method for searching higher symmetries including non autonomous ones for quad graph equations is suggested based on characteristic vector fields

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

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