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 Electronic Journal of Differential Equations , 2005, Abstract: We study the stability characteristics of a time-dependent system of impulsive logistic equations by using discrete modelling.
 Wei Zhu Discrete Dynamics in Nature and Society , 2012, DOI: 10.1155/2012/390691 Abstract: The consensus problem for discrete time second-order multiagent systems with time delay is studied. Some effective methods are presented to deal with consensus problems in discrete time multiagent systems. A necessary and sufficient condition is established to ensure consensus. The convergence rate for reaching consensus is also estimated. It is shown that arbitrary bounded time delay can safely be tolerated. An example is presented to illustrate the theoretical result. 1. Introduction The study of information flow and interaction among multiple agents in a group plays an important role in understanding the coordinated movements of these agents. As a result, a critical problem for coordinated control is to design appropriate protocols and algorithms such that the group of agents can reach consensus on the shared information in the presence of limited and unreliable information exchange as well as communication time delays. In multiagent systems, communication time delays between agents are inevitable due to various reasons. For instance, they may be caused by finite signal transmission speeds, traffic congestions, packet losses, and inaccurate sensor measurements. In addition, in practical engineering applications, the agents in multiagent systems transmit sampled information by using sensors or communication network, and the coordination control algorithms are proposed based on the discrete time sampled data to achieve the whole control object. The typical discrete-time consensus control strategy was provided by Jadbabaie et al. [1], which is a simplified Vicsek model [2]. Recently, the consensus analysis of the discrete time first order multiagent systems with or without communication time delays has been studied extensively, see [3–6], to name a few. While it has been realized that modeling more complex practical processes needs the use of double integrator dynamics, as a result, cooperative control for multiple agents with double-integrator dynamics has become an active area of research. Compared with the first-order consensus, Ren and Atkins [7] show that the existence of a directed spanning tree is a necessary rather than a sufficient condition to reach second-order consensus. Therefore, the extension of consensus algorithms from first order to second order is nontrivial. In recent years, more attention has been paid to the consensus problem of multiagent systems with continuous time second-order systems and much progress has been made, some important works include [8–13]. But there has been little attention to the consensus of discrete time
 Advances in Difference Equations , 2008, Abstract: We aim to study robust impulsive synchronization problem for uncertain discrete dynamical networks. For the discrete dynamical networks with unknown but bounded network coupling, we will design some robust impulsive controllers which ensure that the state of a discrete dynamical network asymptotically synchronize with an arbitrarily assigned state of an isolate node of the network. Three representative examples are also worked through to illustrate our results.
 Advances in Difference Equations , 2008, DOI: 10.1155/2008/184275 Abstract: We aim to study robust impulsive synchronization problem for uncertain discrete dynamical networks. For the discrete dynamical networks with unknown but bounded network coupling, we will design some robust impulsive controllers which ensure that the state of a discrete dynamical network asymptotically synchronize with an arbitrarily assigned state of an isolate node of the network. Three representative examples are also worked through to illustrate our results.
 中国物理快报 , 2009, Abstract: The chaotification of discrete Hopfield neural networks is studied with impulsive control techniques. No matter whether the original systems are stable or not, chaotification theorems for discrete Hopfield neural networks are derived, respectively. Finally, the effectiveness of the theoretical results is illustrated by some numerical examples.
 Computer Science , 2014, Abstract: This paper presents a novel compositional approach to distributed coordination module (CM) synthesis for multiple discrete-event agents in the formal languages and automata framework. The approach is supported by two original ideas. The first is a new formalism called the Distributed Constraint Specification Network (DCSN) that can comprehensibly describe the networking constraint relationships among distributed agents. The second is multiagent conflict resolution planning, which entails generating and using AND/OR graphs to compactly represent conflict resolution (synthesis-process) plans for a DCSN. Together with the framework of local CM design developed in the authors' earlier work, the systematic approach supports separately designing local and deconflicting CM's for individual agents in accordance to a selected conflict resolution plan. Composing the agent models and the CM's designed furnishes an overall nonblocking coordination solution that meets the set of inter-agent constraints specified in a given DCSN.
 Advances in Difference Equations , 2010, DOI: 10.1155/2010/278240 Abstract: This paper studies the problem of stabilization with optimal performance for dissipative DIHS (discrete-time impulsive hybrid systems). By using Lyapunov function method, conditions are derived under which the DIHS with zero inputs is GUAS (globally uniformly asymptotically stable). These GUAS results are used to design feedback control law such that a dissipative DIHS is asymptotically stabilized and the value of a hybrid performance functional can be minimized. For the case of linear DIHS with a quadratic supply rate and a quadratic storage function, sufficient and necessary conditions of dissipativity are expressed in matrix inequalities. And the corresponding conditions of optimal quadratic hybrid performance are established. Finally, one example is given to illustrate the results.
 Advances in Difference Equations , 2010, Abstract: This paper studies the problem of stabilization with optimal performance for dissipative DIHS (discrete-time impulsive hybrid systems). By using Lyapunov function method, conditions are derived under which the DIHS with zero inputs is GUAS (globally uniformly asymptotically stable). These GUAS results are used to design feedback control law such that a dissipative DIHS is asymptotically stabilized and the value of a hybrid performance functional can be minimized. For the case of linear DIHS with a quadratic supply rate and a quadratic storage function, sufficient and necessary conditions of dissipativity are expressed in matrix inequalities. And the corresponding conditions of optimal quadratic hybrid performance are established. Finally, one example is given to illustrate the results.
 Yinshan Chang Mathematics , 2013, DOI: 10.1214/EJP.v20-3176 Abstract: The loop clusters of a Poissonian ensemble of Markov loops on a finite or countable graph have been studied in \cite{Markovian-loop-clusters-on-graphs}. In the present article, we study the loop clusters associated with a rotation invariant nearest neighbor walk on the discrete circle $G^{(n)}$ with $n$ vertices. We prove a convergence result of the loop clusters on $G^{(n)}$, as $n\rightarrow\infty$, under suitable condition of the parameters. These parameters are chosen in such a way that the rotation invariant nearest neighbor walk on $G^{(n)}$, as $n\rightarrow\infty$, converges to a Brownian motion on circle $\mathbb{S}^{1}=\mathbb{R}/\mathbb{Z}$ with certain drift and killing rate. In the final section, we show that several limit results are predicted by Brownian loop-soup on $\mathbb{S}^{1}$.
 Discrete Dynamics in Nature and Society , 2013, DOI: 10.1155/2013/767526 Abstract: By piecewise Euler method, a discrete Lotka-Volterra predator-prey model with impulsive effect at fixed moment is proposed and investigated. By using Floquets theorem, we show that a globally asymptotically stable pest-eradication periodic solution exists when the impulsive period is less than some critical value. Further, we prove that the discrete system is permanence if the impulsive period is larger than some critical value. Finally, some numerical experiments are given. 1. Introduction Impulsive equations are found in almost every domain of applied science, such as population dynamics, ecology, biological systems, and optimal control. In recent years, the theory of impulsive differential equations has been an object of active research (see [1–4] and reference therein) since it is much richer than the corresponding theory of differential equations without impulsive effects. It is well known that continuous-time dynamic systems play an important role in control theory, population dynamics, and so on. But in applications of continuous-time dynamic systems to some practical problems, such as computer simulation, experimental, or computational purposes, it is usual to formulate a discrete-time system which is a version of the continuous-time system. In some sense, the discrete time model inherits the dynamical characteristics of the continuous-time systems. We refer to [4–16] for related discussions of the importance and the need for discrete-time analogs to reflect the dynamics of their continuous-time counterparts. Nevertheless, the discrete-time version can but not always preserve the dynamics of its initial version because the theory of difference equations is a lot richer than the corresponding theory of differential equations as pointed out in [17, 18]. Therefore, it is important to study the dynamics of its initial version alone. Due to the above facts, we construct the following discrete impulsive Lotka-Volterra predator-prey model concerning integrated pest management by piecewise Euler method: where is the intrinsic growth rate of pest, is the coefficient of intraspecific competition, is the per-capita rate of predation of the predator, is the death rate of predator, denotes the product of the per-capita rate of predation and the rate of conversing pest into predator, and is the period of the impulsive effect. represents the fraction of pest (predator) which dies due to the pesticide, and is the release amount of predator at , . That is, we can use a combination of biological (periodic releasing natural enemies) and chemical (spraying
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