Impulsive Control And Synchronization Analysis For Some Kinds Of Discrete-time Dynamical Networks

Dynamical networks, which are consisted of a large number of organic nodes and parallel connections, have kinds of mathematical frameworks with respect to different time domains, i.e., continuous-time and discrete-time. Recently, the discrete-time dynamical networks have attracted much research interest among distinct scenarios including basic sciences, engineering and even social sciences due to their potential prospects of simulation and computation. Actually, the study of discrete-time dynamical networks is originated from analysis of continuous-time counterpart. The primitive aims of this study, on the one hand, reveal the mechanism causing distinct dynamical behaviors in theory, and on the other hand, are to help us to implement computer-based simulation and computation with the offer of better reliability.As we known, synchronization is a kind of desired dynamical behavior among the dynamical behaviors of discrete-time dynamical networks. The study of synchronization is one of the most important topics, which can be explained many phenomena in the natural world and the field of engineering. However, in practice, the synchronization phenomena are often subjected to abrupt changes at some certain moments as a result of unexpected external or internal effects.In the view of modeling, these effects can be presented as impulsive phenomena. From the perspective of synchronization performance, the higher impulsive gains, i.e., desynchronizing impulses, are regarded as perturbations, and even can destroy the synchronization. Conversely, the lower impulsive gains, i.e.,synchronizing impulses, can be considered as potential controllers in promoting synchronization.Motivated by the above discussion, this paper is aimed to deal with the synchronization problems for the discrete-time dynamical networks with some kinds of typical impulsive effects. Furthermore, sereval novel impulsive control strategies have been proposed to investigated the synchronization problems forthe addressed networks. The compendious frame and innovation of this thesis are as follows:(1) Exponential synchronization of impulsive discrete-time dynamical networks with time-varying delay. The exponential synchronization of discrete-time dynamical networks with time-varying delay is investigated.Generally, the concept of ”average impulsive interval” is derived from the”average dwell time” in switching systems, and is often used to describe the bound of the synchronizing and desynchronizing impulsive intervals in many impulsive continuous-time systems. Frist, we introduce the average impulsive interval into impulsive discrete-time systems. By utilizing the Lyapunov function method and the Razumikhin technique, several sufficient criteria are proved to be effective in dealing with synchronizing and desynchronizing impulses respectively. The derived results can also show that the criteria are less conservative than the results considering the upper or lower of impulsive intervals. Two numerical simulation examples are provided to demonstrate the effectiveness of the proposed approach.(2) Synchronization of discrete-time dynamical networks with delayed heterogeneous impulses. The synchronization problem is investigated for discrete-time complex networks with delayed heterogeneous impulses.Heterogeneous impulses are a form of inhomogeneous impulse, which means that the impulsive strengths depend on both time and space domains. In order to more appropriately describe impulsive effects, both heterogeneous impulsive effects and impulsive input delays are considered. Employing the Razumikhin theorem and by constructing a timedependent Lyapunov function, several sufficient criteria are derived to ensure exponential synchronization of the discretetime complex networks with delayed heterogeneous impulsive effects. The derived criteria are formulated in terms of linear matrix inequalities, in which the impulsive intervals, impulsive input delays and impulsive strengths play an important role. Finally, an example is delivered to illustrate the effectiveness of the proposed synchronization criteria.(3) Synchronization of stochastic discrete-time dynamical networks with partial mixed impulsive effects. The synchronization problem is studied for a class of stochastic discrete-time complex networks with partial mixed impulsive effects. Partial mixed impulses can be regarded as local and time-varying impulses, which means that impulses are not only injected into a fraction of nodes in networks but also contain synchronizing and desynchronizing impulses at the same time. In order to handle the case of discretetime form, several mathematical techniques are proposed to tackle mixed impulsive effects in discrete-time dynamical systems. Based on the variation of parameters formula, several sufficient criteria are derived to ensure that synchronization of the addressed networks is achieved in mean square. The obtained criteria not only rely on the strengths of mixed impulses and the impulsive intervals, but also can reduce conservativeness. Finally, a numerical example is presented to show the effectiveness of our results for neural networks(4) Delayed impulsive synchronization of discrete-time dynamical networks with distributed delays. A novel delayed impulsive control strategy is proposed for synchronization of discrete-time dynamical networks with distributed delays. Different from the existing results, the involving time delays include distributed delays and impulsive input delays. Employing the Razumikhin theorem and the mathematical induction method,several sufficient criteria are derived in terms of algebraic conditions, which depend on impulsive input delays and impulsive control gains. Meanwhile,the derived criteria also reveal the relationship between the bounds of impulsive intervals and impulsive input delays. Finally, two examples are given to illustrate the effectiveness of the proposed approach.(5) Impulsive synchronization of discrete-time neural networks with partial state saturation. A novel impulsive control strategy is proposed for synchronization of delayed coupled discrete-time neural networks. The proposed impulsive controller is subjected to partial state saturation, which means that state saturations are only considered for a fraction of neuronsin impulsive controllers. Utilizing the Razumikhin-type techniques, several sufficient criteria are derived to ensure the synchronization of addressed networks under partial saturated impulsive control. In order to calculate the impulsive control gain matrix, the obtained criteria are transformed into the feasibility problem for the admissible upper bound of impulsive intervals or a prescribed parameter related on impulsive control gains. Subsequently,the applicability of the criteria is verified through an example.(6) Consensus in networks of discrete-time multi-agent: low-gain distributed impulsive strategy. The novel distributed impulsive consensus protocols whose magnitudes can be to approach zero or a given bound,have been proposed for consensus problems of a class of linear discretetime multi-agent systems that concerns processes by which the consensus may be not maintained under the usual feedback protocols. Utilizing the Lyapunov technique, the low-gain and low-high-gain theory, a parametric discretetime Riccati equation based strategy has been established for the sake of design impulsive consensus protocols, which require the solution of the addressed discrete-time Riccati equation for the proper value of parameters, and thereby extending recent results related on low-gain and low-high-gain feedback to apply impulsive actuators. The derived criteria also show that these consensus protocols, not only rely to the low-gain and low-high-gain parameters, but also depend on the upper bound of impulsive intervals. Moreover, two algorithms have been presented to derive impulsive control gain matrices. Subsequently, the applicability of proposed strategies is given through numerical examples.