复杂多个体时滞网络系统脉冲一致性的动力学与控制
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摘要
复杂多个体网络系统的合作与协调控制已成为当今国际上一项极其重要且富有挑战性的前沿研究领域之一,它引起了诸如生物学、物理学、应用数学、信息科学、计算机科学和控制科学等众多学科的广泛关注.一致性问题作为表征多个体网络系统在局部个体之间相互动态作用之下系统整体涌现的动力学行为,是研究多个体动态网络系统合作与协调控制问题一个重要的切入点.本文从动力学与控制的角度出发,主要研究复杂多个体时滞网络系统的脉冲一致性及其相关问题.主要工作概括为以下四个方面：
一.具有任意时滞的脉冲微分方程的稳定性理论.利用Lyapunov函数方法和Razumikhin技术,给出了一类具有任意时滞的非线性脉冲微分方程全局指数稳定的准则判据.其突出的特征就是去掉了一般性的限制条件,即时滞小于所有的脉冲间隔.此外,还得到了关于脉冲时滞动力系统的推广的Halanay不等式及其向量形式,这里删除了不含脉冲的时滞动力系统是稳定的预先假设.这样就可以通过脉冲去稳定一个不稳定的甚至是混沌的动力系统.这为实际中多个体网络系统的一致性、耦合振子的同步以及混沌时滞系统的控制与同步提供了理论基础.
二.多个体有向时滞网络系统的平均一致性问题.基于时滞动力系统的脉冲控制理论,提出了在通讯时滞环境中三种一致性协议.第一个协议是具有脉冲效应的一致性协议,结果表明通过设计协议中合适的脉冲增益和脉冲间隔,可使整个网络全局指数地达到一致.第二个是分布式δ-脉冲一致性协议,它能够处理具有随机切换拓扑网络的一致性问题.第三个是对任意通讯时滞都适用的协议,可以根据实际的需要来调节协议中时滞的大小.
三.基于脉冲控制下一般复杂动态网络的同步动力学.首先设计合理的脉冲控制器,然后应用脉冲时滞动力系统推广的Halanay不等式,给出了一般复杂时滞动态网络的全局同步化准则.这些准则能够提供一个新的和有效的控制方法来同步一个任意给定的时滞动态网络到一个期望的同步态,即便原来的网络本身是不同步的.这种同步态可以根据控制策略的目标而选取整个网络状态的权重平均,结果表明网络中不同的节点对网络的同步化影响大小是不一样的.
四.混沌时滞系统的脉冲控制与同步.基于时滞动力系统的脉冲控制理论,首先给出了一类混沌时滞系统脉冲控制和同步的判据；其次把推广的向量形式的Halanay微分不等式应用到保密通讯系统；最后研究了具有脉冲连接的时滞神经网络的指数稳定性问题,并把所获得的结果应用到时滞Hopfield神经网络,数值模拟进一步验证了理论结果的正确性.
The study of cooperative behavior and coordinated control in complex net-worked multi-agent systems has currently become one of the most challenging area of research from various fields such as biology or ecology, physics, applied mathematics, information science, computer science and control theory. This dissertation is mainly concerned with the issues of impulsive consensus prob-lems in complex delayed networked multi-agent systems and the relevant issues from the view of dynamics and control. In details, the fundamental contributions of these works are summarized in the following four aspects:
1. Stability theory of impulsive differential equations with any time delays. By utilizing Lyapunov function methods combined with the Razumikhin tech-nique, several criteria on exponential stability are derived analytically, which are substantially the extension and generalization of the corresponding results in re-cent literatures. Compared with some existing works, a distinctive feature of this work is to address exponential stability problems for any time delays, since the restrictive condition that the time delays are less than the length of all the im-pulsive intervals is actually removed here. Moreover, the previous results on Ha-lanay inequality for impulsive delayed dynamical systems are extended. In con-trast to the previous Halanay differential inequality, the primary contribution of this work is to remove the restrictive condition of a priori stability assumption for the corresponding delayed dynamical systems without impulses, so the result can be usually used to stabilize an unstable delayed dynamical system via impulses. Therefore, our work substantially extends the famous Halanay differential in-equality, which will play an important role in stability problems for consensus in delayed networked multi-agent systems, synchronization in complex delayed dynamical networks, control and synchronization in chaotic delayed systems.
2. Average consensus problems in directed delayed networked multi-agent systems. By employing the impulsive control theory on delayed dynamical sys- tems, three consensus protocols of the networks with communication delays are proposed. The first is a simple distributed consensus protocol with impulsive effects. And it is shown that a directed delayed networked multi-agent system can achieve average consensus globally exponentially by designing suitable im-pulsive gain and impulsive interval. The second is a distributedδ-consensus protocol in directed networks of dynamic agents having communication delays with stochastic switching communication graphs. The third is a distributed im-pulsive consensus protocol which is valid for any communication delays. Here, the communication delays can be adjusted according to practical demands.
3. Impulsive synchronization seeking in general complex delayed dynam-ical networks with nonsymmetrical coupling. Based on the extended Halanay differential inequality, some criteria for global exponential synchronization of the impulsive controlled delayed dynamical networks are derived analytically. The main contributions of the work indicate two aspects:On the one hand, these criteria can provide an effective impulsive control scheme to synchronize an arbi-trary given delayed dynamical networks to a desired synchronization state even if the original given networks may be asynchronous itself. On the other hand, the controlled synchronization state can be selected as any arbitrary weighted average of all the states in the networks for purpose of practical control strategy, which reveals the contributions and influences of various nodes in synchroniza-tion seeking processes of the dynamical networks.
4. Impulsive control and synchronization of chaotic delayed systems. By using the impulsive control theory on delayed dynamical systems, some sim-ple yet generic criteria to guarantee impulsive stabilization and synchronization for a class of chaotic delayed systems are analytically derived. Subsequently, the generalized Halanay differential inequality in vector form may be applied to chaos-based secure communication systems with transmission delay, where a driving system and a response system are employed. Finally, a model of re-current neural networks with time-varying delays in the presence of impulsive connectivity among the neurons is addressed. Moreover, numerical simulations are worked out to further illustrate our theoretical results.
一.具有任意时滞的脉冲微分方程的稳定性理论.利用Lyapunov函数方法和Razumikhin技术,给出了一类具有任意时滞的非线性脉冲微分方程全局指数稳定的准则判据.其突出的特征就是去掉了一般性的限制条件,即时滞小于所有的脉冲间隔.此外,还得到了关于脉冲时滞动力系统的推广的Halanay不等式及其向量形式,这里删除了不含脉冲的时滞动力系统是稳定的预先假设.这样就可以通过脉冲去稳定一个不稳定的甚至是混沌的动力系统.这为实际中多个体网络系统的一致性、耦合振子的同步以及混沌时滞系统的控制与同步提供了理论基础.
二.多个体有向时滞网络系统的平均一致性问题.基于时滞动力系统的脉冲控制理论,提出了在通讯时滞环境中三种一致性协议.第一个协议是具有脉冲效应的一致性协议,结果表明通过设计协议中合适的脉冲增益和脉冲间隔,可使整个网络全局指数地达到一致.第二个是分布式δ-脉冲一致性协议,它能够处理具有随机切换拓扑网络的一致性问题.第三个是对任意通讯时滞都适用的协议,可以根据实际的需要来调节协议中时滞的大小.
三.基于脉冲控制下一般复杂动态网络的同步动力学.首先设计合理的脉冲控制器,然后应用脉冲时滞动力系统推广的Halanay不等式,给出了一般复杂时滞动态网络的全局同步化准则.这些准则能够提供一个新的和有效的控制方法来同步一个任意给定的时滞动态网络到一个期望的同步态,即便原来的网络本身是不同步的.这种同步态可以根据控制策略的目标而选取整个网络状态的权重平均,结果表明网络中不同的节点对网络的同步化影响大小是不一样的.
四.混沌时滞系统的脉冲控制与同步.基于时滞动力系统的脉冲控制理论,首先给出了一类混沌时滞系统脉冲控制和同步的判据；其次把推广的向量形式的Halanay微分不等式应用到保密通讯系统；最后研究了具有脉冲连接的时滞神经网络的指数稳定性问题,并把所获得的结果应用到时滞Hopfield神经网络,数值模拟进一步验证了理论结果的正确性.
The study of cooperative behavior and coordinated control in complex net-worked multi-agent systems has currently become one of the most challenging area of research from various fields such as biology or ecology, physics, applied mathematics, information science, computer science and control theory. This dissertation is mainly concerned with the issues of impulsive consensus prob-lems in complex delayed networked multi-agent systems and the relevant issues from the view of dynamics and control. In details, the fundamental contributions of these works are summarized in the following four aspects:
1. Stability theory of impulsive differential equations with any time delays. By utilizing Lyapunov function methods combined with the Razumikhin tech-nique, several criteria on exponential stability are derived analytically, which are substantially the extension and generalization of the corresponding results in re-cent literatures. Compared with some existing works, a distinctive feature of this work is to address exponential stability problems for any time delays, since the restrictive condition that the time delays are less than the length of all the im-pulsive intervals is actually removed here. Moreover, the previous results on Ha-lanay inequality for impulsive delayed dynamical systems are extended. In con-trast to the previous Halanay differential inequality, the primary contribution of this work is to remove the restrictive condition of a priori stability assumption for the corresponding delayed dynamical systems without impulses, so the result can be usually used to stabilize an unstable delayed dynamical system via impulses. Therefore, our work substantially extends the famous Halanay differential in-equality, which will play an important role in stability problems for consensus in delayed networked multi-agent systems, synchronization in complex delayed dynamical networks, control and synchronization in chaotic delayed systems.
2. Average consensus problems in directed delayed networked multi-agent systems. By employing the impulsive control theory on delayed dynamical sys- tems, three consensus protocols of the networks with communication delays are proposed. The first is a simple distributed consensus protocol with impulsive effects. And it is shown that a directed delayed networked multi-agent system can achieve average consensus globally exponentially by designing suitable im-pulsive gain and impulsive interval. The second is a distributedδ-consensus protocol in directed networks of dynamic agents having communication delays with stochastic switching communication graphs. The third is a distributed im-pulsive consensus protocol which is valid for any communication delays. Here, the communication delays can be adjusted according to practical demands.
3. Impulsive synchronization seeking in general complex delayed dynam-ical networks with nonsymmetrical coupling. Based on the extended Halanay differential inequality, some criteria for global exponential synchronization of the impulsive controlled delayed dynamical networks are derived analytically. The main contributions of the work indicate two aspects:On the one hand, these criteria can provide an effective impulsive control scheme to synchronize an arbi-trary given delayed dynamical networks to a desired synchronization state even if the original given networks may be asynchronous itself. On the other hand, the controlled synchronization state can be selected as any arbitrary weighted average of all the states in the networks for purpose of practical control strategy, which reveals the contributions and influences of various nodes in synchroniza-tion seeking processes of the dynamical networks.
4. Impulsive control and synchronization of chaotic delayed systems. By using the impulsive control theory on delayed dynamical systems, some sim-ple yet generic criteria to guarantee impulsive stabilization and synchronization for a class of chaotic delayed systems are analytically derived. Subsequently, the generalized Halanay differential inequality in vector form may be applied to chaos-based secure communication systems with transmission delay, where a driving system and a response system are employed. Finally, a model of re-current neural networks with time-varying delays in the presence of impulsive connectivity among the neurons is addressed. Moreover, numerical simulations are worked out to further illustrate our theoretical results.
引文
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