测度链上脉冲系统的稳定性及控制问题
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摘要
为统一连续和离散分析学,Stefan Hilger于1988年在其博士论文中建立了测度链理论.测度链上的动力系统理论为人们同时研究连续和离散系统提供了统一的框架.提到动力系统,现实世界中许多物理过程会在离散时刻遭受状态的突变,通过脉冲微分方程建立的模型可有效地反映此类物理现象的本质.近年来,脉冲微分系统理论得到了广泛的研究.从建模的角度来看,通过可以综合连续和离散时间空间的动力系统为此类状态突变的物理过程建立数学模型更具现实意义.
本文研究测度链上脉冲系统的稳定性及控制问题.主要目标分别是分析测度链上线性时变脉冲系统的状态能控能观性,建立测度链上有无时滞脉冲系统的稳定性准则,并将所得结论和解决问题所用的方法应用于实际中.本文的主要内容包括以下四个部分.
第二章考虑测度链上线性时变脉冲系统的能控能观性.通过参数变异法建立了测度链上线性时变脉冲系统解的表达形式,该结论在本节中起着重要作用.分别建立了该类脉冲系统状态能控能观的充分和必要条件.所得结论用于讨论相应的时不变脉冲系统的能控能观性.结果显示当测度链退化为实数空间时,我们的结论包含现有文献中关于连续脉冲系统的能控能观性结论,并且包含现有测度链上线性系统的能控能观性的结论.
第三章研究了测度链上非线性脉冲系统的双测度稳定性.通过测度链上的数学归纳法建立了新的比较定理,并且利用其通过比较方法建立了测度链上脉冲系统双测度稳定的充分条件.所得结论用于研究具体的一类测度链上脉冲系统的渐近稳定性.在本章的第二部分,利用Lyapunov直接方法得到了测度链上脉冲系统的(h0,h)-(一致)稳定性、(h0,h)-(一致)渐近稳定性和(h0：h)-不稳定性准则.通过算例验证了所得结论的有效性.
在第四章,研究了测度链上非线性脉冲泛函系统的指数稳定性.为了建立一般测度链上脉冲泛函系统的指数稳定性准则,首先考虑了离散脉冲时滞系统的指数稳定性问题.通过Lyapunov-Razumikhin方法建立了离散脉冲时滞系统的全局指数稳定性准则.然后,通过Lyapunov泛函方法获得了离散脉冲时滞系统指数稳定的充分条件.由所得离散系统的相关结论和研究方法,建立了测度链上脉冲泛函系统的指数稳定性准则.得到了两个Razumikhin-型全局指数稳定的结论,其中一个结论为无脉冲扰动的泛函系统保持指数稳定性提供了充分条件,另一结论则可用来作为设计脉冲控制器稳定泛函系统的充分条件.所得结论用来研究测度链上具体几类非线性脉冲时滞系统的指数稳定性问题.最后,本章通过Lyapunov泛函方法研究了测度链上脉冲泛函系统的指数稳定性.所得结论分别说明,可以通过选择合适的脉冲控制实现不稳定系统的指数稳定性,脉冲控制可以有效地控制稳定系统实现其指数稳定,以及动态性能良好的脉冲系统可以抵抗何种强度的脉冲扰动.最后,通过数值仿真实验验证了所得结论.
在第五章,上述所得的稳定性结论应用于具体的几类控制问题.首先,设计了一致及非一致非线性脉冲控制器来稳定连续和离散混沌系统.具作者所知,离散时滞复杂动态网络的脉冲指数同步问题首次在本文中进行了研究.通过第四章所得Razumikhin-型的稳定性结论,设计了脉冲控制器实现离散时滞复杂动态网络的指数同步.最后,提出了测度链上复杂动态网络模型,并设计了一类混合脉冲控制器实现测度链上线性和一类非线性复杂动态网络与目标状态的一致.结果显示通过对网络的一小部分节点施加脉冲控制即可有效地实现测度链上动态网络与目标状态的一致.本章所得结论均通过数值仿真实例进行了验证.
Stefan Hilger introduced the theory of time scales in his PhD thesis in1988in order to unify the continuous and discrete analysis. The theory of dynamic systems on time scales provides us a framework to study the continuous and discrete system simultaneously. In terms of dynamic systems, various physical processes undergo abrupt changes of their states at discrete moments, this kind of systems can be modeled by impulsive differential equations. Recently, the theory of impulsive systems has been investigated extensively. From the modeling point of view, it is perhaps more reasonable to model evolution process subjected to impulsive perturbations by a dynamic system which incorporates both continuous and discrete time space, namely impulsive systems on time scales.
Stability and control problems of impulsive systems on time scales are inves-tigated. The objectives of this thesis are to analyze the state controllability and observability of linear impulsive time-varying systems on time scales, to estab-lish stability criteria for nonlinear impulsive systems with and without delay on time scales, and to apply the theory and method to real-world applications. The contents of this thesis consist of four parts which are listed as follows.
Chapter2considers controllability and observability of linear impulsive time-varying systems on time scales. By employing the method of parameter variation, the form of corresponding system's solution is obtained, which plays an important role in this chapter. Sufficient and necessary conditions are established for the linear impulsive time-varying systems on time scales, respectively. These results are then used to discuss the systems'time-invariant counterparts. It is shown that when the time scale reduce to the real numbers, our results contains the existing results about the linear impulsive continuous systems, and some other results about linear systems without impulses on time scales can be our results' special cases.
In Chapter3, stability in terms of two measures is investigated for the non-linear impulsive systems on time scales. A new comparison result is obtained by using the mathematical induction method on time scales, which is used to establish the sufficient conditions for the (h0,h)-stability of nonlinear impulsive systems on time scales by comparison method. The asymptotic stability of a class of nonlinear impulsive systems on time scales is discussed by using the obtained (h0, h)-stability criteria. In the second part of this chapter, we use Lyapunov direct method to study stability in terms of two measures for nonlinear impulsive systems on time scales.(h0,h)-(uniform) stability,(h0,h)-(uniform) asymptotic stability,(h0,h)-nstability criteria are established, respectively. Some examples are also discussed to illustrate our theoretical results.
Exponential stability problem is considered in Chapter4about nonlinear impulsive functional systems. In order to establish exponential stability criteria for impulsive functional systems on general time scales, the exponential stability problem of impulsive discrete delay systems is concerned. By using Lyapunov-Razumikhin method, the global exponential stability criteria are obtained. Then, Lyapunov functional method is employed to construct sufficient conditions about the exponential stability for impulsive discrete delay systems. Inspired by the results obtained about discrete systems and the method used, the exponential stability criteria about impulsive functional systems on time scales are estab-lished. Two Razumikhin-type global exponential stability results are derived, one of which provides sufficient conditions for maintaining the global exponential sta-bility property of the trivial solution of the functional system without impulsive perturbations, while another of which can be used to impulsively stabilize func-tional systems. These results are also used to discuss the stability problem of some special cases of nonlinear impulsive systems on time scales. The last part of this chapter proposes the Lyapunov functional method in the content of impulsive exponential stabilization of functional systems. Several results are constructed to show that an unstable system can be exponentially stabilized by appropriate se-quence of impulses, impulses can contribute to make a stable system exponentially stable, and to what extent can a well-behaved system preserve its stability prop-erties under impulsive perturbations, respectively. Some examples with numerical simulations are exploited to demonstrate the effectiveness of our results.
In Chapter5, several applications of the results obtained in previous parts are concerned. Uniform and nonuniform impulsive controllers are designed to sta-bilize the continuous and discrete chaos systems. As far as the author knows, the impulsive synchronization problem of discrete delay complex dynamic networks is firstly considered in this thesis. By using the Razumikhin-type stability re- sults derived in Chapter4, the impulsive controller is constructed to realize the exponential synchronization of the discrete delay complex dynamic networks. Fi-nally, the complex dynamic networks on time scales is introduced and a hybrid impulsive controller is designed to consensus the networks on time scales to the objective states. It is shown that the state-coupled networks on time scales can be effectively forced to the goal trajectory by adding impulsive control to a small pro-portion of the networks'nodes. Numerical examples and simulations are presented throughout this chapter to illustrate the results.
本文研究测度链上脉冲系统的稳定性及控制问题.主要目标分别是分析测度链上线性时变脉冲系统的状态能控能观性,建立测度链上有无时滞脉冲系统的稳定性准则,并将所得结论和解决问题所用的方法应用于实际中.本文的主要内容包括以下四个部分.
第二章考虑测度链上线性时变脉冲系统的能控能观性.通过参数变异法建立了测度链上线性时变脉冲系统解的表达形式,该结论在本节中起着重要作用.分别建立了该类脉冲系统状态能控能观的充分和必要条件.所得结论用于讨论相应的时不变脉冲系统的能控能观性.结果显示当测度链退化为实数空间时,我们的结论包含现有文献中关于连续脉冲系统的能控能观性结论,并且包含现有测度链上线性系统的能控能观性的结论.
第三章研究了测度链上非线性脉冲系统的双测度稳定性.通过测度链上的数学归纳法建立了新的比较定理,并且利用其通过比较方法建立了测度链上脉冲系统双测度稳定的充分条件.所得结论用于研究具体的一类测度链上脉冲系统的渐近稳定性.在本章的第二部分,利用Lyapunov直接方法得到了测度链上脉冲系统的(h0,h)-(一致)稳定性、(h0,h)-(一致)渐近稳定性和(h0：h)-不稳定性准则.通过算例验证了所得结论的有效性.
在第四章,研究了测度链上非线性脉冲泛函系统的指数稳定性.为了建立一般测度链上脉冲泛函系统的指数稳定性准则,首先考虑了离散脉冲时滞系统的指数稳定性问题.通过Lyapunov-Razumikhin方法建立了离散脉冲时滞系统的全局指数稳定性准则.然后,通过Lyapunov泛函方法获得了离散脉冲时滞系统指数稳定的充分条件.由所得离散系统的相关结论和研究方法,建立了测度链上脉冲泛函系统的指数稳定性准则.得到了两个Razumikhin-型全局指数稳定的结论,其中一个结论为无脉冲扰动的泛函系统保持指数稳定性提供了充分条件,另一结论则可用来作为设计脉冲控制器稳定泛函系统的充分条件.所得结论用来研究测度链上具体几类非线性脉冲时滞系统的指数稳定性问题.最后,本章通过Lyapunov泛函方法研究了测度链上脉冲泛函系统的指数稳定性.所得结论分别说明,可以通过选择合适的脉冲控制实现不稳定系统的指数稳定性,脉冲控制可以有效地控制稳定系统实现其指数稳定,以及动态性能良好的脉冲系统可以抵抗何种强度的脉冲扰动.最后,通过数值仿真实验验证了所得结论.
在第五章,上述所得的稳定性结论应用于具体的几类控制问题.首先,设计了一致及非一致非线性脉冲控制器来稳定连续和离散混沌系统.具作者所知,离散时滞复杂动态网络的脉冲指数同步问题首次在本文中进行了研究.通过第四章所得Razumikhin-型的稳定性结论,设计了脉冲控制器实现离散时滞复杂动态网络的指数同步.最后,提出了测度链上复杂动态网络模型,并设计了一类混合脉冲控制器实现测度链上线性和一类非线性复杂动态网络与目标状态的一致.结果显示通过对网络的一小部分节点施加脉冲控制即可有效地实现测度链上动态网络与目标状态的一致.本章所得结论均通过数值仿真实例进行了验证.
Stefan Hilger introduced the theory of time scales in his PhD thesis in1988in order to unify the continuous and discrete analysis. The theory of dynamic systems on time scales provides us a framework to study the continuous and discrete system simultaneously. In terms of dynamic systems, various physical processes undergo abrupt changes of their states at discrete moments, this kind of systems can be modeled by impulsive differential equations. Recently, the theory of impulsive systems has been investigated extensively. From the modeling point of view, it is perhaps more reasonable to model evolution process subjected to impulsive perturbations by a dynamic system which incorporates both continuous and discrete time space, namely impulsive systems on time scales.
Stability and control problems of impulsive systems on time scales are inves-tigated. The objectives of this thesis are to analyze the state controllability and observability of linear impulsive time-varying systems on time scales, to estab-lish stability criteria for nonlinear impulsive systems with and without delay on time scales, and to apply the theory and method to real-world applications. The contents of this thesis consist of four parts which are listed as follows.
Chapter2considers controllability and observability of linear impulsive time-varying systems on time scales. By employing the method of parameter variation, the form of corresponding system's solution is obtained, which plays an important role in this chapter. Sufficient and necessary conditions are established for the linear impulsive time-varying systems on time scales, respectively. These results are then used to discuss the systems'time-invariant counterparts. It is shown that when the time scale reduce to the real numbers, our results contains the existing results about the linear impulsive continuous systems, and some other results about linear systems without impulses on time scales can be our results' special cases.
In Chapter3, stability in terms of two measures is investigated for the non-linear impulsive systems on time scales. A new comparison result is obtained by using the mathematical induction method on time scales, which is used to establish the sufficient conditions for the (h0,h)-stability of nonlinear impulsive systems on time scales by comparison method. The asymptotic stability of a class of nonlinear impulsive systems on time scales is discussed by using the obtained (h0, h)-stability criteria. In the second part of this chapter, we use Lyapunov direct method to study stability in terms of two measures for nonlinear impulsive systems on time scales.(h0,h)-(uniform) stability,(h0,h)-(uniform) asymptotic stability,(h0,h)-nstability criteria are established, respectively. Some examples are also discussed to illustrate our theoretical results.
Exponential stability problem is considered in Chapter4about nonlinear impulsive functional systems. In order to establish exponential stability criteria for impulsive functional systems on general time scales, the exponential stability problem of impulsive discrete delay systems is concerned. By using Lyapunov-Razumikhin method, the global exponential stability criteria are obtained. Then, Lyapunov functional method is employed to construct sufficient conditions about the exponential stability for impulsive discrete delay systems. Inspired by the results obtained about discrete systems and the method used, the exponential stability criteria about impulsive functional systems on time scales are estab-lished. Two Razumikhin-type global exponential stability results are derived, one of which provides sufficient conditions for maintaining the global exponential sta-bility property of the trivial solution of the functional system without impulsive perturbations, while another of which can be used to impulsively stabilize func-tional systems. These results are also used to discuss the stability problem of some special cases of nonlinear impulsive systems on time scales. The last part of this chapter proposes the Lyapunov functional method in the content of impulsive exponential stabilization of functional systems. Several results are constructed to show that an unstable system can be exponentially stabilized by appropriate se-quence of impulses, impulses can contribute to make a stable system exponentially stable, and to what extent can a well-behaved system preserve its stability prop-erties under impulsive perturbations, respectively. Some examples with numerical simulations are exploited to demonstrate the effectiveness of our results.
In Chapter5, several applications of the results obtained in previous parts are concerned. Uniform and nonuniform impulsive controllers are designed to sta-bilize the continuous and discrete chaos systems. As far as the author knows, the impulsive synchronization problem of discrete delay complex dynamic networks is firstly considered in this thesis. By using the Razumikhin-type stability re- sults derived in Chapter4, the impulsive controller is constructed to realize the exponential synchronization of the discrete delay complex dynamic networks. Fi-nally, the complex dynamic networks on time scales is introduced and a hybrid impulsive controller is designed to consensus the networks on time scales to the objective states. It is shown that the state-coupled networks on time scales can be effectively forced to the goal trajectory by adding impulsive control to a small pro-portion of the networks'nodes. Numerical examples and simulations are presented throughout this chapter to illustrate the results.
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