Stability Analysis And Stabilization Control For Several Classes Of Time-Scale Systems

Stability theory plays a dominant role in the research field of system theory and system engineering.Up to now,fruitful approaches have been proposed for stability analysis of different kinds of systems,such as the matrix eigenvalue method for lin-ear time-invariant systems,the Lyapunov's second method for nonlinear systems,the Razumikhin and Lyapunov-Krasovskii stability theorems for time-delay systems.It should be noted that the majority of the existing stability results are about continu-ous or discrete systems,and the existing stability criteria are only effective for one of them.Moreover,due to the complexity and diversity of some natural phenomena,it is no longer effective to build related models based on continuous and discrete systems.Therefore,it is interesting and necessary to study systems on time scales.The main contents can be summarized as follows:1.By introducing the time-scale type uniformly stable function,uniformly asymptotically stable function and uniformly exponentially stable function,a less conservative Lyapunov inequality is proposed,whose right side is allowed to be pos-itive or non-negative at some time.Based on this inequality,some necessary and sufficient conditions for asymptotic stability,exponential stability,uniformly expo-nential stability of linear time-varying systems on time scales are obtained.2.By introducing the time-scale type uniformly asymptotically stable function,the definitions of uniform convergence set and overshoot,two stability results about nonlinear time-delay systems are proposed.First,a less conservative Razumikhin theorem is presented,in which the time-scale time derivative of Lyapunov function can be positive or non-negative at some time.Second,a conservative Lyapunov-Krasovskii theorem is provided by requiring the time-scale time derivative of relating Lyapunov functional to be negative,and then,a more relaxed stability theorem is proposed,in which the time-scale time derivative of Lyapunov functional is allowed to be non-negative at some time point.3.By introducing the time-scale type uniformly asymptotically stable function,a new Lyapunov-Krasovskii type theorem is provided for uniform asymptotic stabil-ity and uniform exponential stability of delay impulsive systems on time scales.It is shown that this Lyapunov-Krasovskii stability criterion does not require the time derivative of the Lyapunov functional to be necessarily non-positive on each impul-sive interval.Then,asymptotic stability problems of delay impulsive systems on time scales are investigated based on the Razumikhin approach.The advantage of this method is that the length of each impulse interval does not depend on the time delay,which results in the fact that the state trajectory may not decrease instantly and sharply at each impulsive point.4.By considering the stabilizing and destabilizing behaviors of switching sig-nals,a more general stability criterion is proposed for nonlinear switched systems on time scales,in which the value of Lyapunov function at some switching instants may not necessarily decrease.Then,based on the discretized Lyapunov function technique,a sufficient condition is derived for stability analysis of linear switched systems with all subsystems unstable on a special kind of time scale.5.By constructing a new Wirtinger-based inequality on time scales,an im-proved and simple synchronization criteria is proposed for complex dynamical net-works with discrete time delays on time scales.Furthermore,based on an effective pinning impulsive control strategy,a finite-time synchronization theorem is provided for nonlinear complex dynamical networks on time scales.