Xinzhi Liu

Nonlinear integral equations in abstract spaces

Preface. 1. Preliminaries. 2. Nonlinear Integral Equations in Banach Spaces. 3. Nonlinear Integro-Differential Equations in Banach Spaces. 4. Nonlinear Impulsive Integral Equations in Banach Spaces. References.

Robust impulsive synchronization of uncertain dynamical networks

This paper studies robust impulsive synchronization of uncertain dynamical networks. By utilizing the concept of impulsive control and the stability results for impulsive systems, several criteria for robust local and robust global impulsive synchronization are established for complex dynamical networks, in which the network coupling functions are unknown but bounded. Three examples are also worked through for illustrating the main results.

Stability of a class of linear switching systems with time delay

We consider a switching system composed of a finite number of linear delay differential equations (DDEs). It has been shown that the stability of a switching system composed of a finite number of linear ordinary differential equations (ODEs) may be achieved by using a common Lyapunov function method switching rule. We modify this switching rule for ODE systems to a common Lyapunov functional method switching rule for DDE systems and show that it stabilizes our model. Our result uses a Riccati-type Lyapunov functional under a condition on the time delay.

Uniform asymptotic stability of impulsive delay differential equations

Abstract In this paper, criteria on uniform asymptotic stability are established for impulsive delay differential equations using Lyapunov functions and Razumikhin techniques. It is shown that impulses do contribute to yield stability properties even when the underlying system does not enjoy any stability behavior. Some examples are also discussed to illustrate the theorems.

Input-to-state stability of impulsive and switching hybrid systems with time-delay

This paper investigates input-to-state stability (ISS) and integral input-to-state stability (iISS) of impulsive and switching hybrid systems with time-delay, using the method of multiple Lyapunov-Krasovskii functionals. It is shown that, even if all the subsystems governing the continuous dynamics, in the absence of impulses, are not ISS/iISS, impulses can successfully stabilize the system in the ISS/iISS sense, provided that there are no overly long intervals between impulses, i.e., the impulsive and switching signal satisfies a dwell-time upper bound condition. Moreover, these impulsive ISS/iISS stabilization results can be applied to systems with arbitrarily large time-delays. Conversely, in the case when all the subsystems governing the continuous dynamics are ISS/iISS in the absence of impulses, the ISS/iISS properties can be retained if the impulses and switching do not occur too frequently, i.e., the impulsive and switching signal satisfies a dwell-time lower bound condition. Several illustrative examples are presented, with their numerical simulations, to demonstrate the main results.

Stability analysis in terms of two measures

Part 1 Basic theory: definitions of stability basic Lyapunov theory comparison method converse theorem boundedness and Lagrange stability invariance principle. Part 2 Refinements: several Lyapunov functions perturbations of Lyapunov functions method of vector Lyapunov functions perturbed systems integral stability method of higher derivatives cone-valued Lyapunov functions. Part 3 Extensions: delay differential equations impulsive differential systems stabilization of control systems impulsive integro-differential systems discrete systems random differential systems dynamic systems on time scales. Part 4 Applications: holomorphic mechanical systems motion of winged aircraft models from economics motion of a length-varying pendulum population models angular motion of rigid bodies.

Stability results for impulsive differential systems with applications to population growth models

This paper establishes some stability criteria for impulsive differential systems. It is shown that impulses do contribute to yield stability properties even when the corresponding differential system without impulses does not enjoy any stability behavior. As an application, these results are applied to some population growth models

Permanence of population growth models with impulsive effects

This paper establishes criteria for permanence of populations which undergo impulsive effects at fixed times between intervals of continuous evolution governed by a differential system. It is also shown that suitable impulses may prevent the extinction or unbounded growth of populations whose evolutions are otherwise governed solely by a differential system. Examples are provided to demonstrate the application of the results obtained.

Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft

Necessary and sufficient conditions for impulsive controllability of linear dynamical systems are obtained, which provide a novel approach to problems that are basically defined by continuous dynamical systems, but on which only discrete-time actions are exercised. As an application, impulsive maneuvering of a spacecraft is discussed.

Global asymptotic stability of high-order Hopfield type neural networks with time delays

Abstract This paper studies the problem of global asymptotic stability of a class of high-order Hopfield type neural networks with time delays. By utilizing Lyapunov functionals, we obtain some sufficient conditions for the global asymptotic stability of the equilibrium point of such neural networks in terms of linear matrix inequality (LMI). Numerical examples are given to illustrate the advantages of our approach.