Asymptotic behavior of nonlinear systems with impulses: Application to Hopfield neural networks

In this paper, we introduce a new type of stability for nonlinear impulsive systems of differential equations, namely practical h-stability. By using the Lyapunov stability theory, some sufficient conditions which guarantee practical h-stability are established. Our original results generalize well-known fundamental stability results, practical stability, practical exponential stability and practical asymptotic stability for nonlinear time-varying impulsive systems. Then two classes of nonlinear impulsive systems, namely perturbed and cascaded impulsive systems, are discussed. Furthermore, the problem of practical h-stabilization for certain classes of nonlinear impulsive systems is considered. Finally, two numerical examples are given to show the effectiveness of our theoretical results.

This article is concerned with the practical stability performance of nonlinear impulsive stochastic functional differential systems driven by G-Brownian motion (G-ISFDSs). Comparing with traditional Lyapunov stability theory, practical stability can portray qualitative behavior and quantitative properties of suggested systems. By employing G-Itô formula, Lyapunov-Razumikhin approach and stochastic analysis theory, some novel conditions for pth moment practical exponential stability and quasi sure global practical uniform exponential stability of G-ISFDSs are established. The obtained results show that impulses may influence dynamic behavior of the addressed system. Two numerical examples are given to verify the validity of our developed results.

SummaryThis paper is concerned with the problem of practical exponential stability for hybrid impulsive stochastic functional differential systems with delayed impulses, which comprise three classes of systems: the systems with unstable continuous stochastic dynamics and stable discrete dynamics, the systems with stable continuous stochastic dynamics and unstable discrete dynamics, and the systems where both the continuous stochastic dynamics and the discrete dynamics are stable. By using the Lyapunov‐Razumikhin approach, several new sufficient conditions for the practical exponential stability are established for each class of systems. It shows that the stabilizing and destabilizing delayed impulses that satisfy some conditions on their frequency and amplitude can stabilize the systems with unstable continuous stochastic dynamics in the practical exponential stability sense and ensure the practical exponential stability of the systems with stable continuous stochastic dynamics, respectively. Conversely, if the continuous stochastic dynamics are practically exponentially stable and the delayed impulses are stabilizing, then the systems can be practically exponentially stable regardless of the restrictions on the delayed impulses frequency and amplitude. Finally, two numerical examples are presented to illustrate the efficiency of the results.

Mean-square exponential stability of impulsive conformable fractional stochastic differential system with application on epidemic model

Exponential stability of non-linear neutral stochastic delay differential system with generalized delay-dependent impulsive points

Stability and boundedness of nonlinear impulsive systems in terms of two measures via perturbing Lyapunov functions

Stability of nonlinear variable-time impulsive differential systems with delayed impulses

Almost sure exponential stability and stochastic stabilization of stochastic differential systems with impulsive effects

Stability and boundedness criteria of nonlinear impulsive systems employing perturbing Lyapunov functions

formula omitted]-stability theorems of nonlinear impulsive functional differential systems

1R11. Qualitative Methods in Nonlinear Dynamics: Novel Approaches to Liapunov’s Matrix Functions. Pure and Applied Mathematics, Vol 246. - AA Martynyuk (Stability of Processes Dept, Inst of Mech, Natl Acad of Sci, Kiev, Ukraine). Marcel Dekker, New York. 2002. 301 pp. ISBN 0-8247-0735-4. $150.00. Reviewed by RA Ibrahim (Dept of Mech Eng, Wayne State Univ, 5050 Anthony Wayne Dr, Rm 2119 Engineering Bldg, Detroit MI 48202).This addition to the series of pure and applied mathematics monographs deals with the modern theory of dynamics of continuous, discrete-time, and impulsive nonlinear systems using Liapunov matrix-valued functions. It is known that this theory is originally rooted in the developments of Poincare´’s and Liapunov’s ideas for treating nonlinear systems of differential equations. The book is devoted to introduce mathematical theorems for analyzing Liapunov matrix-valued functions in five chapters. The first chapter introduces the mathematical statements of qualitative methods of the general equations of continuous nonlinear systems. The definitions of various types of stability are introduced for nonlinear non-autonomous systems. Scalar, vector, and matrix-valued Liapunov functions, and the comparison principle were introduced to allow the estimation of the distance from every point of the system integral curve to the origin when the time changes from the fixed value. Other stability theorems, based on the work of the author and others, are stated with their proofs. Some methods for analyzing continuous nonlinear systems of hierarchical structure are presented in Chapter 2. These methods are supported by an example of third-order systems. Some stability theorems of systems with regular hierarchy subsystems, large systems, and their extension to overlapping decomposition are discussed. The problem of poly-stability of nonlinear systems with separable motion is analyzed as an application of the matrix-valued function. Chapter 2 includes the concepts of integral and Lipschitz stability based on the use of the principle of comparison with a matrix-valued Liapunov function. Chapter 3 presents the qualitative analysis of discrete-time systems that model mechanical systems with impulse control, digital computing devices, population dynamics, chaotic dynamics of economical systems, and many others. These systems are usually described in terms of difference equations whose stability conditions are defined in terms of the matrix-valued functions method. Chapter 4 introduces the stability of nonlinear dynamical systems subjected to impulsive perturbations. The impulsive system of differential equations are stated for general class of dynamical systems. The stability definitions presented in Chapter 2 for ordinary differential equations are adapted for the impulsive systems. Conditions and definitions of uniqueness, continuity, boundedness, and stability of solutions of impulsive systems are presented. Chapter 5 culminates the theorems and general results presented in the first four chapters by introducing some applications. They include numerical algorithms of constructing a point network supported by illustrative examples. The oscillations and stability of coupled mechanical systems are demonstrated for three pendulums through elastic springs and coupled two non-autonomous parametric oscillators. Qualitative Methods in Nonlinear Dynamics: Novel Approaches to Liapunov’s Matrix Functions is recommended to researchers who are studying the mathematical stability theory of dynamical systems. The author is commended for introducing illustrative examples from different applications to support the idea of Liapunov’s matrix functions.

Since the quantum system, a classical example of complex-valued system, is one of the foci of ongoing research, in this paper, the issue of existence and uniqueness of solutions to nonlinear impulsive differential systems defined in complex fields, to be brief, complex-valued nonlinear impulsive differential systems, is addressed. The existence and uniqueness conditions of solutions of such systems are established by fixed point theory.

Exponential stability of nonlinear time-delay systems with delayed impulse effects

A novel finite-time stability criteria and controller design for nonlinear impulsive systems

In this paper we develop Lyapunov and invariant set stability theorems for non-linear impulsive dynamical systems. Furthermore, we generalize dissipativity theory to non-linear dynamical systems with impulsive effects. Specifically, the classical concepts of system storage functions and supply rates are extended to impulsive dynamical systems providing a generalized hybrid system energy interpretation in terms of stored energy, dissipated energy over the continuous-time system dynamics and dissipated energy over the resetting instants. Furthermore, extended Kalman‐Yakubovich‐Popov conditions in terms of the impulsive system dynamics characterizing dissipativeness via system storage functions are derived. Finally, the framework is specialized to passive and non-expansive impulsive systems to provide a generalization of the classical notions of passivity and non-expansivity for non-linear impulsive systems. These results are used in the second part of this paper to develop extensions of the small gain and positivity theorems for feedback impulsive systems as well as to develop optimal hybrid feedback controllers.