Stability Of Two Classes Of Impulsive Delayed Differential Systems And Neural Networks

As a class of hybrid system, discontinuous system is an evergreen hot topic in the field of nonlinear systems. Impulsive differential system is one kind of special discontinuous systems, which has been widely applied in engineering fields to describe the instantaneous features of control systems, including satellite orbit technology, industrial robots technology, etc. In the past several decades, impulsive differential systems have attracted increasing attention because they provide a mathematical modelling in many scientific and engineering fields, such as economics, physics, biomedical engineering and communication engineering. Furthermore, since neural networks have been applied in many fields, including image processing, pattern recognition, associative memory, signal processing, and secure communication etc, they are also the hot topic of nonlinear systems and have been in-depth studied by more and more researchers in recent years.In this dissertation, based on the presented theories of the impulsive control differential systems, the novel theories for these delayed impulsive systems are further investigated. Moreover, these presented theories are applied to research memristive neural networks systems, which are hot topics recently. Two main aspects are studied in this dissertation. On the one hand, the stability theories of impulsive delayed differential system are intensively studied. On the other hand, the theories of impulsive differential system are used to discuss the stability problems of neural dynamic systems. These two main research contents are correlative. Both of them are the important and hot topic in the field of nonlinear sciences.In Chapter Two, the stability of linear and nonlinear impulsive delayed differential systems with delayed impulses is mainly discussed. Since delays may exist in the transmission of impulses, it is necessary to consider delayed impulses when we research impulsive delayed differential systems. A mathematical modelling of impulsive delayed differential system with delayed impulses is firstly constructed. Then, by utilizing Lyapunov function, Razumikhin technique and other analysis methods, the sufficient criteria of uniform stability and exponential stability for linear and nonlinear impulsive delayed differential systems with delayed impulses are obtained. Finally, two numerical examples are illustrated to show the effectiveness of the theoretical analysis in this chapter.In Chapter Three, the stability of linear and nonlinear impulsive delayed differential systems with unfixed impulse moments is mainly studied. The impulse does not always occur at the fixed-time point, and it may occur in a little range of time, which is called impulse time window in this Chapter. Based on the definition of impulse time window, a novel differential system, i.e., impulsive delayed differential system with impulse time windows, is firstly constructed, in which an impulse can stochastically occur in a small time interval. Then, by using Lyapunov function, integral method and mathematical induction method, the sufficient conditions for the linear and nonlinear impulsive delayed differential systems with impulse time windows are presented. Finally, three numerical examples are illustrated to show the effectiveness of the theoretical analysis in this chapter.In the fourth Chapter, the periodic solution’s existence and stability problems of the impulsive delayed neural networks with periodic coefficients are mainly investigated. By using the integral average form of periodic function and some related inequalities derived by Lisena, combined with contracting mapping principle, fixed point theorem and impulsive differential inequalities, the periodic solution’s existence, uniqueness and global exponential stability of the impulsive delayed neural networks with periodic coefficients are researched. And several sufficient criteria are obtained. Finally, one numerical example is illustrated to show the effectiveness of the theoretical analysis in this chapter.Chapter Five mainly deals with the robust stability of memristive delayed recurrent neural networks with impulses disturbance. And stabilizing unstable memristive delayed neural network is another important research content of this Chapter. Memristive neural networks can be implemented by VLSI circuits, and it may undergo abrupt unexpected perturbations, such as sudden noise, voltage instantaneous change, switching action etc, which can be seen as impulse disturbance phenomena. Therefore, it is necessary to study the memristive neural networks with impulsive effects. Basing on the framework of Filippovs’ s solution and differential inclusion theorem, a generalized impulsive delayed memristive recurrent neural network model is firstly formulated in this Chapter. Then, taking advantage of impulsive differential inequality, Lyapunov function, Lyapunov-Krasovskii-type functional and LMI method, some exponential stability criteria are obtained, which implies that the initial stable systems have the capacity of resisting disturbance under some proper conditions. Furthermore, hybrid impulsive and adaptive feedback controllers are applied to control memristive delayed neural network. An impulsive controlled memristive neural network model with time-varying delays is formulated and researched. By utilizing Lyapunov function, mathematical induction method, Razumikhin technique and linear matrix inequality, sufficient conditions of exponential stability for the impulsive controlled memristive delayed neural network are derived. Finally, three numerical examples are illustrated to show the effectiveness of the theoretical analysis in this chapter.In the final Chapter, the conclusions of this paper and some future possible research contents are given. In this dissertation, we mainly study the stability for impulsive differential system with delayed impulses, impulsive delayed differential system with impulse time windows, impulsive delayed neural networks with periodic coefficients, memristive delayed recurrent neural networks with impulses disturbance and impulsive controlled memristive neural networks. The research contents of this dissertation have certain theoretical value and practical guiding significance in the field of impulsive differential systems and neural networks, which enrich the theoretical studies of nonlinear systems. In the future, we can further investigate dynamical qualities of impulsive delayed differential systems with unfixed impulse moments, impulsive memristive delayed neural networks and complex-valued memristive delayed neural networks, which has significative exploration and contribution for the theories of nonlinear systems.