Impulsive And Switching Control Of Several Special Classes Of Delayed Neural Networks

In the past decades, various types of delayed neural networks,such as delayed cellular neural network, delayed Hopfield neural network, delayed Cohen-Grossberg neural network, delayed BAM neural network and delayed memristive neural network have been extensively studied and many excellent results related to the subject have been obtained. The extensive study of this area is mainly due to its wide application in pattern recognition, signal processing, associative memory, financial industry, and optimization and so on. As we all know, the dynamical properties of neural networks, such as stability, bifurcation and chaos, play very important roles in the design of neural network. Among of them, stability issue is the main concern which has been extensively studied in the recent years. However, sometimes the delayed neural network may not achieve a stable state, and then controller has to be added to the network. There are many effective control strategies have been proposed, such as intermittent feedback control, pinning control, impulsive control, switching control, adaptive control. A significant trend of controlled stability of delayed neural networks is that a designed controller should reduce the control cost and be applicable in practice. Among the proposed control schemes, impulsive control and switching control is attractive because it needs small control gains and acts only at discrete times; thus control cost and the amount of transmitted information can be reduced drastically. Due to these advantages, in this dissertation we focus on the stability problem of delayed neural networks via impulsive control and switching control. Drawing on past research, many novel results are obtained in this dissertation and the main contributions and originality contained are as follows:â‘  Stability of the inertial delayed BAM neural network via impulsive controlThe issues of stability of inertial BAM neural networks with time-varying delays via impulsive control are investigated. First, by choosing a proper variable substitution, the original inertial delayed BAM neural networks can be rewritten as first-order differential equations. Second, based on the Lyapunov functional method and the comparison principle for impulsive systems, we derive some sufficient conditions guaranteeing the exponential stability of the system via impulsive control. For different variable transformation, the first-order differential equations obtained from the original system are different, and the corresponding designed impulsive controllers are different. Third, based on the optimization method, the more effective and less conservative impulsive controller is designed to ensure the stability of inertial delayed BAM neural networks. Finally, two numerical examples are proposed to illustrate our theoretical results.â‘¡ stability of periodic solution of delayed Cohen-Grossberg neural networks via impulsive controlThe issues of existence, uniqueness and global exponential stability of periodic solution of delayed Cohen-Grossberg neural networks under impulsive control are investigated. Some novel delay-independent criteria are obtained by using contraction mapping theorem, Lyapunov function and comparison principle for impulsive systems. The novelty of the obtained results is that the systems may be originally unstable or divergent, but they will admit a periodic solution which is globally stable via impulsive control. Finally,simulation results demonstrate the effectiveness of the proposed results.â‘¢ Stability of delayed memristive neural networks with time-varying impulsesThe issues of the stability problem of the memristive delayed neural networks with time-varying impulses are investigated. An impulsive sequence is said to be destabilizing if the impulsive effects can suppress the stability of dynamical systems. Conversely, an impulsive sequence is said to be stabilizing if it can enhance the stability of dynamical systems. Different from the existing results, here, we consider the impulses including the stabilizing and destabilizing impulses we call time-varying impulses. By utilizing the theories of set-valued maps and differential inclusion, the Lyapunov method and comparison principle for impulsive systems, the stability of delayed memristive neural networks under time-varying impulses is analyzed and several easy to verify conditions are derived in the framework of dwell times. It is shown that the time intervals between two stabilizing impulses instants are required to be confined by the upper bound and the lower bounds for destabilizing impulsive intervals is defined to guarantee the global exponential stability. The effectiveness of the proposed results is illustrated by an example.â‘£ Stability of switched neural network with time-varying delayThe issues of designing appropriate switching law to stabilize the switched neural network with time-varying delay are investigated. There are three cases are considered in this paper: all subsystems are unstable, all subsystems are stable, stable subsystems and unstable subsystems coexist in the switched systems. Different from existing results in the literature considering that the switching behaviors may cause the state divergence which is going against the stability of the switched systems, here, we consider the stabi-lization effect of switching behaviors. Based on this idea, the stability problems of the switched system under three different cases are studied. Firstly, by utilizing the discretized Lyapunov function and the extended comparison principle from impulsive systems, sufficient conditions for the stability of switched delayed neural networks with all subsystems being unstable are obtained, these criteria are simple in form and easy to verify in practice. Secondly, the discretized Lyapunov function approach is generalized to the cases when all subsystems are stable. By using the mathematical induction method, the stability of switched delayed neural networks with all stable subsystems is analyzed and several easy to verify conditions are derived to ensure the exponential stability of the systems with higher convergence rate. Thirdly, combining the results of the above two cases,the stability problem of the switched delayed neural networks with stable subsystems and unstable subsystems is investigated. The main idea is to use the stabilization property of switching behaviors and stable subsystems to compensate the divergence effect caused by unstable subsystems. By using the discretized Lyapunov function approach and the extended comparison principle for impulsive systems, sufficient condition for switched neural networks with time-varying delay is proposed. Due to the stabilization effect of switching behavior, the switched systems can put up with longer running time of the unstable subsystems compare with the previous results on this problem. Finally, the effectiveness of the proposed results is illustrated by several examples.