Xiaodi Li

Impulsive Control for Existence, Uniqueness, and Global Stability of Periodic Solutions of Recurrent Neural Networks With Discrete and Continuously Distributed Delays

In this paper, a class of recurrent neural networks with discrete and continuously distributed delays is considered. Sufficient conditions for the existence, uniqueness, and global exponential stability of a periodic solution are obtained by using contraction mapping theorem and stability theory on impulsive functional differential equations. The proposed method, which differs from the existing results in the literature, shows that network models may admit a periodic solution which is globally exponentially stable via proper impulsive control strategies even if it is originally unstable or divergent. Two numerical examples and their computer simulations are offered to show the effectiveness of our new results.

Stabilization of Delay Systems: Delay-Dependent Impulsive Control

The stabilization problem of delay systems is studied under the delay-dependent impulsive control. The main contributions of this technical note are that, for one thing, it shows that time delays in impulse term may contribute to the stabilization of delay systems, that is, a control strategy which does not work without delay feedback in impulse term can be activated to stabilize some unstable delay systems if there exist some time delay feedbacks; for another, it shows the robustness of impulsive control, that is, the designed control strategy admits the existence of some time delays in impulse term which may do harm to the stabilization. In this technical note, from impulsive control point of view we firstly propose an impulsive delay inequality. Then we apply it to the delay systems which may be originally unstable, and derive some delay-dependent impulsive control criteria to ensure the stabilization of the addressed systems. The effectiveness of the proposed strategy is evidenced by two illustrative examples.

An Impulsive Delay Inequality Involving Unbounded Time-Varying Delay and Applications

In this paper, a new impulsive delay inequality that involves unbounded and nondifferentiable time-varying delay is presented. As an application, some sufficient conditions ensuring stability and stabilization of impulsive systems with unbounded time-varying delay are derived. Some numerical examples are given to illustrate the results. Especially, a stabilizing memoryless controller for a second-order time-varying system with unbounded time-varying delay is proposed.

Impulsive differential equations

This paper deals with the periodic solutions problem for impulsive differential equations. By using Lyapunovs second method and the contraction mapping principle, some conditions ensuring the existence and global attractiveness of unique periodic solutions are derived, which are given from impulsive control and impulsive perturbation points of view. As an application, the existence and global attractiveness of unique periodic solutions for Hopfield neural networks are discussed. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed results.

Effect of delayed impulses on input-to-state stability of nonlinear systems

Abstract This paper studies the input-to-state stability (ISS) and integral input-to-state stability (iISS) of nonlinear systems with delayed impulses. By using Lyapunov method and the analysis technique proposed by Hespanha etxa0al. (2005), some sufficient conditions ensuring ISS/iISS of the addressed systems are obtained. Those conditions establish the relationship between impulsive frequency and the time delay existing in impulses, and reveal the effect of delayed impulses on ISS/iISS. An example is provided to illustrate the efficiency of the obtained results.

Exponential synchronization of chaotic neural networks with mixed delays and impulsive effects via output coupling with delay feedback

In this paper, we deal with the exponential synchronization problem for a class of chaotic neural networks with mixed delays and impulsive effects via output coupling with delay feedback. The mixed delays in this paper include time-varying delays and unbounded distributed delays. By using a Lyapunov-Krasovskii functional, a drive-response concept and a linear matrix inequality (LMI) approach, several sufficient conditions are established that guarantee the exponential synchronization of the neural networks. Also, the estimation gains can be easily obtained. Finally, a numerical example and its simulation are given to show the effectiveness of the obtained results.

An impulsive delay differential inequality and applications

An impulsive delay differential inequality is formulated in this paper. An estimate of the rate of decay of solutions to this inequality is obtained. It can be applied to the study of dynamical behavior of delay differential equations from the impulsive control point of view. As an application, we consider a class of impulsive control systems with time-varying delays and establish a sufficient condition to guarantee the global exponential stability. It is shown that, via proper impulsive control law, a linear delay differential system can be exponentially stabilized even if it is initially unstable. A numerical example is given to demonstrate the effectiveness of the development method.

Sufficient Stability Conditions of Nonlinear Differential Systems Under Impulsive Control With State-Dependent Delay

In this technical note we study the delayed impulsive control of nonlinear differential systems, where the impulsive control involves the delayed state of the system for which the delay is state-dependent. Since the state dependence of the delay makes the impulsive transients dependent on the historical information of the states, which means that it is hard to know exactly a priori how far in the history the information is needed, the main challenge is how to determine the historical states. We resolve this challenge and establish some sufficient conditions for local stability of nonlinear differential systems with state-dependent delayed impulsive control based on impulsive control theory. Two examples are given to show the effectiveness of the proposed approach.

Impulsive control of unstable neural networks with unbounded time-varying delays

This paper considers the impulsive control of unstable neural networks with unbounded time-varying delays, where the time delays to be addressed include the unbounded discrete time-varying delay and unbounded distributed time-varying delay. By employing impulsive control theory and some analysis techniques, several sufficient conditions ensuring μ-stability, including uniform stability, (global) asymptotical stability, and (global) exponential stability, are derived. It is shown that an unstable delay neural network, especially for the case of unbounded time-varying delays, can be stabilized and has μ-stability via proper impulsive control strategies. Three numerical examples and their simulations are presented to demonstrate the effectiveness of the control strategy.

Exponential Synchronization of Neural Networks via Feedback Control in Complex Environment

The problem of exponential synchronization for neural networks is investigated via feedback control in complex environment. By constructing suitable Lyapunov-Krasovskii functionals and applying the piecewise analytic method, some sufficient criteria for exponential synchronization of the addressed neural networks are established in terms of linear matrix inequalities (LMIs). The feedback control in complex environment includes the delayed aperiodically intermittent control and dynamic output feedback control. Moreover, the delayed aperiodically intermittent dynamic output feedback controller is designed based on the established LMIs. A numerical example and its numerical simulations are finally presented to show the effectiveness of obtained theoretical results.