Lyapunov-Razumikhin Method Research Articles

Event-triggered impulsive control for nonlinear stochastic delayed systems and complex networks

In this paper, we probe the pth moment exponential stability (p−ES) of stochastic delayed systems subject to event-triggered delayed impulsive control (ETDIC), where the impulsive intensities are assumed to be positive random variables. Based on event-triggered mechanism (ETM) in the sense of expectation, some new sufficient conditions are developed to ensure the stability of the addressed system with Zeno-free behavior. The Lyapunov–Razumikhin method is adopted to handle the time-varying delay in continuous dynamics, and the concept of average random impulsive estimation (ARIE) is introduced to reduce the design requirements of the controller. Especially, the proposed ETDIC strategy not only generates the impulse time sequence according to the predesigned ETM, but also removes the limitations on the size of time delays. Furthermore, the ETM serves as a solution for synchronization problems in complex neural networks. Finally, two examples are given to illustrate the effectiveness of our conclusions.

Event-triggered protocol-based adaptive impulsive control for delayed chaotic neural networks.

This article focuses on the synchronization problem of delayed chaotic neural networks via adaptive impulsive control. An adaptive impulsive gain law in a discrete-time framework is designed. The delay is handled skillfully by using the Lyapunov-Razumikhin method. To improve the flexibility of impulsive control, an event-triggered impulsive strategy to determine when the impulsive instant happens is designed. Additionally, it is proved that the event-triggered impulsive sequence cannot result in the occurrence of Zeno behavior. Some criteria are derived to guarantee synchronization for delayed chaotic neural networks. Eventually, an illustrative example is presented to empirically validate the effectiveness of the suggested strategy.

Event-triggered hybrid impulsive control for synchronization of fractional-order multilayer signed networks under cyber attacks

In this paper, we consider the exponential bipartite synchronization (EBS) problem of fractional-order multilayer signed networks with time-varying delays (FO-MSNT) under random cyber attacks. In contrast to the existing literature, the proposed hybrid event-triggered controller combines the advantages of feedback controller and impulsive controller, and the event-triggered condition is constructed by applying the network topology and the Lyapunov function of the subsystem, rather than the state function of the subsystem. Based on the Lyapunov–Razumikhin method and the graph theory, some sufficient conditions for achieving EBS of FO-MSNT under cyber attacks which are related to the topology of networks, the event-triggered parameters, the order of fractional derivative and the signal sent by the enemy are obtained. Furthermore, fractional-order coupled Chua’s circuits model and fractional-order power systems built on MSNT are established and the EBS issues under cyber attacks are analyzed. Numerical examples and simulations are provided to show the validity of our theories.

Local Synchronization for Delayed Complex Dynamical Networks via Self-Triggered Impulsive Control Involving Delays.

This article investigates the local synchronization for delayed complex dynamical networks (CDNs) under self-triggered impulsive control (STIC) approaches involving delays. With the help of Lyapunov-Razumikhin methods and comparison principle, some design criteria of STIC strategies ensuring local synchronization for delayed CDNs with delayed impulses are provided, and Zeno behavior can be avoided. Compared with the existing results on synchronization of CDNs under STIC, in this article, time delays in both continuous and discrete system dynamics are well considered. Moreover, the proposed self-triggered mechanism (STM) is an explicit expression, under which the next triggering instant can be derived directly, with simple structure and easy implementation. Finally, two numerical examples are provided to validate the proposed theoretical criteria.

Event-triggered impulsive control for stability of stochastic delayed complex networks under deception attacks

In this paper, we consider the stability of stochastic complex networks with time-varying delays (SCNTD) under deception attacks, adopting the event-triggered impulsive control (ETIC). A new event-triggered mechanism (ETM) which was used to determine the impulsive instance is given using the topology of networks and the Lyapunov functions of subsystems. For stochastic complex networks, to avoid the Zeno behavior, a positive minimum inter-event time (MIET) is guaranteed. Combining the graph theory and the Lyapunov–Razumikhin method, several criteria for achieving pth moment exponential stability (pMES) of SCNTD under deception attacks which are related to the topology of networks, the event-triggered parameters and the signal sent by the enemy are obtained. Finally, the theoretical results apply to the stochastic oscillator systems, single-link robot arms systems, and Chua’s circuit systems. The stability of these systems under deception attacks is considered separately, and to show the validity of our theories, numerical simulations are provided.

Uniform asymptotic stability of q -deformed conformable fractional systems with delay and application

In this article, we initiate the study of new concepts of conformable \(q\)-fractional calculus. The conformable fractional \(q\)-derivative and \(q\)-integral are defined and their fundamental theorems are also proved. The uniform asymptotic stability of the \(q\)-deformed conformable fractional system with constant delay is investigated by using the Lyapunov-Razumikhin method. For application, a new asymptotic stability necessary condition for the conformable \(q\)-fractional linear system with constant delay is obtained in term of linear matrix inequality (LMI). A numerical example is demonstrated for the results given to illustrate the effectiveness.

Stability and Feedback Control of Volterra Type Systems with Time Delay

New results for stability and feedback control of time delay systems were proposed. These results were obtained by using Lyapunov Razumikhin method to approximate the stability of the uncontrolled Volterra type system with delay and designing a state feedback controller using a model transformation technique, the Lyapunov matrix equation and the Razumikhin approach for the stabilization of the controlled Volterra type system with delay. Examples are given to illustrate the effectiveness of the theoretical results.

Observer-based adaptive ISMC for connected vehicles against cyber-attacks

This paper concentrates on decentralized variable structure adaptive integral sliding mode control (ISMC) for connected vehicles (CV) against cyber-attacks. First of all, a novel vehicle leader-follower model is established by jointly considering the cyber-attacks which involve replay attack and DoS attack simultaneously. Secondly, a decentralized control scheme is designed on the basis of the variable structure observer and adaptive integral sliding mode technique. The Lyapunov-Razumikhin method is adopted such that the state estimation error of CV system is asymptotically converged to zero, which can overcome the replay attack, DoS attack, and measurement delays, then a corrective signal is designed for the deception attack to eliminate the influence of deception attack on the CV system. Furthermore, the robust asymptotic stability of CV system is guaranteed by exploiting the emerging technique of robust control for time-delay systems. Meanwhile, the string stability of CV is guaranteed, even under the condition of heterogeneous vehicles. Finally, extensive simulations are conducted to verify the theoretical analysis and demonstrate the effectiveness and superiority of the proposed method.

Global finite-time stability of delayed quaternion-valued neural networks based on a class of extended Lyapunov-Razumikhin methods.

In this paper, a class of global finite-time stability problem for quaternion-valued neural networks with time-varying delays are investigated by adopting an extended modification Lyapunov-Razumikhin (L-R) method and a new upper bounds estimation of system solution in terms of convergence rate was obtained. Firstly, a new extended method of L-R is proposed to solve the general difficulty to find a proper Lyapunov functional. Then, a new suitable controller is designed, the new conditions of inequalities global finite-time stability are obtained via combining with the former proposed L-R method in the separated real-valued system. Finally, for purpose of verifying the availability of the theorem presented, two given illustrative examples are shown.

Improved Design of Nonlinear Control Systems with Time Delay

It is well known that time delay in nonlinear control systems may lead to the deterioration or even destabilization of the closed-loop systems. Therefore, specific analysis techniques and design methods are needed to be developed for nonlinear control systems in the presence of time delay. This chapter aims to give a broad overview of the stability and control of nonlinear time-delay systems. Firstly, we present some motivations and a comprehensive survey for the study of time-delay systems. Then, a brief overview of some control approaches is provided, specifically, the Lyapunov-Krasoviskii functional method for high-order polynomial uncertainties nonlinear time-delay systems, and nonlinear time-delay systems with nonlinear input, the Lyapunov-Razumikhin method for triangular structure nonlinear time-delay systems, dynamic gain control for full state time-delay systems. Finally, an application in chemical reactor systems is provided and some related open problems are discussed.

An asymptotic state estimator design and synchronization criteria for fractional order time‐delayed genetic regulatory networks

AbstractThis paper mainly investigates the asymptotic state estimator design and impulsive controlled synchronization for fractional‐order time‐delayed genetic regulatory networks (FOTDGRNs). Different from the existing state estimator results, the asymptotic state estimator design of FOTDGRNs is studied by using a novel algebraic method, fractional‐order Lyapunov–Razumikhin method, and some famous inequality techniques. Afterward, a suitable impulsive controller is designed for the global asymptotic synchronization criteria for addressing master–slave systems. At last, the paper comes up with two numerical cases to justify the applicability of our theoretical outcomes.

New Fundamental Results on the Continuous and Discrete Integro-Differential Equations

This work studies certain perturbed and un-perturbed nonlinear systems of continuous and discrete integro-delay differential equations (IDDEs). Using the Lyapunov–Krasovskii functional (LKF) method and the Lyapunov–Razumikhin method (LRM), uniform asymptotic stability (UAS), uniform stability (US), integrability and boundedness of solutions as well as exponential stability (ES) and instability of solutions are discussed. In this paper, five new theorems and a corollary are given and three numerical applications are provided with their simulations. With this work, we aim to make new contributions to the theory of the continuous and discrete integro-differential equations.

Asymptotic and finite-time cluster synchronization of neural networks via two different controllers

<p style='text-indent:20px;'>In this paper, by using a pinning impulse controller and a hybrid controller respectively, the research difficulties of asymptotic synchronization and finite time cluster synchronization of time-varying delayed neural networks are studied. On the ground of Lyapunov stability theorem and Lyapunov-Razumikhin method, a novel sufficient criterion on asymptotic cluster synchronization of time-varying delayed neural networks is obtained. Utilizing Finite time stability theorem and hybrid control technology, a sufficient criterion on finite-time cluster synchronization is also obtained. In order to deal with time-varying delay and save control cost, pinning pulse control is introduced to promote the realization of asymptotic cluster synchronization. Following the idea of pinning control scheme, we design a progressive hybrid control to promote the realization of finite time cluster synchronization. Finally, an example is given to illustrate the theoretical results.</p>

Robust Stability Analysis of Time-Varying Delay Systems

The generalized Myshkis problem consists in analysis of robust stability of time-delay system under variation of its parameters within a set in an infinite-dimensional space. In our paper, we consider a linear system, where the delay is a bounded function under an additional restriction on its growth rate. Some sufficient conditions of solvability of the generalized Myshkis problem are expressed in the form of quadratic convex optimization task. The new approach is based on a modification of the Lyapunov-Razumikhin method.

Impulsive effects on fractional order time delayed gene regulatory networks: Asymptotic stability analysis

This paper address the fractional-order gene regulatory networks (FOGRNs) with both time delays and impulsive effects. Inspired by the integer-order gene regulatory network models, a general class of fractional-order gene regulatory network model is further investigated via fractional-order Lyapunov-Razumikhin stability theory. Firstly, impulsive effects, feedback regulation, and translation time delays are taken well into account and effective analysis techniques are used to reflect the system’s practically dynamic behavior. Secondly, the existence and uniqueness of the equilibrium point of the fractional-order gene regulatory networks are derived based on the Banach contraction mapping principle. Besides, we derived analytically the condition under which the solution of this network is bounded through generalized Gronwall inequality. Thirdly, some novel delay-free sufficient conditions are derived to ensure the global asymptotic stability of the considered model with the help of the fractional-order Lyapunov-Razumikhin method and properties of Mittag-Leffler functions. Finally, an example with numerical simulation is provided to illustrate the validity and effectiveness of the proposed results.

Finite-time stability of impulsive pantograph systems with applications

This paper is mainly concerned with the finite-time stability of impulsive pantograph systems. By proposing a novel Razumikhin condition, and combining Lyapunov-Razumikhin method with average impulsive interval approach, we derive some criteria to ensure that the addressed impulsive pantograph systems are finite-time stable. Also, based on this Razumikhin condition, new Lyapunov-based conditions are obtained for the finite-time stability and the global power stability of nonlinear pantograph systems. The validity of our results is finally illustrated by three examples.

Lyapunov-Razumikhin techniques for state-dependent delay differential equations

We present Lyapunov stability and asymptotic stability theorems for steady state solutions of general state-dependent delay differential equations (DDEs) using Lyapunov-Razumikhin methods. Our results apply to DDEs with multiple discrete state-dependent delays, which may be nonautonomous for the Lyapunov stability result, but autonomous (or periodically forced) for the asymptotic stability result. Our main technique is to replace the DDE by a nonautonomous ordinary differential equation (ODE) where the delayed terms become source terms in the ODE. The asymptotic stability result is based on a contradiction argument together with the Arzelà-Ascoli theorem. This approach alleviates the need to construct auxiliary functions to ensure the asymptotic contraction, which is a feature of other Lyapunov-Razumikhin asymptotic stability results of which we are aware. We apply our results to a state-dependent model equation which includes Hayes equation as a special case, to directly establish asymptotic stability in parts of the stability domain together with lower bounds on the size of the basin of attraction.

An application of Lyapunov–Razumikhin method to behaviors of Volterra integro-differential equations

An application of Lyapunov–Razumikhin method to behaviors of Volterra integro-differential equations

Exponential Stability of Impulsive Stochastic Delay System Based on Razumikhin Method and Its Application to Chaos Control

In this paper, the exponential stability of a stochastic delay system with impulsive signal is considered, and stability theorem of this system is proposed based on the Lyapunov–Razumikhin method; the convergence rate is also given, which gives theoretical foundation to chaos control and synchronization using the impulsive method. Finally, the classic delay chaos system with white noise and impulsive signal is employed to verify the feasibility and effectiveness of our theorem.

Self-triggered impulsive control of nonlinear time delay systems: Application to chemotherapeutic dose-regimen design

This paper proposes a self-triggered impulsive control for nonlinear time-delay systems, where the time instant of the next impulsive input is calculated based on the last measurement and the values of the systems’ parameters. Contrary to event-triggered scheme where the actuation is discrete but a continuous monitoring of the system’s states is necessary, in the self-triggered approach, both sampling and actuation are performed at distinct moments of time. Utilizing Lyapunov–Razumikhin method and by upper-bounding the system’s trajectory, the global asymptotic stability of the system’s equilibrium is verified when the rate of change in the Lyapunov function is exponential. In the general case, the global ultimate boundedness of the system’s trajectory is shown where the ultimate bound can be set arbitrarily small. As an application, for the first time, the problem of dose regimen design is formulated in the sampled data framework. Then, based on the obtained theoretical results, the appropriate regimen is suggested. In particular, time-triggered and self-triggered therapy protocols for docetaxel, a phase specific chemotherapeutic drug which is administered intravenously, are proposed. Clinical constraints such as maximum tolerated dose, discontinuous drug administration, and intermittent measurements are met in the proposed therapy protocols. According to in-silico results, both proposed self-triggered and time-triggered dose regimens outperform the traditional weekly fixed dose administration. Finally, the robustness of the proposed schemes to parameter uncertainties is evaluated through an extensive set of simulations.