Halanay-type Inequality Research Articles

Stability and synchronization of octonion-valued neural networks with leakage and mixed delays on time scales

There has been a great deal of interest in the last few years for neural networks (NNs) with values in multidimensional domains. The most popular models are complex-valued neural networks (CVNNs), followed by quaternion-valued neural networks (QVNNs), and, more recently, by Clifford-valued neural networks (ClVNNs). However, also very recently, a different type of NNs were put forward, namely octonion-valued neural networks (OVNNs). OVNNs are defined on the 8D octonion algebra, and they are not a special type of ClVNNs, because Clifford numbers are associative, whereas octonions are not. Moreover, beside the complex and quaternion algebras, the only other normed division algebra over the reals is the algebra of octonions, which makes OVNNs a direct generalization of CVNNs and QVNNs from this point of view, raising interest for applications handling high-dimensional data. On the other hand, systems defined on time scales were proposed as a generalization of both discrete time and continuous time systems, or any type of hybrid combination between the two. Finally, time delays appear as a consequence of implementing NNs in real life circuits. Taking all these into consideration, this paper studies the fundamental properties of exponential stability and exponential synchronization for OVNNs with leakage and mixed delays defined on time scales. In order to avoid the problems raised by the non-associativity of the octonion algebra, the OVNN model is decomposed into a real-valued one. Then, two different Lyapunov-type functionals are defined and the particularities of time scale calculus are used in order to deduce sufficient conditions expressed as scalar and linear matrix inequalities (LMIs) for the exponential stability of the proposed models, based on Halanay-type inequalities suitable for time scale systems. Afterwards, a state feedback controller is used to deduce sufficient criteria given as scalar inequalities and LMIs for the exponential synchronization of the same type of models. The generality of the model, given by the definition on the algebra of octonions, the use of different types of delays, and the definition on time scales represent an important advantage. Also, for less general models, it is possible to particularize the obtained results. With the aim to demonstrate each of the paper’s four theorems, four numerical examples are provided.

Generalized Halanay-Type Inequalities for Finite-Time Stabilization of Delayed Systems

Generalized Halanay-Type Inequalities for Finite-Time Stabilization of Delayed Systems

Event-Based Output Quantized Synchronization Control for Multiple Delayed Neural Networks.

This article concentrates on the global exponential synchronization problem of multiple neural networks with time delay by the event-based output quantized coupling control method. In order to reduce the signal transmission cost and avoid the difficulty of obtaining the systems' full states, this article adopts the event-triggered control and output quantized control. A new dynamic event-triggered mechanism is designed, in which the control parameters are time-varying functions. Under weakened coupling matrix conditions, by using a Halanay-type inequality, some simple and easily verified sufficient conditions to ensure the exponential synchronization of multiple neural networks are presented. Moreover, the Zeno behaviors of the system are excluded. Some numerical examples are given to verify the effectiveness of the theoretical analysis in this article.

Synchronization for stochastic large-scale systems via intermittent delay discrete observation control

This paper proposes a novel intermittent delay discrete observation control for synchronization of stochastic large-scale systems (LSs) with semi-Markov jump. Different from the existing intermittent delay control, our proposed control strategy is based on discrete-time state observations rather than continuous-time state observations during the work time. And by applying the method of average control rate, the control strategy becomes more practical. Then we establish a lower conservative differential inequality, which is significantly conducive to analyze the synchronization of LSs with multiple delays. Based on the generalized Halanay-type inequality, Lyapunov method and stochastic analysis techniques, several sufficient conditions are derived. Finally, we apply the theoretical results to Chua’s circuit with numerical examples being given.

Stability of Delayed Reaction-Diffusion Neural-Network Models With Hybrid Impulses via Vector Lyapunov Function.

This article focuses on stability analysis of delayed reaction-diffusion neural-network models with hybrid impulses based on the vector Lyapunov function. First, several properties of a vector Halanay-type inequality are given to be the key ingredient for the stability analysis. Then, the Krasovskii-type theorems are established for sufficient conditions of exponential stability, which removes the common threshold of impulses in each neuron subsystem at every impulse time. It shows that the stability of neural networks can be retained with hybrid impulses involved in neural networks, and the synchronization of neural networks can be achieved by designing an impulsive controller, which allows the existence of impulsive perturbation in some nodes and time. Finally, the effectiveness of theoretical results is verified by numerical examples with a successful application to image encryption.

Adaptive control for output projective synchronization of fractional-order hybrid coupled neural networks with mismatched dimensions

This paper is concerned with output projective synchronization (OPS) for fractional-order Hybrid Coupled neural networks (FOHCNNs) with mismatched dimensions. Distinguishing from the traditional projective synchronization, the concept of OPS for fractional-order systems is given. Moreover, a new fractional-order Halanay-type inequality is presented by the properties of the Mittag-Leffler function. Then, several new sufficient conditions to ensure OPS of FOHCNNs with mismatched dimensions are obtained via a new effective adaptive controller. In particular, several adaptive control strategies are given to ensure the output synchronization, output anti-synchronization, and stabilization of FOHCNNs with mismatched dimensions. Finally, a numerical example is given to show the effectiveness of the theoretical results.

A discrete Gronwall–Halanay-type inequality with infinite delay and its applications to difference equations

We establish a discrete Gronwall–Halanay-type inequality with infinite delay, which is not covered in the existing literature. As an application, a new criterion is obtained for the asymptotic stability of the zero solutions for a class of Volterra difference equations. A concrete example is also given to illustrate the efficiency of the general results.

Fractional-Order Vectorial Halanay-Type Inequalities With Applications for Stability and Synchronization Analyses

The Halanay inequality is widely used in various time-delayed dynamical systems analyses and its vectorial form has become available recently. In this article, the integer-order vectorial Halanay-type inequality is further extended to fractional-order ones in both time-invariant and time-varying forms. It is shown that the fractional-order vectorial Halanay-type inequalities hold under the derived conditions in the form of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> -matrices. In addition, the time-invariant inequalities are applied to analyzing the stability and synchronization of fractional-order systems with two numerical examples to substantiate the theoretical results.

Event-Triggered Stabilization of Linear Time-Delay Systems via Halanay-Type Inequality

Event-Triggered Stabilization of Linear Time-Delay Systems via Halanay-Type Inequality

Stability of stochastic delayed multi-links complex network with semi-Markov switched topology: A time-varying hybrid aperiodically intermittent control strategy

This paper is concerned with the stability analysis of stochastic delayed multi-links complex networks with semi-Markov switched topology (SDMNST). It ought to be highlighted that a time-varying hybrid aperiodically intermittent control (THAIC) strategy, that the control gain is a sign-indefinite function and can be unbounded, is proposed. In the light of a major consideration for both stabilizing control and destabilizing input, the THAIC is capable of being manipulated to a wider range of intermittent control signals. Meanwhile, our proposed control strategy can not only deal with the problem that the intermittent control signals subject to external interferences, but also effectively avoid the possible damage of actuators caused by the sudden and rapid switching of control gain from a negative constant value to zero. The concept “average intermittent control gain” is introduced to quantify the gain of THAIC. As a generalization of the existing Halanay-type inequalities, several novel differential inequalities with time-varying and unbounded coefficients are established, based on which and Lyapunov method as well as graph theory, less conservative stability criteria of SDMNST are rigorously demonstrated. Subsequently, the theoretical results are applied to spring-mass-damper systems and numerical simulations are exploited with the design of control gain being shown.

New Criteria for Synchronization of Multilayer Neural Networks via Aperiodically Intermittent Control.

In this paper, the globally asymptotic synchronization of multi-layer neural networks is studied via aperiodically intermittent control. Due to the property of intermittent control, it is very hard to deal with the effect of time-varying delays and ascertain the control and rest widths for intermittent control. A new lemma with generalized Halanay-type inequalities are proposed first. Then, by constructing a new Lyapunov–Krasovskii functional and utilizing linear programming methods, several useful criteria are derived to ensure the multilayer neural networks achieve asymptotic synchronization. Moreover, an aperiodically intermittent control is designed, which has no direct relationship with control widths and rest widths and extends existing aperiodically intermittent control techniques, the control gains are designed by solving the linear programming. Finally, a numerical example is provided to confirm the effectiveness of the proposed theoretical results.

Input-to-state stability for time-varying delayed systems in Halanay-type inequality forms

This paper studies the input-to-state stability (ISS) for time-varying delayed systems (TVDS) in Halanay-type inequality forms. The time-delay in TVDS is allowed to be time-varying and unbounded. By introducing the notion of a uniform M-matrix, exponential ISS theorems are established respectively for continuous-time, discrete-time, and zero-order TVDS. The convergence rates of exponential ISS and ISS gains and their relation are subsequently estimated. These ISS theorems are less conservative and generalize the results of stability and ISS for Halanay-type inequalities in the literature. Moreover, necessary conditions of ISS are given for TVDS in Halanay-type equality forms. By specializing the ISS results to linear time-invariant delayed systems, the necessary and sufficient conditions of ISS are derived respectively. Three examples are given throughout the paper to illustrate the theoretical results.

Input-to-state stability of delayed systems with bounded-delay impulses

&lt;abstract&gt;&lt;p&gt;This paper considers the input-to-state stability (ISS) of delayed systems with bounded-delay impulses, where the delays in impulses are arbitrarily large but bounded. A novel Halanay-type inequality with delayed impulses and external inputs is proposed to deeply evaluate the effects of delayed impulses on ISS of delayed systems. Then, we obtain some delay-independent ISS criteria for the addressed delayed systems by using Lyapunov method. Particularly, by applying a new analysis technique, the current study enriches the Halanay-type inequalities and further improve the results derived in &lt;sup&gt;[&lt;xref ref-type="bibr" rid="b1"&gt;1&lt;/xref&gt;]&lt;/sup&gt;. Two illustrative examples are presented to validate theoretical results.&lt;/p&gt;&lt;/abstract&gt;

Asymptotic and finite-time synchronization of fractional-order multiplex networks with time delays by adaptive and impulsive control

In this article, we analyze the asymptotic synchronization of fractional-order multiplex networks with time delays via adaptive control, where intra-layer and inter-layer transmission delays are taken into consideration. By introducing a new fractional-order Halanay-type inequality and utilizing graph-theoretic method and Lyapunov method, we formulate some sufficient conditions to ensure the asymptotic synchronization of fractional-order multiplex networks with time delays. Furthermore, in the case that there is no delay in intra-layer and inter-layer transmissions, we investigate the finite-time synchronization of fractional-order multiplex networks by impulsive control. Meanwhile, the upper bound of the settling time for finite-time synchronization is reckoned. In addition, the results herein are applied to coupled Chua’s circuits on fractional-order multiplex networks with time delays. Finally, some numerical simulations are carried out to demonstrate the effectiveness of the developed results.

Dynamical and static multisynchronization analysis for coupled multistable memristive neural networks with hybrid control

This paper investigates the dynamical multisynchronization (DMS) and static multisynchronization (SMS) problems for a class of delayed coupled multistable memristive neural networks (DCMMNNs) via a novel hybrid controller which includes delayed impulsive control and state feedback control. Based on the state–space partition method and the geometrical properties of the activation function, each subnetwork has multiple locally exponential stable equilibrium states. By employing a new Halanay-type inequality and the impulsive control theory, some new linear matrix inequalities (LMIs)-based sufficient conditions are proposed. It is shown that the delayed impulsive control with suitable impulsive interval and allowable time-varying delay can still guarantee the DMS and SMS of DCMMNNs. Finally, a numerical example is presented to illustrate the effectiveness of the hybrid controller.

Quantization synchronization of chaotic neural networks with time delay under event-triggered strategy.

This paper shows solicitude for the quantization synchronization of delayed chaotic master and slave neural networks under an dynamic event-triggered strategy. In virtue of a generalized Halanay-type inequality, a theoretical criterion for quasi-synchronization of master and slave neural networks is derived. Meanwhile, we can obtain an exact upper bound of synchronization error by using this criterion. Compared with output feedback controller with event triggering and quantization, the case where the controller only affected by quantization is also considered. Then, we exclude the Zeno behavior of the event-triggered controller. A sufficient criterion for the existence of the quantized output feedback controllers is also provided. A numerical example is cited to illustrate the efficiency of our theoretical criteria. In addition, some experiments of secure image communication are conducted under quasi-synchronization of master and slave neural networks.

Exponential synchronization of fractional-order reaction-diffusion coupled neural networks with hybrid delay-dependent impulses

This paper is concentrated on exploring the exponential synchronization of reaction-diffusion coupled neural networks with fractional-order and impulses. Firstly, an extended Halanay-type inequality is established to cope with the hybrid delay-dependent impulsive problem by utilizing the mathematical induction. Furthermore, a direct error method is introduced by constructing Lyapunov function for the addressed networks to investigate the exponential synchronization under impulsive effects. By utilizing the technique of average impulsive interval and strength, some sufficient synchronization criteria are derived, which are closely associated with time delay and the commensurate order for fractional-order systems. Lastly, three numerical examples are presented to demonstrate the correctness for established results.

Matrix Measure Strategies for Stabilization of Delayed Inertial Neural Networks via Intermittent Control

In this paper, stabilization control of a class of delayed inertial neural networks (INN) is investigated. Employing matrix measure method and two Halanay-type inequalities, some succinct stabilization criteria in terms of algebraic inequalities are derived for the INN with time-varying delays under periodically intermittent control (PIMC) strategy. Moreover, more precise results are obtained to stabilize the INN with time-invariant delays by using comparison principle. Specifically, the criteria of matrix measure form proposed in this paper can be converted into LMI-type condition for the case of 2-norm, which provides a bridge between the matrix-measure method and the Lyapunov function method. Finally, two numerical examples validate the efficacy of the derived results. The comparative research shows that the proposed methods generalize and develop some known results.

Stabilization of Highly Nonlinear Stochastic Coupled Systems via Periodically Intermittent Control

This article considers the stabilization of highly nonlinear stochastic coupled systems (HNSCSs) with time delay via periodically intermittent control. This article is motivated by that known differential inequalities to deal with periodically intermittent control do not work for HNSCSs, since the coefficients of the system do not satisfy the linear growth condition. In order to cope with this problem, a novel Halanay-type inequality is established to handle periodically intermittent control, which generalizes previous results. Then, based on this differential inequality, the graph theory, and the Lyapunov method, two main theorems are shown, whose conditions indicate how the control duration, the control gain, and the coupling strength affect the realization of the stability. Then, the theoretical results are applied to the modified van der Pol–Duffing oscillators. Finally, corresponding simulation results are presented to illustrate the effectiveness of the theoretical results.

Event-Based Synchronization for Multiple Neural Networks With Time Delay and Switching Disconnected Topology.

This article discusses the synchronization problem for a class of multiple delayed neural networks (MDNNs) with a directed switching topology by using an event-triggering strategy. First, a new differential inequality with delay is shown, which is a generalization of Halanay-type inequalities. Then, the sufficient conditions of event-based synchronization (quasisynchronization) for MDNN with sequentially connected topology are obtained by using this inequality and the iterative method. Meantime, we prove that Zeno behavior can be avoided under the designed event-triggering rules. As an extension, MDNN with jointly connected topology is also discussed. Finally, a numerical example is listed to illustrate the results in theory analysis.