Wellposedness of abstract differential equations with state‐dependent delay

We study the existence and uniqueness of solutions, and the wellposedness of a general class of second order abstract differential equations with state‐dependent delay. Some examples related to partial differential equations with state‐dependent delay are presented.     

Multiple‐interval‐dependent robust stability analysis for uncertain stochastic neural networks with mixed‐delays

This article deals with the problem of robust stochastic asymptotic stability for a class of uncertain stochastic neural networks with distributed delay and multiple time‐varying delays. It is noted that the reciprocally convex approach has been intensively used in stability analysis for time‐delay systems in the past few years. We will extend the approach from deterministic time‐delay systems to stochastic time‐delay systems. And based on the new technique dealing with matrix cross‐product and multiple‐interval‐dependent Lyapunov–Krasovskii functional, some novel delay‐dependent stability criteria with less conservatism and less decision variables for the addressed system are derived in terms of linear matrix inequalities. At last, several numerical examples are given to show the effectiveness of the results. © 2014 Wiley Periodicals, Inc. Complexity 21: 147–162, 2015     

State estimation of memristor‐based recurrent neural networks with time‐varying delays based on passivity theory

This article deals with the state estimation problem of memristor‐based recurrent neural networks (MRNNs) with time‐varying delay based on passivity theory. The main purpose is to estimate the neuron states, through available output measurements such that for all admissible time delay, the dynamics of the estimation error is passive from the control input to the output error. Based on the Lyapunov–Krasovskii functional (LKF) involving proper triple integral terms, convex combination technique, and reciprocal convex technique, a delay‐dependent state estimation of MRNNs with time‐varying delay is established in terms of linear matrix inequalities (LMIs). The information about the neuron activation functions and lower bound of the time‐varying delays is fully used in the LKF. Then, the desired estimator gain matrix is accomplished by solving LMIs. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed theoretical results. © 2013 Wiley Periodicals, Inc. Complexity 19: 32–43, 2014     

Improved H∞ performance analysis of uncertain Markovian jump systems with overlapping time‐varying delays

In this article, we study the problem of robust H_∞ performance analysis for a class of uncertain Markovian jump systems with mixed overlapping delays. Our aim is to present a new delay‐dependent approach such that the resulting closed‐loop system is stochastically stable and satisfies a prescribed H_∞ performance level χ. The jumping parameters are modeled as a continuous‐time, finite‐state Markov chain. By constructing new Lyapunov‐Krasovskii functionals, some novel sufficient conditions are derived to guarantee the stochastic stability of the equilibrium point in the mean‐square. Numerical examples show that the obtained results in this article is less conservative and more effective. The results are also compared with the existing results to show its conservativeness. © 2016 Wiley Periodicals, Inc. Complexity 21: 460–477, 2016     

Exponential stabilization and L2‐gain for uncertain switched nonlinear systems with interval time‐varying delay

This paper is concerned with the exponential stabilization and L₂‐gain for a class of uncertain switched nonlinear systems with interval time‐varying delay. Based on Lyapunov–Krasovskii functional method, novel delay‐dependent sufficient conditions of exponential stabilization for a class of uncertain switched nonlinear delay systems are developed under an average dwell time scheme. Then, novel criteria to ensure the exponential stabilization with weighted L₂‐gain performance for a class of uncertain switched nonlinear delay systems are established. Furthermore, an effective method is proposed for the designing of a stabilizing feedback controller with L₂‐gain performance. Finally, some numerical examples are given to illustrate the effectiveness of the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Robust stability results for nonlinear Markovian jump systems with mode‐dependent time‐varying delays and randomly occurring uncertainties

This article is concerned with the robust stability analysis for Markovian jump systems with mode‐dependent time‐varying delays and randomly occurring uncertainties. Sufficient delay‐dependent stability results are derived with the help of stability theory and linear matrix inequality technique using direct delay‐decomposition approach. Here, the delay interval is decomposed into two subintervals using the tuning parameter η such that , and the sufficient stability conditions are derived for each subintervals. Further, the parameter uncertainties are assumed to be occurring in a random manner. Numerical examples are given to validate the derived theoretical results. © 2015 Wiley Periodicals, Inc. Complexity 21: 50–60, 2016     

Slowly oscillating periodic solutions for the Nicholson's blowflies equation with state‐dependent delay

In this paper, we study the dynamics of a Nicholson's blowflies equation with state‐dependent delay. For the constant delay, it is known that a sequence of Hopf bifurcation occurs at the positive equilibrium as the delay increases and global existence of periodic solutions has been established. Here, we consider the state‐dependent delay instead of the constant delay and generalize the results on the existence of slowly oscillating periodic solutions under a set of mild conditions on the parameters and the delay function. In particular, when the positive equilibrium gets unstable, a global unstable manifold connects the positive equilibrium to a slowly oscillating periodic orbit. Copyright © 2017 John Wiley & Sons, Ltd.     

Periodic solutions of abstract functional differential equations with state‐dependent delay

In this paper, we are concerned with the existence of solutions of systems determined by abstract functional differential equations with infinite and state‐dependent delay. We establish the existence of mild solutions and the existence of periodic solutions. Our results are based on local Lipschitz conditions of the involved functions. We apply our results to study the existence of periodic solutions of a partial differential equation with infinite and state‐dependent delay. Copyright © 2016 John Wiley & Sons, Ltd.     

DELAY DEPENDENT ROBUST DISSIPATIVE CONTROL FOR A CLASS OF NONLINEAR TIME-DELAY SYSTEMS

We discuss the delay dependent robust dissipative problem for a class of time-delay systems with nonlinear perturbations. And the sufficient conditions for the delay dependent robust dissipation and quadratic stability are given by the linear matrix inequalities (LMIs), then the time-delay bound and the delay dependent robust dissipative state feedback controller are presented.     

On second order differential equations with state-dependent delay

We study existence and uniqueness of solutions for a general class of second order abstract differential equations with state-dependent delay. Some examples related to partial differential equations with state dependent delay are presented.     

一类非线性时滞系统的时滞相关镇定及跟踪控制问题

本文讨论了一类满足Lipschitz条件的非线性时滞系统的镇定与跟踪控制问题.基于非线性状态反馈控制器,利用Lyapunov-Krasovskii泛函和矩阵理论,得到了系统时滞相关全局渐近镇定的新判据,并且保证了输出和状态跟踪控制的误差全局渐近收敛于零.本文推广了文献[9]所得到的结论.因此,本文所研究的模型及所给出的判定条件更具有一般性和实用性.     

Robust consensus of nonlinear multi‐agent systems via reliable control with probabilistic time delay

In this paper, the consensus problem of uncertain nonlinear multi‐agent systems is investigated via reliable control in the presence of probabilistic time‐varying delay. First, the communication topology among the agents is assumed to be directed and fixed. Second, by introducing a stochastic variable which satisfies Bernoulli distribution, the information of probabilistic time‐varying delay is equivalently transformed into the deterministic time‐varying delay with stochastic parameters. Third, by using Laplacian matrix properties, the consensus problem is converted into the conventional stability problem of the closed‐loop system. The main objective of this paper is to design a state feedback reliable controller such that for all admissible uncertainties as well as actuator failure cases, the resulting closed‐loop system is robustly stable in the sense of mean‐square. For this purpose, through construction of a suitable Lyapunov–Krasovskii functional containing four integral terms and utilization of Kronecker product properties along with the matrix inequality techniques, a new set of delay‐dependent consensus stabilizability conditions for the closed‐loop system is obtained. Based on these conditions, the desired reliable controller is designed in terms of linear matrix inequalities which can be easily solved by using any of the effective optimization algorithms. Moreover, a numerical example and its simulations are included to demonstrate the feasibility and effectiveness of the proposed control design scheme. © 2016 Wiley Periodicals, Inc. Complexity 21: 138–150, 2016     

‐type stability criteria in division regions for multitime‐scale networks with delays

This paper is concerned with new results on ‐type stability criteria in division regions for competitive neural networks with different time scales. Under the decomposition of state space, both the neural activity levels (the short‐term memory) and the synaptic modifications (the long‐term memory), are taken into account in constructing division regions that allow the coexistence of equilibrium points. Meanwhile, novel delay‐dependent multistability and monostability criteria are established in division regions that depend on divisions in index set of neurons and boundedness of unsupervised synaptic variables. The attained results show the effects of self‐interactions of neurons and Hebbian learning behavior on the multistable convergence of the networks. Finally, numerical simulations will illustrate multistable neuron activity and synaptic dynamics of multitime‐scale competitive networks. Copyright © 2011 John Wiley & Sons, Ltd.     

Synchronization of nonlinear master–slave systems under input delay and slope‐restricted input nonlinearity

This article addresses the synchronization of nonlinear master–slave systems under input time‐delay and slope‐restricted input nonlinearity. The input nonlinearity is transformed into linear time‐varying parameters belonging to a known range. Using the linear parameter varying (LPV) approach, applying the information of delay range, using the triple‐integral‐based Lyapunov–Krasovskii functional and utilizing the bounds on nonlinear dynamics of the nonlinear systems, nonlinear matrix inequalities for designing a simple delay‐range‐dependent state feedback control for synchronization of the drive and response systems is derived. The proposed controller synthesis condition is transformed into an equivalent but relatively simple criterion that can be solved through a recursive linear matrix inequality based approach by application of cone complementary linearization algorithm. In contrast to the conventional adaptive approaches, the proposed approach is simple in design and implementation and is capable to synchronize nonlinear oscillators under input delays in addition to the slope‐restricted nonlinearity. Further, time‐delays are treated using an advanced delay‐range‐dependent approach, which is adequate to synchronize nonlinear systems with either higher or lower delays. Furthermore, the resultant approach is applicable to the input nonlinearity, without using any adaptation law, owing to the utilization of LPV approach. A numerical example is worked out, demonstrating effectiveness of the proposed methodology in synchronization of two chaotic gyro systems. © 2015 Wiley Periodicals, Inc. Complexity 21: 220–233, 2016     

Dynamics of a ratio‐dependent stage‐structured predator‐prey model with delay

In this paper, we investigate the dynamics of a time‐delay ratio‐dependent predator‐prey model with stage structure for the predator. This predator‐prey system conforms to the realistically biological environment. The existence and stability of the positive equilibrium are thoroughly analyzed, and the sufficient and necessary conditions for the stability and instability of the positive equilibrium are obtained for the case without delay. Then, the influence of delay on the dynamics of the system is investigated using the geometric criterion developed by Beretta and Kuang. 26 We show that the positive steady state can be destabilized through a Hopf bifurcation and there exist stability switches under some conditions. The formulas determining the direction and the stability of Hopf bifurcations are explicitly derived by using the center manifold reduction and normal form theory. Finally, some numerical simulations are performed to illustrate and expand our theoretical results.     

Robust sampled‐data control of uncertain switched neutral systems with probabilistic input delay

This article focuses on the robust sampled‐data control for a class of uncertain switched neutral systems based on the average dwell‐time approach. In particular, the system is considered with probabilistic input delay using sampled state vectors, which are described by the stochastic variables with a Bernoulli distributed white sequence and time‐varying norm‐bounded uncertainties. By constructing a novel Lyapunov–Krasovskii functional which involves the lower and upper bounds of the delay, a new set of sufficient conditions are derived in terms of linear matrix inequalities for ensuring the robust exponential stability of the uncertain switched neutral system about its equilibrium point. Moreover, based on the stability criteria, a state feedback sampled‐data control law is designed for the considered system. Finally, a numerical example based on the water‐quality dynamic model for the Nile River is given to illustrate the effectiveness of the proposed design technique. © 2015 Wiley Periodicals, Inc. Complexity 21: 308–318, 2016     

A delay decomposition approach for robust dissipativity and passivity analysis of neutral‐type neural networks with leakage time‐varying delay

This article presents the robust dissipativity and passivity analysis of neutral‐type neural networks with leakage time‐varying delay via delay decomposition approach. Using delay decomposition technique, new delay‐dependent criteria ensuring the considered system to be ‐γ dissipative are established in terms of strict linear matrix inequalities. A new Lyapunov–Krasovskii functional is constructed by dividing the discrete and neutral delay intervals into m and l segments, respectively, and choosing different Lyapunov functionals to different segments. Further, the dissipativity behaviors of neural networks which are affected due to the sensitiveness of the time delay in the leakage term have been taken into account. Finally, numerical examples are provided to show the effectiveness of the proposed method. © 2015 Wiley Periodicals, Inc. Complexity 21: 248–264, 2016     

Structured populations with distributed recruitment: from PDE to delay formulation

In this work, first, we consider a physiologically structured population model with a distributed recruitment process. That is, our model allows newly recruited individuals to enter the population at all possible individual states, in principle. The model can be naturally formulated as a first‐order partial integro‐differential equation, and it has been studied extensively. In particular, it is well posed on the biologically relevant state space of Lebesgue integrable functions. We also formulate a delayed integral equation (renewal equation) for the distributed birth rate of the population. We aim to illustrate the connection between the partial integro‐differential and the delayed integral equation formulation of the model utilising a recent spectral theoretic result. In particular, we consider the equivalence of the steady state problems in the two different formulations, which then lead us to characterise irreducibility of the semigroup governing the linear partial integro‐differential equation. Furthermore, using the method of characteristics, we investigate the connection between the time‐dependent problems. In particular, we prove that any (non‐negative) solution of the delayed integral equation determines a (non‐negative) solution of the partial differential equation and vice versa. The results obtained for the particular distributed states at birth model then lead us to present some very general results, which establish the equivalence between a general class of partial differential and delay equation, modelling physiologically structured populations. Copyright © 2016 John Wiley & Sons, Ltd.     

Global stability and the Hopf bifurcation for some class of delay differential equation

In this paper, we present an analysis for the class of delay differential equations with one discrete delay and the right‐hand side depending only on the past. We extend the results from paper by U. Fory? (Appl. Math. Lett. 2004; 17 (5):581–584), where the right‐hand side is a unimodal function. In the performed analysis, we state more general conditions for global stability of the positive steady state and propose some conditions for the stable Hopf bifurcation occurring when this steady state looses stability. We illustrate the analysis by biological examples coming from the population dynamics. Copyright © 2007 John Wiley & Sons, Ltd.