General Time-varying Delays Research Articles

Adaptive Exponential Synchronization of Complex Networks With Nondifferentiable Time-Varying Delay.

In recent years, the adaptive exponential synchronization (AES) problem of delayed complex networks has been extensively studied. Existing results rely heavily on assuming the differentiability of the time-varying delay, which is not easy to verify in reality. Dealing with nondifferentiable delay in the field of AES is still a challenging problem. In this brief, the AES problem of complex networks with general time-varying delay is addressed, especially when the delay is nondifferentiable. A delay differential inequality is proposed to deal with the exponential stability of delayed nonlinear systems, which is more general than the widely used Halanay inequality. Next, the boundedness of the adaptive control gain is theoretically proved, which is neglected in much of the literature. Then, the AES criteria for networks with general delay are established for the first time by using the proposed inequality and the boundedness of the control gain. Finally, an example is given to demonstrate the effectiveness of the theoretical results.

Improved delay-dependent stability of superlinear hybrid stochastic systems with general time-varying delays

In the recent paper (Fei et al., 2019), the study of delay-dependent stability of hybrid stochastic differential delay equations (SDDEs) was generalized to superlinear ones (namely, do not satisfy the usual linear growth condition). However, the theory developed there could not be applied to hybrid SDDEs with non-differentiable time delays, or whose drift coefficients miss the key decomposition in (Fei et al., 2019) (see Assumption 1 below). This paper therefore is to deal with these two challenging problems so that the delay-dependent stability criteria derived in (Fei et al., 2019) could be improved. The decomposition scheme is modified in order to include more general hybrid SDDEs. The differentiability assumption on time-varying delays is replaced by a relatively weaker one. Also the Lyapunov functional used in this paper is modulated to adapt to these new changes. Finally, two interesting examples, an application to mosquito model, and design of nonlinear delay feedback control, respectively, are given to demonstrate the effectiveness of our new theory.

Stability and Synchronization of Nonautonomous Reaction-Diffusion Neural Networks With General Time-Varying Delays.

This article investigates the stability and synchronization of nonautonomous reaction-diffusion neural networks with general time-varying delays. Compared with the existing works concerning reaction-diffusion neural networks, the main innovation of this article is that the network coefficients are time-varying, and the delays are general (which means that fewer constraints are posed on delays; for example, the commonly used conditions of differentiability and boundedness are no longer needed). By Green's formula and some analytical techniques, some easily checkable criteria on stability and synchronization for the underlying neural networks are established. These obtained results not only improve some existing ones but also contain some novel results that have not yet been reported. The effectiveness and superiorities of the established criteria are verified by three numerical examples.

Leader-Following consensus of nonlinear multi-agent systems with hybrid delays: Distributed impulsive pinning strategy

The leader-following consensus problem of a type of nonlinear multi-agent systems with both general time-varying delay and distributed time-varying delay is investigated in this paper. To realize the global and exponential consensus, a kind of effective distributed impulsive pinning control strategy is proposed. Adequate conditions are derived to assure the consensus by applying the average impulsive interval, the comparison principle with hybrid time-varying delays, and the method of parametric variation. Considering that the impulse could not only be beneficial but also be adverse to the final consensus, the positive and negative impulsive effects are studied in detail, respectively, and the larger ranges of impulsive effects are obtained. Correspondingly, by considering the impulsive effects in different situations, different convergence rates are accurately calculated. In addition, the validity of the control strategy proposed in this paper is verified based on the simulation examples.

Pinning synchronization of complex networks with time-varying outer coupling and nonlinear multiple time-varying delay coupling

This article studies the pinning control problem for complex networks with time-varying outer coupling and nonlinear multiple time-varying delay coupling. The design aim is to provide appropriate pinning feedback controllers such that the nodes converge to the consistent state as well as the equilibrium point, periodic orbit or chaotic orbit of the nonlinear part of the node dynamic. Based on Lyapunov function theory, appropriate positive-definite functions are constructed, and sufficient synchronization criteria for two kinds of complex networks with different time-varying delay coupling are obtained respectively. Meanwhile, the boundary of the time-varying outer coupling and its influence on the synchronization are investigated. The theoretical results are extended to complex networks with general time-varying delay coupling. Simulation results on complex networks consisting of six nonidentical Chua’s circuits with time-varying outer coupling and nonlinear multiple time-varying delay coupling are presented to demonstrate the effectiveness of the proposed approaches.

Delayed finite-dimensional observer-based control of 1-D parabolic PDEs

In our recent paper a constructive method for finite-dimensional observer-based control of 1-D linear heat equation was suggested. In the present paper we aim to extend this method to the case of input/output general time-varying delays or sawtooth delays (that correspond to network-based control). We assume known measurement delays and, for the first time under observer-based control of PDEs, unknown input delays. We use a modal decomposition approach, and consider boundary or non-local sensing together with non-local actuation, or Dirichlet actuation with non-local sensing. The dimension of the controller is equal to the number of unstable modes, whereas the observer may have a larger dimension N. Under the Dirichlet actuation we present two methods: a direct one that manages with time-varying input and output delays, and a dynamic-extension-based one that treats constant input and time-varying output delays. To compensate the fast-varying output delay (without any constraints on the delay derivative) that appears in the infinite-dimensional part of the closed-loop system, we combine Lyapunov functionals with Halanay’s inequality. For the slowly-varying output delay (with the delay derivative smaller than d<1), we suggest a direct Lyapunov method. We provide LMIs for finding N and upper bounds on the delays that preserve the exponential stability.

Synchronization of Complex Networks With Nondifferentiable Time-Varying Delay.

In this article, we investigate the synchronization of complex networks with general time-varying delay, especially with nondifferentiable delay. In the literature, the time-varying delay is usually assumed to be differentiable. This assumption is strict and not easy to verify in engineering. Until now, the synchronization of networks with nondifferentiable delay through adaptive control remains a challenging problem. By analyzing the boundedness of the adaptive control gain and extending the well-known Halanay inequality, we solve this problem and establish several synchronization criteria for networks under the centralized adaptive control and networks under the decentralized adaptive control. Particularly, the boundedness of the centralized adaptive control gain is theoretically proved. Numerical simulations are provided to verify the theoretical results.

Pole assignment stabilization for a class of switched nonlinear time-varying delay systems

This paper investigates the pole assignment stabilization with memory state feedback controllers for switched nonlinear time-varying delay systems. More general time-varying delays which depend on the subsystem number are considered. Based on novel common Lyapunov functions (CLFs), the aggregation techniques, the proprieties and the Borne–Gentina criterion, new algebraic stabilization criteria are established. The proposed design can guarantee the stability of the closed-loop systems under arbitrary switching. It allows to avoid researching a common Lyapunov function (CLF), considered a difficult matter. Four illustrate examples were finally utilized to exhibit the effectiveness of the obtained results.

Finite-time stability analysis for neutral-type neural networks with hybrid time-varying delays without using Lyapunov method

In this paper, we study the finite-time stability problem of neutral-type neural networks with various hybrid time-varying delays. These delays include general time-varying delays, finite distributed time-varying delays, neutral-type time-varying delays and infinite distributed time-varying delays. Based on the definition of finite-time stability instead of Lyapunov function method, with the aid of inequality techniques, some simple and novel sufficient conditions are derived to guarantee the finite-time stability of our proposed network model. The deduction process is simple and easy to understand. Finally, three simulation examples are given to show the effectiveness of our main results.

Robust sliding mode control of general time-varying delay stochastic systems with structural uncertainties

This paper presents a new robust sliding mode control (SMC) method with well-developed theoretical proof for general uncertain time-varying delay stochastic systems with structural uncertainties and the Brownian noise (Wiener process). The key features of the proposed method are to apply singular value decomposition (SVD) to all structural uncertainties and to introduce adjustable parameters for control design along with the SMC method. It leads to a less-conservative condition for robust stability and a new robust controller for the general uncertain stochastic systems via linear matrix inequality (LMI) forms. The system states are able to reach the SMC switching surface as guaranteed in probability 1. Furthermore, it is theoretically proved that the proposed method with the SVD and adjustable parameters is less conservatism than the method without the SVD. The paper is mainly to provide all strict theoretical proofs for the method and results.