Delay-dependent Asymptotic Stability Research Articles

Some novel results for DNNs via relaxed Lyapunov functionals

<abstract><p>The focus of this paper was to explore the stability issues associated with delayed neural networks (DNNs). We introduced a novel approach that departs from the existing methods of using quadratic functions to determine the negative definite of the Lyapunov-Krasovskii functional's (LKFs) derivative $ \dot{V}(t) $. Instead, we proposed a new method that utilizes the conditions of positive definite quadratic function to establish the positive definiteness of LKFs. Based on this approach, we constructed a novel the relaxed LKF that contains delay information. In addition, some combinations of inequalities were extended and used to reduce the conservatism of the results obtained. The criteria for achieving delay-dependent asymptotic stability were subsequently presented in the framework of linear matrix inequalities (LMIs). Finally, a numerical example confirmed the effectiveness of the theoretical result.</p></abstract>

Observer-based state estimation for memristive neural networks with time-varying delay

This paper investigates the observer-based state estimation of memristive neural networks (MNNs) with time-varying delay. In order to obtain the accurate state information of the switching MNNs system, a new full-order state observer based on system measurement output is developed such that the reconstruction of states is achieved. By applying set-valued maps and differential inclusions, some sufficient conditions for delay-independent and delay-dependent asymptotic stability are obtained. The validity of theoretical results is verified via numerical examples.

Improved Results on Delay-Dependent Robust H ∞ Control of Uncertain Neutral Systems with Mixed Time-Varying Delays

In this paper, the problem of the delay-dependent robust H ∞ control for a class of uncertain neutral systems with mixed time-varying delays is studied. Firstly, a robust delay-dependent asymptotic stability criterion is shown by linear matrix inequalities (LMIs) after introducing a new Lyapunov–Krasovskii functional (LKF). The LKF including vital terms is expected to obtain results of less conservatism by employing the technique of various efficient convex optimization algorithms and free matrices. Then, based on the obtained criterion, analyses for uncertain systems and H ∞ controller design are presented. Moreover, on the analysis of the state-feedback controller, different from the traditional method which multiplies the matrix inequality left and right by some matrix and its transpose, respectively, we can obtain the state-feedback gain directly by calculating the LMIs through the toolbox of MATLAB in this paper. Finally, the feasibility and validity of the method are illustrated by examples.

The Lyapunov-Razumikhin theorem for the conformable fractional system with delay

<abstract><p>This paper explicates the Razumikhin-type uniform stability and a uniform asymptotic stability theorem for the conformable fractional system with delay. Based on a Razumikhin-Lyapunov functional and some inequalities, a delay-dependent asymptotic stability criterion is in the term of a linear matrix inequality (LMI) for the conformable fractional linear system with delay. Moreover, an application of our theorem is illustrated via a numerical example.</p></abstract>

New results on stability for non‐linear Markov switched stochastic functional differential systems

For Markov switched stochastic functional differential systems (SFDSs), asymptotic property is one of the most desired issues. Recently, a new class of delay-dependent asymptotic stability for non-linear Markov switched SFDSs was investigated. However, the existing references do not take the convergent speed and non-autonomous factor into consideration. Therefore, by means of multiple Lyapunov–Krasovskii functionals, this study is devoted to examine the exponential stability for highly non-linear autonomous Markov switched SFDSs and the exponential stability, polynomial stability and polynomial growth at most for highly non-linear non-autonomous systems, where all the criteria rely on the intervals lengths of continuous delays. In addition, the properties of boundedness and asymptotic stability are as well explored.

Delay-dependent-stability of stochastic delay coupled systems on networks with regime-switching-diffusions

In this paper, delay-dependent-stability of stochastic delay coupled systems on networks with regime-switching-diffusions (SDCSRD) is investigated. Here, regime-switching-diffusions and time-varying delay are both considered into stochastic coupled systems on networks. Combining Lyapunov functional method with graph theory, several delay-dependent-stability criteria are derived to ensure H∞-stability and asymptotical stability in Lp of SDCSRD. Particularly, as a concrete application of the theoretical results, we study the delay-dependent asymptotical stability in mean square of stochastic delay coupled oscillators with regime-switching-diffusions. Ultimately, two numerical examples are provided to exhibit the effectiveness of the results.

Delay dependent asymptotic mean square stability analysis of the stochastic exponential Euler method

This paper is concerned with the delay dependent stability of the stochastic exponential Euler method for stochastic delay differential equations and stochastic delay partial differential equations. By using root locus technique, the necessary and sufficient condition of the numerical delay dependent stability of the method is derived for a class of stochastic delay differential equations and it is shown that the stochastic exponential Euler method can fully preserve the asymptotic mean square stability of the underlying system. Furthermore, we investigate the delay dependent stability of the semidiscrete and fully discrete systems for a linear stochastic delay partial differential equation. The necessary and sufficient condition for the delay dependent stability of the semidiscrete system based on the standard central difference scheme in space is given. Based on this condition, the delay dependent stability of the fully discrete system by using the stochastic exponential Euler method in time is studied. It is shown that the fully discrete scheme can inherit the delay dependent stability of the semidiscrete system completely. At last, some numerical experiments are given to validate our theoretical results.

A Novel Delay-Dependent Asymptotic Stability Conditions for Differential and Riemann-Liouville Fractional Differential Neutral Systems with Constant Delays and Nonlinear Perturbation

The novel delay-dependent asymptotic stability of a differential and Riemann-Liouville fractional differential neutral system with constant delays and nonlinear perturbation is studied. We describe the new asymptotic stability criterion in the form of linear matrix inequalities (LMIs), using the application of zero equations, model transformation and other inequalities. Then we show the new delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral system with constant delays. Furthermore, we not only present the improved delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral system with single constant delay but also the new delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral equation with constant delays. Numerical examples are exploited to represent the improvement and capability of results over another research as compared with the least upper bounds of delay and nonlinear perturbation.

Delay-dependent criterion for asymptotic stability of a class of fractional-order memristive neural networks with time-varying delays

The Lyapunov–Krasovskii functional approach is an important and effective delay-dependent stability analysis method for integer order system. However, it cannot be applied directly to fractional-order (FO) systems. To obtain delay-dependent stability and stabilization conditions of FO delayed systems remains a challenging task. This paper addresses the delay-dependent stability and the stabilization of a class of FO memristive neural networks with time-varying delay. By employing the FO Razumikhin theorem and linear matrix inequalities (LMI), a delay-dependent asymptotic stability condition in the form of LMI is established and used to design a stabilizing state-feedback controller. The results address both the effects of the delay and the FO. In addition, the upper bound of the absolute value of the memristive synaptic weights used in previous studies are released, leading to less conservative conditions. Three numerical simulations illustrate the theoretical results and show their effectiveness.

Delay dependent stability of stochastic split-step θ methods for stochastic delay differential equations

In this paper, the delay dependent asymptotic mean square stability of the stochastic split-step θ method for a scalar linear stochastic delay differential equation with real coefficients is studied. The full stability region of this method is given by using root locus technique. The necessary and sufficient condition with respect to the equation coefficients, time stepsize and method parameter θ is derived. It is proved that the stochastic split-step backward Euler can preserve the asymptotic mean square stability of the underlying system completely. In addition, the numerical stability regions of the stochastic split-step θ method and the stochastic θ method are compared with each other. At last, we validate our conclusions by numerical experiments.

Decentralized H∞ Control of Interconnected Systems with Time-varying Delays

This paper focuses on the problem of delay dependent stability/stabilization of interconnected systems with time-varying delays. The approach is based on a new Lyapunov-Krasovskii functional. A decentralized delay-dependent stability analysis is performed to characterize linear matrix inequalities (LMIs) based on the conditions under which every local subsystem of the linear interconnected delay system is asymptotically stable. Then we design a decentralized state-feedback stabilization scheme such that the family of closedloop feedback subsystems enjoys the delay-dependent asymptotic stability for each subsystem. The decentralized feedback gains are determined by convex optimization over LMIs. All the developed results are tested on a representative example and compared with some recent previous ones.

Stability Analysis of Nonlinear Time–Delayed Systems with Application to Biological Models

Abstract In this paper, we analyse the local stability of a gene-regulatory network and immunotherapy for cancer modelled as nonlinear time-delay systems. A numerically generated kernel, using the sum-of-squares decomposition of multivariate polynomials, is used in the construction of an appropriate Lyapunov–Krasovskii functional for stability analysis of the networks around an equilibrium point. This analysis translates to verifying equivalent LMI conditions. A delay-independent asymptotic stability of a second-order model of a gene regulatory network, taking into consideration multiple commensurate delays, is established. In the case of cancer immunotherapy, a predator–prey type model is adopted to describe the dynamics with cancer cells and immune cells contributing to the predator–prey population, respectively. A delay-dependent asymptotic stability of the cancer-free equilibrium point is proved. Apart from the system and control point of view, in the case of gene-regulatory networks such stability analysis of dynamics aids mimicking gene networks synthetically using integrated circuits like neurochips learnt from biological neural networks, and in the case of cancer immunotherapy it helps determine the long-term outcome of therapy and thus aids oncologists in deciding upon the right approach.

Decentralised quantised fuzzy control for large-scale T-S discrete delay systems

A class of fuzzy large-scale nonlinear discrete systems with local quantisers is examined. This fuzzy system has unknown-but-bounded couplings and delays. Of interest are the issues of analysis and design related to decentralised fuzzy feedback structure with H∞ measure. It is established that the resulting closed-loop fuzzy system exhibits delay-dependent asymptotic stability with disturbance-attenuation level. In addition, all the computations are performed at the subsystem level using a fuzzy-basis-dependent Lyapunov-Krasovskii functional. Using quantised output measurements, a local procedure is constructed for tuning the quantiser parameters to achieve similar asymptotic stability and guaranteed performance. Special cases of interest are derived and simulation results on typical applications are provided.

Decentralised quantised fuzzy control for large-scale T-S discrete delay systems

A class of fuzzy large-scale nonlinear discrete systems with local quantisers is examined. This fuzzy system has unknown-but-bounded couplings and delays. Of interest are the issues of analysis and design related to decentralised fuzzy feedback structure with H∞ measure. It is established that the resulting closed-loop fuzzy system exhibits delay-dependent asymptotic stability with disturbance-attenuation level. In addition, all the computations are performed at the subsystem level using a fuzzy-basis-dependent Lyapunov-Krasovskii functional. Using quantised output measurements, a local procedure is constructed for tuning the quantiser parameters to achieve similar asymptotic stability and guaranteed performance. Special cases of interest are derived and simulation results on typical applications are provided.

Stability analysis and feedback control of T-S fuzzy hyperbolic delay model for a class of nonlinear systems with time-varying delay

In this paper, a new T-S fuzzy hyperbolic delay model for a class of nonlinear systems with time-varying delay, is presented to address the problems of stability analysis and feedback control. Fuzzy controller is designed based on the parallel distributed compensation (PDC), and with a new Lyapunov function, delay dependent asymptotic stability conditions of the closed-loop system are derived via linear matrix inequalities (LMIs). Besides, considering the differences between the model and the real system, we extent the model to uncertain T-S fuzzy hyperbolic delay model. Based on the uncertain model, a robust $H_{infty}$ fuzzy controller is obtained and stability conditions are developed in terms of LMIs. The main advantage of the control based on T-S fuzzy hyperbolic delay model is that it can achieve small control amplitude via ``soft'' constraint approach. Finally, a numerical example and the Van de Vusse example are given to validate the advantages of the proposed method.

Delay-dependent asymptotic stability criteria for genetic regulatory networks with impulsive perturbations

This paper considers the problem for asymptotic stability of genetic regulatory networks with impulse control using the delay partitioning approach. By using the direct Lyapunov method, a new Lyapunov–Krasovskii functional is introduced based on the decomposition approach. Time delays here are assumed to be time-varying and belong to the given intervals. When dealing with the time derivative of Lyapunov functional, a new tight integral inequality is adopted for bounding the cross terms. Then, a new less conservative delay-dependent stability criterion is formulated in terms of linear matrix inequalities (LMIs), which can be easily solved by the Matlab LMI toolbox. Finally, the proposed method is validated through the numerical simulation, which shows the effectiveness of the presented stability criteria.

Stability analysis of decentralized event-triggered H∞ control using the quadratic convex approach

This paper studies the stability analysis of the decentralized event-triggered H∞ control with communication delays using the quadratic convex approach. Unlike the decentralized event-triggered mechanism (ETM), which only uses the information from the sensor itself by considering the communication topology of the wireless sensor network, a more general decentralized ETM is first proposed by using the information from both the sensor itself and its neighbours. Then, a time-delay system model with parameters of the decentralized ETM, directed graph information, communication delays and external disturbances is presented. In addition, novel delay-dependent asymptotic stability criteria are derived by using the augmented Lyapunov–Krasovski functional (LKF), which contains the cross terms of variables and quadratic terms multiplied by a higher degree scalar function. Unlike some prior results using the first-order convex combination property, our derivation applies the quadratic convex approach with the augmented LKF, which results in less conservatism. Moreover, sufficient conditions for the co-design of the controller and the decentralized ETM are obtained. Finally, numerical examples confirm the effectiveness of the proposed method.

Robust Stability Analysis of Takagi–Sugeno Fuzzy Nonlinear Singular Systems with Time-Varying Delays Using Delay Decomposition Approach

This paper investigates the problem of robust stability analysis for Takagi---Sugeno fuzzy nonlinear singular systems with time-varying delays. The nonlinear functions are assumed to satisfy the Lipschitz conditions. By constructing Lyapunov---Krasovskii functional with different weighted matrices, sufficient delay-dependent asymptotic stability conditions are expressed in terms of linear matrix inequalities. Further, delay decomposition approach is used to derive less conservative results. The effectiveness of the derived theoretical result is shown through numerical examples.

Asymptotic Stability Criteria for Genetic Regulatory Networks with Time-Varying Delays and Reaction–Diffusion Terms

This paper investigates the asymptotic stability problem for delayed genetic regulatory networks with reaction---diffusion terms under both Dirichlet boundary conditions and Neumann boundary conditions. First, by constructing a new Lyapunov---Krasovskii functional and using Jensen's inequality, Wirtinger's inequality, Green's second identity and the reciprocally convex approach, we establish delay-dependent asymptotic stability criteria that do not require a restriction of the upper bounds of the delays' derivatives being less than 1. Thus, the stability criteria that we establish are less conservative than the existing criteria and extend the range of applications of the theoretical results. In addition, it is shown that the obtained criterion under Dirichlet boundary conditions retains the information about the reaction---diffusion terms, while these do not exist in the criterion under Neumann boundary conditions. It is then theoretically presented that the stability criteria established in this paper are less conservative than the existing ones. Finally, numerical examples are given to illustrate the effectiveness of the theoretical results.

Improved Delay-Dependent Stability Analysis for Neural Networks with Interval Time-Varying Delays

The problem of delay-dependent asymptotic stability analysis for neural networks with interval time-varying delays is considered based on the delay-partitioning method. Some less conservative stability criteria are established in terms of linear matrix inequalities (LMIs) by constructing a new Lyapunov-Krasovskii functional (LKF) in each subinterval and combining with reciprocally convex approach. Moreover, our criteria depend on both the upper and lower bounds on time-varying delay and its derivative, which is different from some existing ones. Finally, a numerical example is given to show the improved stability region of the proposed results.