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In this study, a stability analysis and H ∞ performance are derived for the bilateral teleoperation systems with time-varying delay. A new model transformation is firstly applied by employing a three-term approximation for the delayed state with the scaled small-gain theorem (SSG). By using appropriate Lyapunov–Krasovskii functional (LKF) and a generalized free-matrix-based integral inequality, new stability analysis and H ∞ performance criteria are proposed in terms of linear matrix inequalities. Finally, the engineering relevance of theory has been highlighted with different examples. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the global exponential stability of a class of inertial Cohen-Grossberg neural networks with parameter uncertainties and time-varying delays. By constructing a modified delaydependent Lyapunov-Krasovskii functional, delay-dependent criteria stated with simple algebraic inequalities are given in order to ensure the global exponential stability for the addressed neural network model. In sharp contrast to the existed reduced order method used to and delay-independent criteria derived for the neural networks with inertial terms, the model proposed and results established of this paper are more general and rigorous. Finally, numerical examples with simulations are presented to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this work, the issue of synchronization control for complex dynamical networks with time-varying coupling delay based on sampled-data is addressed. First, the sampled-data control is transformed into the bounded sampling period with time-varying delay. As a result, this model is converted to the investigation for synchronization of complex networks with multiple time-varying delays. Second, an appropriate Lyapunov-Krasovkii functional (LKF) is constructed, which composites of double and triple integral terms in quadratic form. Third, we propose a new integral inequality including a reciprocally convex technique which leads to a better condition. Fourth, the solution for the controller gain matrix is obtained by solving linear matrix inequalities (LMIs) with available software such that the synchronization error system is exponentially stable. Finally, the effectiveness and less conservatism of our proposed method are demonstrated via numerical examples.

In this paper we investigate the presence of oscillations and simple dynamics in x'(t) = -?x(t) + f(x(t))g(x(t - t)). Specifically, we prove that the global attraction towards a nontrivial equilibrium is reduced to the nonexistence of solutions of a certain system of inequalities. Our results cover situations in which the global attraction critically depends on the delay. From an applied point of view, this equation naturally appears in many biological situations, e.g., epidemic models with awareness and the production of platelets. In epidemiology, it is becoming clear that the different types of population awareness and the time delays of individuals' responses to available information about the disease play a critical role in its spread. Using our theoretical results, we describe qualitative properties of the behavioral responses that prevent the presence of sustainable oscillations in the number of infected individuals. Regarding physiological models, we discuss the influence of some biological parameters in certain anomalies in the production of platelets. [ABSTRACT FROM AUTHOR]

Summary: This article addresses the issue of delay‐dependent H∞ filtering design for TakagiŮSugeno fuzzy time‐varying delay systems using the input–output approach. A three‐term approximation model has been used to transform the original system into two interconnected subsystems. Since the nonquadratic Lyapunov–Krasovskii functional requires to deal with the membership function's (MF) time derivative, upper‐bound inequalities have been added to the obtained conditions. Based on the scaled small gain theorem, nonquadratic Lyapunov–Krasovskii functional approach and considering the bounds of the MF time‐derivative, the H∞ full‐ and reduced‐order filters are designed and then formulated in terms of linear matrix inequalities. Finally, illustrative examples are presented to demonstrate the validity of the proposed methods. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with global exponential stability of time-varying delayed fuzzy inertial neural networks. Different from previous approaches of variable transformation, we use non-reduced order method. Different from previous non-reduced order method used to investigate the inertial neural networks without time-varying delays, we take the time-varying delayed effects into account. By constructing a modified delay-dependent Lyapunov functional and inequality technique, delay-dependent criteria stated with simple algebraic inequalities are given in order to ensure the global exponential stability for the addressed delayed fuzzy inertial neural network model. The approach applied can provide a new method to study the fuzzy inertial neural networks with time delays via non-reduced order method. Some previous works in the literature are extend and complement. Finally, numerical examples with simulations are presented to make comparisons between the system with delays and without delays, and further demonstrate the validity and originality of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper mainly studies the periodicity and exponential stability for a class of inertial neural networks (INNs) with time-varying delays. Without utilizing standard reduced-order transformation, by using the continuation theorem and Cauchy–Schwarz inequality, delay-dependent criteria shown by some algebraic inequalities are derived to ensure the existence of periodic solutions. Furthermore, by means of the fundamental inequality and constructing a modified delay-dependent Lyapunov functional, global exponential stability analysis is obtained based on the derived delay-dependent criteria. In comparison with the reduced order approach applied to the INNs and delay-independent criteria provided for the INNs in the existed literatures, the results obtained in this paper are new. Finally, numerical simulations are carried out to verify the main results. [ABSTRACT FROM AUTHOR]

We analyse the stability and attractivity of a class of $n$-dimensional Nicholson systems with constant coefficients and multiple time-varying delays. Delay-independent sufficient conditions on the coefficients are given, for the existence and absolute global exponential stability of a unique positive equilibrium $N^*$, generalizing and improving known results for autonomous systems. We further establish delay-dependent criteria for $N^*$ to be a global attractor of all positive solutions. In the latter case, upper bounds on the size of the delays which do not require an a priori explicit knowledge of the equilibrium $N^*$ are also derived.

This paper discusses the problem of delay-dependent robust filtering design for polytopic systems with a time-varying delay. A new model transformation is firstly applied by employing a three-term approximation for the delayed state, which leads to a smaller approximation error than the two-term approximation. Then, based on the scaled small-gain Theorem combined with an appropriate Lyapunov-Krasovskii Functional, the performance analysis of the filtering error system is examined and then the full- and reduced-order filters are designed in terms of linear matrix inequalities via a simple linearisation technique. Before the end, a sufficient condition is presented to solve the problem of filtering design for a time-delay system without polytopic uncertainties. Finally, illustrative examples are presented to demonstrate the validity of the proposed methods. [ABSTRACT FROM AUTHOR]

Abstract. We analyse the stability and attractivity of a class of n-dimensional Nicholson systems with constant coefficients and multiple time-varying delays. Delayindependent sufficient conditions on the coefficients are given, for the existence and absolute global exponential stability of a unique positive equilibrium N*, generalizing and improving known results for autonomous systems. We further establish delaydependent criteria for N* to be a global attractor of all positive solutions. In the latter case, upper bounds on the size of the delays which do not require an a priori explicit knowledge of the equilibrium N* are also derived. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of robust stabilization and non-fragile robust control for a class of uncertain stochastic nonlinear time-delay systems that satisfy a one-sided Lipschitz condition. The parametric uncertainties are assumed to be real time-varying and norm bounded. Based on the one-sided Lipschitz condition including useful information of the nonlinear part, a new stability criterion for this class of nonlinear systems is provided. A memoryless non-fragile state-feedback controller is designed to guarantee robust stochastic stability of closed-loop systems. The approach of linear matrix inequalities is proposed to solve the robust stability for stochastic nonlinear systems with time-varying delay, and to obtain new delay-dependent sufficient conditions. Numerical examples are given to illustrate the validity and advantages of the proposed theoretical results. [ABSTRACT FROM AUTHOR]

Highlights: [•] The design of state-feedback H ∞ problem is considered. [•] Practical engineering systems are given to illustrate the proposed results. [•] The solution is posed in terms of LMI and such conditions checked via LMI solvers. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of an observer design for stochastic time-delay systems. The system states are unmeasured. We derive delay-dependent LMI criteria by means of the Leibniz-Newton formula, the Itô's differential operator and stochastic Lyapunov stability theory in order to obtain sufficient conditions for the asymptotic stability in the mean square for the closed-loop stochastic time-delay system. The proposed conditions are easily and numerically tractable via a Matlab LMI toolbox. The effectiveness of the control strategy is verified by numerical experiments. [ABSTRACT FROM AUTHOR]

This article is concerned with the problem of H∞ filter design for nonlinear Markovian jump neutral systems through the Takagi-Sugeno fuzzy model approach. By using a novel Markovian switched Lyapunov functional, a delay-dependent bounded real lemma (BRL) is presented in terms of linear matrix inequalities. Based on the derived BRL, both normal H∞ filters and non-fragile H∞ filters are designed, which guarantee that the corresponding filtering error systems are stochastically stable with a specified H∞ performance level. A numerical example is given to demonstrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the stochastic stability problem of neutral-type impulsive neural networks (NINNs) with mixed time-varying delays and Markovian jumping is investigated. By utilizing the Lyapunov–Krasovkii functional approach and linear matrix inequality (LMI) technique, we obtain some novel globally exponentially stable results. Delay-dependent sufficient condition for the above problem is obtained, which is usually less conservative than delay-independent ones. An example is given to show the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

Abstract: This Letter considers a robust stability analysis method for delayed neural networks of neutral type. By constructing a new Lyapunov functional, a novel delay-dependent criterion for the stability is derived in terms of LMIs (linear matrix inequalities). A less conservative stability criterion is derived by using nonlinear properties of the activation function of the neural networks. Two numerical examples are illustrated to show the effectiveness of the proposed method. [Copyright &y& Elsevier]

This brief investigates the problem of mean square exponential stability of uncertain stochastic delayed neural networks (DNNs) with time-varying delay. A novel Lyapunov functional is introduced with the idea of the discretized Lyapunov--Krasovskii functional (LKF) method. Then, a new delay-dependent mean square exponential stability criterion is derived by applying the free-weighting matrix technique and by equivalently eliminating time-varying delay through the idea of convex combination. Numerical examples illustrate the effectiveness of the proposed method and the improvement over some existing methods. [ABSTRACT FROM AUTHOR]

Abstract: This paper focuses on the design problem of a hybrid dynamical output feedback controller for exponential stabilization of a class of uncertain time-varying delay systems with actuator failures. By representing the time-delay system in the descriptor form and using the average dwell time technique, a new delay-dependent stabilization criterion for the existence of hybrid output feedback controllers is derived in terms of matrix inequalities. A cone complementary linearization algorithm is developed to design the hybrid output feedback controller. Finally, the effectiveness and the applicability of the proposed method are demonstrated by using two simulation examples. [Copyright &y& Elsevier]

Abstract: This paper studies the problem of filtering for a class of stochastic systems with Markovian switching and mixed mode-dependent time-varying delays. By introducing slack matrix variables and using Markovian switching Lyapunov functionals, we obtain delay-dependent sufficient conditions which guarantee the existence of filters such that the filtering error system is exponentially mean-square stable and satisfies a prescribed performance level; the conditions are in terms of linear matrix inequalities (LMIs). When these LMIs are feasible, an explicit expression of a desired filter is given. Numerical examples are provided to demonstrate the effectiveness and the reduced conservatism of our results. [Copyright &y& Elsevier]

The problem of absolute stability of Lur’e systems with time-delay is investigated in this paper. By employing a new Lyapunov-Krasovskii functional with the idea of N-segmentation of delay length, less conservative delay-dependent stability criteria are obtained and formulated in the form of linear matrix inequalities (LMIs). Numerical example shows that the results obtained in this paper are better than existing ones. [ABSTRACT FROM AUTHOR]

This paper considers the problem of delay-dependent exponential stability in mean square for stochastic systems with polytopic-type uncertainties and time-varying delay. Applying the descriptor model transformation and introducing free weighting matrices, a new type of Lyapunov-Krasovskii functional is constructed based on linear matrix inequalities (LMIs), and some new delay-dependent criteria are obtained. These criteria include the delay-independent/ratedependent and delay-dependent/rate-independent exponential stability criteria. These new criteria are less conservative than existing ones. Numerical examples demonstrate that these new criteria are effective and are an improvement over existing ones. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the problem of global exponential stability analysis for a class of cellular neural networks with time-varying discrete and distributed delays (DDCNNs). A new delay-dependent sufficient condition is derived for the global exponential stability of the DDCNNs by using the integral inequality method and the newly proposed Lyapunov–Krasovskii functional. The obtained stability condition is less conservative than some of the existing results in the literature. Numerical examples are given to demonstrate the effectiveness and superiority of the proposed results. [Copyright &y& Elsevier]

This letter considers uncertain Lur'e systems of neutral type with sector and slope restrictions. By constructing a new Lyapunov functional, a novel delay-dependent criterion for absolute stability is derived in terms of linear matrix inequalities (LMIs). Two numerical examples are illustrated to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The problem of the delay-dependent nonfragile H∞ observer-based control for a class of neutral systems with time delays is investigated. The additive gain variations under consideration are contained in both the controller gain and the observer gain. Novel delay-dependent criteria are derived to guarantee the stability of the nonfragile H∞ observer-based control system using the Lyapunov function approach combined with linear matrix inequalities (LMI). The controller and observer gains are given from the LMI feasible solutions. Based on the result of this paper, the constraint of matrix equality is not necessary for designing a nonfragile H∞ observer-based control. The computer software Matlab can be applied to solve the proposed problems. Finally, a numerical example is given illustrating the design of the nonfragile H∞ observer-based control. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the mean square exponential stability is investigated for a class of discrete-time stochastic neural networks with time-varying delays and norm-bounded uncertainties. Based on Lyapunov stability theory and stochastic approaches, delay-dependent criteria are derived to ensure the robust exponential stability in the mean square for the addressed system. Meantime, by using the numerically efficient Matlab LMI Toolbox, a example is presented to show the usefulness of the derived LMI-based stability condition. [Copyright &y& Elsevier]

This brief is concerned with asymptotic stability of neural networks with uncertain delays. Two types of uncertain delays are considered: one is constant while the other is time varying. The discretized Lya- punov-Krasovskii functional (LKF) method is integrated with the technique of introducing the free-weighting matrix between the terms of the Leibniz-Newton formula. The integrated method leads to the establishment of new delay-dependent sufficient conditions in form of linear matrix inequalities for asymptotic stability of delayed neural networks (DNNs). A numerical simulation study is conducted to demonstrate the obtained theoretical results, which shows their less conservatism than the existing stability criteria. [ABSTRACT FROM AUTHOR]

Abstract: This paper considers the exponential stability problem for a class of discrete time-varying delay systems with large delay sequences (LDSs). A new method based on a switching technique is presented to solve this problem. A switched delay system, in which one of the discrete subsystems may be unstable, is firstly employed to describe such a system, and then some new concepts about LDSs are introduced. Next, using a novel piecewise Lyapunov functional, explicit delay-dependent conditions are developed to show how long and how frequent the LDSs can be and still maintain the system exponentially stable. Without LDSs, the criterion obtained in this paper includes the established one as a special case. Two numerical examples are given to show the effectiveness of the proposed method. [Copyright &y& Elsevier]

Abstract-The robust state estimation problem for a class of uncertain neural networks with time-varying delay is studied in this paper. The parameter uncertainties are assumed to be norm bounded. Based on a new bounding technique, a sufficient condition is presented to guarantee the existence of the desired state estimator for the uncertain delayed neural networks. The criterion is dependent on the size of the time-varying delay and on the size of the time derivative of the time-varying delay. It. is shown that the design of the robust state estimator for such neural networks can be achieved by solving a linear matrix inequality (LMI), which can be easily facilitated by using some standard numerical packages. Finally, two simulation examples are given to demonstrate the effectiveness of the developed approach. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the problem of delay-dependent stability criteria for a class of constant time-delay neutral systems with time-varying structured uncertainties is investigated. New asymptotic stability criteria are derived in the form of linear matrix inequality. Two numerical examples are given to indicate that the results presented in this paper are effective and less conservative than some existing results. [Copyright &y& Elsevier]

This paper focuses on the design problem of a memoryless state feedback robust H-infinity controller for a class of uncertain neutral systems. By using a newly established integral inequality, a new delay-dependent bounded real lemma for such systems is derived without involving a fixed model transformation. Furthermore, new delay-dependent sufficient conditions for the existence of robust H-infinity controllers are presented in terms of nonlinear matrix inequalities. A design procedure involving an iterative algorithm is also proposed to design such controllers. Numerical examples are given to demonstrate the less conservatism of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some ideas of switched systems into the field of neural networks and a large class of switched recurrent neural networks (SRNNs) with time-varying structured uncertainties and time-varying delay is investigated. Some delay-dependent robust periodicity criteria guaranteeing the existence, uniqueness, and global asymptotic stability of periodic solution for all admissible parametric uncertainties are devised by taking the relationship between the terms in the Leibniz-Newton formula into account. Because free weighting matrices are used to express this relationship and the appropriate ones are selected by means of linear matrix inequalities (LMIs), the criteria are less conservative than existing ones reported in the literature for delayed neural networks with parameter uncertainties. Some examples are given to show that the proposed criteria are effective and are an improvement over previous ones. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the problem of global exponential stability for a general class of stochastic high-order neural networks with mixed time delays and Markovian jumping parameters. The mixed time delays under consideration comprise both discrete time-varying delays and distributed time-delays. The main purpose of this paper is to establish easily verifiable conditions under which the delayed high-order stochastic jumping neural network is exponentially stable in the mean square in the presence of both mixed time delays and Markovian switching. By employing a new Lyapunov–Krasovskii functional and conducting stochastic analysis, a linear matrix inequality (LMI) approach is developed to derive the criteria ensuring exponential stability. Furthermore, the criteria are dependent on both the discrete time-delay and distributed time-delay, and hence less conservative. The proposed criteria can be readily checked by using some standard numerical packages such as the Matlab LMI Toolbox. A simple example is provided to demonstrate the effectiveness and applicability of the proposed testing criteria. [Copyright &y& Elsevier]

Abstract: This Letter concerns with the mean square exponential stability of uncertain stochastic delayed neural networks. By applying Lyapunov functional method, new delay-dependent/independent mean square exponential stability criteria are derived in terms of linear matrix inequalities. Two examples are presented which show our result are less conservative than the existing stability criteria. [Copyright &y& Elsevier]

In this letter, the global asymptotical stability analysis problem is considered for a class of Markovian jumping stochastic Cohen-Grossberg neural networks (CGNNs) with mixed delays including discrete delays and distributed delays. An alternative delay-dependent stability analysis result is established based on the linear matrix inequality (LMI) technique, which can easily be checked by utilizing the numerically efficient Matlab LMI toolbox. Neither system transformation nor free-weight matrix via Newton-Leibniz formula is required. Two numerical examples are included to show the effectiveness of the result. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a delay-dependent approach is developed to deal with the robust stabilization problem for a class of stochastic time-delay interval systems with nonlinear disturbances. The system matrices are assumed to be uncertain within given intervals, the time delays appear in both the system states and the nonlinear disturbances, and the stochastic perturbation is in the form of a Brownian motion. The purpose of the addressed stochastic stabilization problem is to design a memoryless state feedback controller such that, for all admissible interval uncertainties and nonlinear disturbances, the closed-loop system is asymptotically stable in the mean square, where the stability criteria are dependent on the length of the time delay and therefore less conservative. By using Itô''s differential formula and the Lyapunov stability theory, sufficient conditions are first derived for ensuring the stability of the stochastic interval delay systems. Then, the controller gain is characterized in terms of the solution to a delay-dependent linear matrix inequality (LMI), which can be easily solved by using available software packages. A numerical example is exploited to demonstrate the effectiveness of the proposed design procedure. [Copyright &y& Elsevier]

This paper deals with the problem of delay-dependent robust stability for a class of switched Hopfield neural networks with time-varying structured uncertainties and time-varying delay. Some Lyapunov-Krasovskii functionals are constructed and the linear matrix inequality (LMI) approach and free weighting matrix method are employed to devise some delay-dependent stability criteria which guarantee the existence, uniqueness and global exponential stability of the equilibrium point for all admissible parametric uncertainties. By using Leibniz-Newton formula, free weighting matrices are employed to express this relationship, which implies that the new criteria are less conservative than existing ones. Some examples suggest that the proposed criteria are effective and are an improvement over previous ones. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the problem of delay-dependent robust stability for delay neutral type control system with time-varying structured uncertainties and time-varying delay. Some new delay and its derivative dependent criteria are derived and formulated in the form of linear matrix inequalities (LMIs), the new criteria are less conservative than the existing ones. Numerical examples are given to illustrate the proposed method. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the problems of robust stability, robust stabilization and H ∞ control via memoryless state feedback for uncertain discrete-time switched systems with mode-dependent time delays. Based on linear matrix inequalities (LMIs), a descriptor model transformation of the system and a switched Lyapunov function, new delay-dependent criteria are established which are not contained in known literatures. Numerical examples are presented to illustrate the effectiveness of the theoretical results. [Copyright &y& Elsevier]

Abstract: This Letter provides new exponential stability criteria for discrete-time neural networks with variable delays. The main technique is to reduce exponential convergence estimation of the neural network solution to that of one component of the corresponding solution by constructing Lyapunov function based on M-matrix. By introducing the tuning parameter diagonal matrix, the delay-independent and delay-dependent exponential stability conditions have been unified in the same mathematical formula. The effectiveness of the new results are illustrated by three examples. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the problems of robust stability and stabilization via memoryless state feedback for uncertain discrete-time switched systems with mode-dependent time-varying delays. By using a descriptor system method and linear matrix inequality technique, and by introducing a switched Lyapunov functional, we establish some new delay-dependent stability and stabilization criteria for the system. Numerical examples are presented to illustrate the effectiveness of the theoretical results. [Copyright &y& Elsevier]

This brief focuses on the problem of delay-dependent stability analysis of neural networks with variable delay. Two types of variable delay are considered: one is differentiable and has bounded derivative; the other one is continuous and may vary very fast. By introducing a new type of Lyapunov-Krasovskii functional, new delay-dependent sufficient conditions for exponential stability of delayed neural networks are derived in terms of linear matrix inequalities. We also obtain delay-independent stability criteria. Two examples are presented which show our results are less conservative than the existing stability criteria. [ABSTRACT FROM AUTHOR]

The delay-dependent robust stability of uncertain linear neutral systems with delays is investigated. Both discrete-delay-dependent/neutral-delay-independent and neutral-/discrete-delay-dependent stability criteria will be developed. The proposed stability criteria are formulated in the form of linear matrix inequalities and it is easy to check the robust stability of the considered systems. By introducing certain Lyapunov-Krasovskii functional the mathematical development of our result avoids model transformation and bounding for cross terms, which lead to conservatism. Finally, numerical example is given to indicate the improvement over some existing results. [ABSTRACT FROM AUTHOR]

The problem of delay-dependent stability and passivity for linear neutral systems is discussed. By constructing a novel type Lyapunov-krasovskii functional, a new delay-dependent passivity criterion is presented in terms of linear matrix inequalities (LMIs). Model transformation, bounding for cross terms and selecting free weighting matrices [12∼14] are not required in the arguments. Numerical examples show that the proposed criteria are available and less conservative than existing results. [ABSTRACT FROM AUTHOR]

Abstract: This Letter provides new exponential stability criteria for neural networks with variable delays. By applying the properties of M-matrix properties and inequality techniques, simple and improved delay-dependent exponential stability criteria are derived. These criteria extend and improve the existing ones, which is important in the design of neural networks. [Copyright &y& Elsevier]

This paper deals with the problem of robust stability for continuous-time singular systems with state delay and parameter uncertainty. The uncertain singular systems with delay considered in this paper are assumed to be regular and impulse free. By decomposing the systems into slow and fast subsystems,a robust delay-dependent asymptotic stability criteria based on linear matrix inequality is proposed, which is derived by using Lyapunov-Krasovskii functionals, neither model transformation nor bounding for cross terms is required in the derivation of our delay-dependent result. The robust delay-dependent stability criterion proposed in this paper is a sufficient condition. Finally, numerical examples and Matlab simulation are provided to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper focuses on the problem of delay-dependent robust stability of neutral systems with different discrete-and-neutral delays and time-varying structured uncertainties. Some new criteria are presented, in which some free weighting matrices are used to express the relationships between the terms in the Leibniz-Newton formula. The criteria include the information on the size of both neutral-and-discrete delays. It is shown that the present results also include the results for identical discrete-and-neutral delays as special cases. A numerical example illustrates the improvement of the proposed methods over the previous methods and the influences between the discrete and neutral delays. [ABSTRACT FROM AUTHOR]