A novel approach to exponential stability of nonlinear systems with time-varying delays

In this paper, the stability of nonlinear systems with time-varying delays is investigated by means of the concepts of generalized Dahlquist constant, generalized relative Dahlquist constant and relative minimal Lipschitz constant. In detail, two sufficient conditions are derived for the exponential stability of nonlinear systems with time-varying delays and the exponential decay of the solutions is also estimated. Compared with some existing results, our stability conditions are less conservative. Some examples are given to illustrate the effectiveness of the obtained results.     

Robust stability criteria for systems with interval time-varying delay and nonlinear perturbations

This paper considers the robust stability for a class of linear systems with interval time-varying delay and nonlinear perturbations. A Lyapunov-Krasovskii functional, which takes the range information of the time-varying delay into account, is proposed to analyze the stability. A new approach is introduced for estimating the upper bound on the time derivative of the Lyapunov-Krasovskii functional. On the basis of the estimation and by utilizing free-weighting matrices, new delay-range-dependent stability criteria are established in terms of linear matrix inequalities (LMIs). Numerical examples are given to show the effectiveness of the proposed approach.     

Stability radii of positive linear Volterra-Stieltjes equations

We study stability radii of linear Volterra-Stieltjes equations under multi-perturbations and affine perturbations. A lower and upper bound for the complex stability radius with respect to multi-perturbations are given. Furthermore, in some special cases concerning the structure matrices, the complex stability radius can precisely be computed via the associated transfer functions. Then, the class of positive linear Volterra-Stieltjes equations is studied in detail. It is shown that for this class, complex, real and positive stability radius under multi-perturbations or multi-affine perturbations coincide and can be computed by simple formulae expressed in terms of the system matrices. As direct consequences of the obtained results, we get some results on robust stability of positive linear integro-differential equations and of positive linear functional differential equations. To the best of our knowledge, most of the results of this paper are new.     

On the stability and boundedness of solutions of nonlinear vector differential equations of third order

In this paper, by using Liapunov’s second method, we establish some new results for stability and boundedness of solutions of nonlinear vector differential equations of third order. By constructing a Liapunov function, sufficient conditions for stability and boundedness of solutions of equations considered are obtained. Concerning to the subject, some explanatory examples are also given. Our results improve and include a result existing in the literature.     

Robust stability analysis for Markovian jumping interval neural networks with discrete and distributed time-varying delays

This paper investigates robust stability analysis for Markovian jumping interval neural networks with discrete and distributed time-varying delays. The parameter uncertainties are assumed to be bounded in given compact sets. The delay is assumed to be time-varying and belong to a given interval, which means that the lower and upper bounds of interval time-varying delays are available. Based on the new Lyapunov–Krasovskii functional (LKF), some inequality techniques and stochastic stability theory, new delay-dependent stability criteria have been obtained in terms of linear matrix inequalities (LMIs). Finally, two numerical examples are given to illustrate the less conservative and effectiveness of our theoretical results.     

On nonuniform exponential dichotomy of evolution operators in Banach spaces

The problem of nonuniform exponential dichotomy of evolution operators in Banach spaces is considered. Connections between this concept and admissibility of the pair (C ₀,C ₀) are established. Generalizations to the nonuniform case of some results of Van Min, Räbiger and Schnaubelt ([MRS]) are obtained. It is shown that an implication from the uniform case is not true for nonuniform exponential dichotomy. The results are applicable for general time-varying linear equations with unbounded coefficients in Banach spaces.     

Delay-dependent robust stability criteria for uncertain systems with interval time-varying delay

This paper proposes an approach for the robust stability of uncertain systems with interval time-varying delay. The key features of the approach include the introduction of uncorrelated augmented matrix items into the Lyapunov functional and the use of a tighter bounding technology. Unlike existing methodologies, the proposed approach involves neither free weighting matrices nor any model transformation. It can, however, lead to much less conservative stability criteria than the existing ones for the systems under consideration. Numerical examples show that the proposed criteria improve the existing results significantly with much less computational effort.     

Improved delay-dependent robust stability criteria for a class of uncertain mixed neutral and Lur’e dynamical systems with interval time-varying delays and sector-bounded nonlinearity

The previously proposed delay-dependent robust stability criteria for a class of uncertain mixed neutral and Lur’e dynamical systems with interval time-varying delays are refined by eliminating nonlinearity and/or reducing the number of free-weighting matrices. Several robust stability criteria proposed in this paper are simpler and effective. Numerical examples show that the new method is less conservative than some previous approaches.     

A new criterion for global robust stability of interval delayed neural networks

A novel criterion for the global robust stability of Hopfield-type interval neural networks with delay is presented. An example showing the effectiveness of the present criterion is given.     

Parameter-dependent robust stability of uncertain time-delay systems

This paper presents a new delay-dependent and parameter-dependent robust stability criterion for linear continuous-time systems with polytopic parameter uncertainties and time-varying delay in the state. This criterion, expressed as a set of linear matrix inequalities, requires no matrix variable to be fixed for the entire uncertainty polytope, which produces a less conservative stability test result. Numerical examples are given to show the effectiveness of the proposed techniques.     

Sufficient conditions for the asymptotic stability of nonlinear multidelay differential equations with linear parts defined by pairwise permutable matrices

In this paper a generalization of the delayed exponential defined by Khusainov and Shuklin (2003) [1] for autonomous linear delay systems with one delay defined by permutable matrices is given for delay systems with multiple delays and pairwise permutable matrices. Using this multidelay-exponential a solution of a Cauchy initial value problem is represented. By an application of this representation and using Pinto’s integral inequality an asymptotic stability results for some classes of nonlinear multidelay differential equations are proved.     

Further results on robust stability of neutral system with mixed time-varying delays and nonlinear perturbations

This paper studies delay-dependent robust stability problem for neutral system with mixed time-varying delays. The uncertainties under consideration are nonlinear time-varying parameter perturbations and norm-bounded uncertainties, respectively. Based on Lyapunov functional approach and linear matrix inequality technology, some improved delay-dependent stability conditions are derived by introducing free-weighting matrices. Numerical examples are given to demonstrate the effectiveness and less conservativeness of the proposed methods.     

Novel delay-dependent asymptotic stability criteria for neural networks with time-varying delays

The problem of delay-dependent asymptotic stability criteria for neural networks (NNs) with time-varying delays is investigated. An improved linear matrix inequality based on delay-dependent stability test is introduced to ensure a large upper bound for time-delay. A new class of Lyapunov function is constructed to derive a novel delay-dependent stability criteria. Finally, numerical examples are given to indicate significant improvement over some existing results.     

Stability analysis for neutral systems with mixed delays

Relationships between system states contained in the neutral equation are used to address the delay-dependent stability of a neutral system with time-varying state delay. Using linear matrix inequalities, we present a new asymptotic stability criterion, and a new robust stability criterion, for neutral systems with mixed delays. Since the criteria take into account the sizes of the neutral delay, discrete delay and the derivative of discrete delay, they are less conservative than those produced by previous approaches. Numerical examples are presented to demonstrate that these criteria are indeed more effective.     

Exponential stability criteria for discrete-time recurrent neural networks with time-varying delay

In this paper, the robust global exponential stability is investigated for the discrete-time recurrent neural networks (RNNs) with time-varying interval delay. By choosing an augmented Lyapunov–Krasovskii functional, delay-dependent results guaranteeing the global exponential stability and the robust exponential stability of the concerned neural network are obtained. The results are shown to be a generalization of some previous results, and less conservative than the existing works. Two numerical examples are given to demonstrate the applicability of the proposed method.     

Novel robust stability criteria for uncertain systems with time-varying delay

This paper is concerned with the delay-dependent stability and robust stability criteria for linear systems with time-varying delay and norm-bounded uncertainties. Through constructing a general form of Lyapunov–Krasovskii functional, and using integral inequalities, some slack matrices and newly established convex combination condition in the calculation, the delay-dependent stability criteria are derived in terms of linear matrix inequalities. Numerical examples are given to illustrate the improvement on the conservatism of the delay bound over some reported results in the literature.     

Sampled-data control of uncertain switched neutral nonlinear systems under asynchronous switching

The robust exponential stabilization for a class of the uncertain switched neutral nonlinear systems with time-varying delays based on the sampled-data control is investigated in this paper. The closed-loop system with sampled-data control is modeled as a continuous time system with a time-varying piecewise continuous control input delay. Considering the relationship between the sampling period and the dwell time of two switching instants, sampling interval with no switching and sampling interval with one switching are discussed, respectively. By Wirtinger-based inequality, Wirtinger-based double integral inequality, and free-weighting matrix technique, some delay-dependent sufficient conditions are given to guarantee the exponential stability of uncertain switched neutral nonlinear systems under asynchronous switching. In addition, sampled-data controllers can also be designed by special operations of matrices. Finally, two numerical examples are used to show the effectiveness of the approach proposed in this paper.     

Improved results on robust exponential stability criteria for neutral-type delayed neural networks

In this paper, we investigate the problem of robust global exponential stability analysis for a class of neutral-type neural networks. The interval time-varying delays allow for both slow and fast time-varying delays. The values of the time-varying uncertain parameters are assumed to be bounded within given compact sets. Improved global exponential stability condition is derived by employing new Lyapunov-Krasovskii functional and the integral inequality. The developed nominal and robust stability criteria is delay-dependent and characterized by linear-matrix inequalities (LMIs). The developed results are less conservative than previous published ones in the literature, which are illustrated by representative numerical examples.     

Qualitative analysis and applications of a kind of state-dependent impulsive differential equations

This paper analyzes a certain type of impulsive differential equations (IDEs). Several useful theorems for its periodic solutions and their stabilities are given. The key idea is that a periodically time-dependent IDE can be transformed into the state-dependent IDE. As applications of our theory, the optimization problems in population dynamics are studied. That is, the maximum sustainable yields of single population models with periodically impulsive constant harvesting are discussed. Furthermore, we apply these results to the studies of the order-1 periodic solutions and their stability of a single population model with stage structure in which the mature is impulsively proportionally harvested while the immature is impulsively added with the constant.     

Some improved criteria for global robust exponential stability of neural networks with time-varying delays

In this paper, some sufficient conditions for global robust exponential stability of interval neural networks with time-varying delays are presented. It is shown that our results include some counterparts of the previous literatures. On basis of the obtained results, some linear matrix inequality (LMI) criteria are derived. Moreover, three numerical examples and a simulation are given to show the effectiveness of the obtained results.