Stability radii for positive linear time-invariant systems on time scales

We deal with dynamic equations on time scales, where we characterize the positivity of a system. Uniform exponential stability of a system is determined by the spectrum of its matrix. We investigate the corresponding stability radii with respect to structured perturbations and show that, for positive systems, the complex and the real stability radius coincide.     

Moment exponential stability analysis of Markovian jump stochastic differential equations with uncertain transition jump rates

This paper is concerned with the moment exponential stability analysis of Markovian jump stochastic differential equations. The equations under consideration are more general, whose transition jump rates matrix Q is not precisely known. Sufficient conditions for testing the stability of such equations are established, and some numerical examples to illustrate the effectiveness of our results are presented.     

Necessary conditions of exponential stability for a class of linear neutral-type time-delay systems

For a class of linear neutral-type time-delay systems (NTTDSs), this paper will present necessary exponential stability conditions by employing the Lyapunov--Krasovskii functional approach. Since these conditions are represented by the Lyapunov matrix and the neutral coefficient matrix, they not only offer a novel tool for analysing stability of linear NTTDSs by characterising instability domains, but also extend the existing results of the neutral-delay-free systems. As a medium step, the relations between the Lyapunov matrix and the fundamental matrix are characterised. The validation of the obtained results is explained by numerical examples and comparison with some existing results.     

区间时变细胞神经网络的全局鲁棒指数稳定性

研究了一类区间时变扰动、变时滞细胞神经网络的全局鲁棒指数稳定性问题．利用Leibniz-Newton公式对原系统进行模型变换,并分析了变换模型和原始模型的等价性．基于变换模型,运用线性矩阵不等式的方法,通过选择适当的Lyapunov-Krasovskii泛函,推导了该系统全局鲁棒指数稳定的时滞相关的充分条件．通过数值实例将所得结果与前人的结果相比较,表明了本文所提出的稳定判据具有更低的保守性．     

Robustness of stability of time-varying index-1 DAEs

We study exponential stability and its robustness for time-varying linear index-1 differential-algebraic equations. The effect of perturbations on the leading coefficient matrix is investigated. An appropriate class of allowable perturbations is introduced. Robustness of exponential stability with respect to a certain class of perturbations is proved in terms of the Bohl exponent and perturbation operator. Finally, a stability radius involving these perturbations is introduced and investigated. In particular, a lower bound for the stability radius is derived. The results are presented by means of illustrative examples.     

Stability and stabilization of stochastic systems with multiplicative noise

In this paper, stability and stabilization of linear stochastic time-invariant systems are studied based on spectrum technique. Firstly, the relationship among mean square exponential stability, asymptotical mean square stability, second-order moment exponential stability and the spectral location of the systems is revealed with the help of a spectrum operator L _A,C . Then, we focus on almost sure exponential stability and stochastic stabilization. A criterion on almost sure exponential stability based on spectrum technique is obtained. Sufficient conditions for mean square exponentially stability and asymptotic mean square stability are given via linear matrix inequality approach and some numerical examples to illustrate the effectiveness of our results are presented.     

New results on robust exponential stability of integral delay systems

The robust exponential stability of integral delay systems with exponential kernels is investigated. Sufficient delay-dependent robust conditions expressed in terms of linear matrix inequalities and matrix norms are derived by using the Lyapunov–Krasovskii functional approach. The results are combined with a new result on quadratic stabilisability of the state-feedback synthesis problem in order to derive a new linear matrix inequality methodology of designing a robust non-fragile controller for the finite spectrum assignment of input delay systems that guarantees simultaneously a numerically safe implementation and also the robustness to uncertainty in the system matrices and to perturbation in the feedback gain.     

Exponential stability and stabilisation of fuzzy time-varying delay systems

This article presents new exponential stability and stabilisation conditions for Takagi–Sugeno fuzzy time-varying delay systems. Based on an improved Lyapunov–Krasovskii functional, the exponential stability and stabilisation conditions are derived in terms of linear matrix inequalities, which allows one to compute simultaneously the two bounds that characterise the exponential stability rate of the solution. Finally, some numerical examples are given to illustrate the effectiveness of the proposed conditions.     

Globally asymptotical stability of discrete-time analog neuralnetworks

Some globally asymptotical stability criteria for the equilibrium states of a general class of discrete-time dynamic neural networks with continuous states are presented using a diagonal Lyapunov function approach. The neural networks are assumed to have the asymmetrical weight matrices throughout the paper. The resulting criteria are described by the diagonal stability of some matrices associated with the network parameters. Some novel stability conditions represented by either the existence of the positive diagonal solutions of the Lyapunov equations or some inequalities are given. Using the equivalence between the diagonal stability and the Schur stability for a nonnegative matrix, some simplified global stability conditions are also presented. Finally, some examples are provided for demonstrating the effectiveness of the global stability conditions presented.     

Reduced-order dynamic compensator design for stability robustnessof linear discrete-time systems

A reduced-order dynamic compensator design is presented with stability robustness for linear discrete systems, by including a stability robustness component in addition to the standard quadratic state and control terms in the performance criterion. The robustness component is based on an unstructured perturbation stability bound for time varying perturbations. The controller design is developed by the parameter optimization technique and involves the solution of five algebraic matrix equations, four of which are discrete-time Lyapunov matrix equations     

Global exponential stability of recurrent neural networks forsolving optimization and related problems

Global exponential stability is a desirable property for dynamic systems. The paper studies the global exponential stability of several existing recurrent neural networks for solving linear programming problems, convex programming problems with interval constraints, convex programming problems with nonlinear constraints, and monotone variational inequalities. In contrast to the existing results on global exponential stability, the present results do not require additional conditions on the weight matrices of recurrent neural networks and improve some existing conditions for global exponential stability. Therefore, the stability results in the paper further demonstrate the superior convergence properties of the existing neural networks for optimization.     

On the stability of strategies in competitive systems

This work discusses in some details the mathematical properties of competitive systems. It is demonstrated how an optimal strategy of any dynamic matrix game can be derived analytically. The number of various pure strategies contributing to the optimal strategy is found by analysing the properties of the gain matrix. A relationship between the stability and fitness of equilibrium states is established. It is shown that the fitness of the system can be expressed in terms of the eigenvalue spectrum of the system's stability matrix. The methods developed are applied to a few examples.     

p-th moment exponential stability of stochastic differential equations with impulse effect

The p-th moment exponential stability of stochastic differential equations with impulse effect is addressed.By employing the method of vector Lyapunov functions,some sufficient conditions for the p-th moment exponential stability are established.In addition,the usual restriction of the growth rate of Lyapunov function is replaced by the condition of the drift and diffusion coefficients to study the p-th moment exponential stability.Several examples are also discussed to illustrate the effectiremess of the r...     

Further results on exponential stability of functional differential equations

General non-linear functional differential equations are considered. New explicit criteria for the exponential stability are presented. The stability criteria given in this paper include many existing results as particular cases. In particular, they unify, generalise and improve some ones published recently in [Ngoc, P. H. A. (2012). On exponential stability of non-linear differential systems with time-varying delay. Applied Mathematics Letters, 25(9), 1208–1213 and Ngoc, P. H. A. (2013b). Novel criteria for exponential stability of functional differential equations. Proceedings of the American Mathematical Society, 141(9), 3083–3091]. Two examples are given to show the effectiveness and advantage of the obtained results.     

A less conservative stability test for second-order linear time-varying vector differential equations

System stability and stability bounds play an essential role in control theory. This note is concerned with the exponential stability of a class of second-order linear time-varying vector differential equations with real piecewise continuous coefficient matrices. A less conservative explicit condition for stability of such a system is derived using the matrix measure theory and a more accurate upper bound for the decay exponent of its stable solution is established. Examples are included for illustration.     

New sufficient conditions for the stability of slowly varyinglinear systems

The frozen-time approach is used to state some new sufficient conditions for the stability of linear time-varying systems. An upper bound on the norm of the time derivative of system matrix which, under different assumptions on frozen-time system eigenvalues, guarantees asymptotic stability or exponential stability of the system is established     

变时滞分布参数系统的全局指数稳定性

基于比较原理,利用推广的向量Hanalay微分不等式,Dini导数,结合Green公式及不等式分析技术,研究几类变时滞分布参数控制系统所导出的滑动模运动方程的全局指数稳定性问题,在仅要求系数矩阵是个M-矩阵的条件下,获得了几类滑动模运动方程全局指数稳定性的充分条件,建立了滑动模运动方程全局指数稳定性定理.推广和改进了前人的结论.并为研究时滞分布参数系统的变结构控制问题奠定了基础.     

Stochastically exponential stability and stabilization of uncertain linear hyperbolic PDE systems with Markov jumping parameters

This paper is concerned with the problem of robustly stochastically exponential stability and stabilization for a class of distributed parameter systems described by uncertain linear first-order hyperbolic partial differential equations (FOHPDEs) with Markov jumping parameters, for which the manipulated input is distributed in space. Based on an integral-type stochastic Lyapunov functional (ISLF), the sufficient condition of robustly stochastically exponential stability with a given decay rate is first derived in terms of spatial differential linear matrix inequalities (SDLMIs). Then, an SDLMI approach to the design of robust stabilizing controllers via state feedback is developed from the resulting stability condition. Furthermore, using the finite difference method and the standard linear matrix inequality (LMI) optimization techniques, recursive LMI algorithms for solving the SDLMIs in the analysis and synthesis are provided. Finally, a simulation example is given to demonstrate the effectiveness of the developed design method.     

pth moment and almost sure exponential stability of impulsive neutral stochastic functional differential equations with Markovian switching

In this paper, the problems on the pth moment and the almost sure exponential stability for a class of impulsive neutral stochastic functional differential equations with Markovian switching are investigated. By using the Lyapunov function, the Razumikhin-type theorem and the stochastic analysis, some new conditions about the pth moment exponential stability are first obtained. Then, by using the Borel–Cantelli lemma, the almost sure exponential stability is also discussed. The results generalise and improve some results obtained in the existing literature. Finally, two examples are given to illustrate the obtained results.     

Global exponential stability analysis for neutral delay-differential systems: an LMI approach

For neutral differential systems with time-varying or constant delays, the problems of determining the global exponential stability and estimating the exponential convergence rate are investigated in this paper. By using the Lyapunov method, some useful criteria of global exponential stability for the systems are derived. These stability conditions are formulated as linear matrix inequalities (LMIs) which can easily be solved by various convex optimization algorithms. Numerical examples are given to illustrate the application of the proposed method.