Abstract:
In this paper, we consider the problem of robust stability for a class of linear systems with interval time-varying delay under nonlinear perturbations using Lyapunov-Krasovskii (LK) functional approach. By partitioning the delay-interval into two segments of equal length, and evaluating the time-derivative of a candidate LK functional in each segment of the delay-interval, a less conservative delay-dependent stability criterion is developed to compute the maximum allowable bound for the delay-range within which the system under consideration remains asymptotically stable. In addition to the delay-bi-segmentation analysis procedure, the reduction in conservatism of the proposed delay-dependent stability criterion over recently reported results is also attributed to the fact that the time-derivative of the LK functional is bounded tightly using a newly proposed bounding condition without neglecting any useful terms in the delay-dependent stability analysis. The analysis, subsequently, yields a stable condition in convex linear matrix inequality (LMI) framework that can be solved non-conservatively at boundary conditions using standard numerical packages. Furthermore, as the number of decision variables involved in the proposed stability criterion is less, the criterion is computationally more effective. The effectiveness of the proposed stability criterion is validated through some standard numerical examples.

Abstract:
We establish some algebraic results on the zeros of some exponential polynomials and a real coefficient polynomial. Based on the basic theorem, we develop a decomposition technique to investigate the stability of two coupled systems and their discrete versions, that is, to find conditions under which all zeros of the exponential polynomials have negative real parts and the moduli of all roots of a real coefficient polynomial are less than 1. 1. Introduction For an ordinary (delay) differential equation, the trivial solution is asymptotically stable if and only if all roots of the corresponding characteristic equation of the linearized system have negative real parts while the moduli of all roots of a real coefficient polynomial less than 1 mean the trivial solution is asymptotically stable for the difference equation. However, it is difficult to obtain the expression of the characteristic equation corresponding to the linearized systems. Special cases of the characteristic equation have been discussed by many authors. For example, Bellman and Cooke [1], Boese [2], Kuang [3], and Ruan and Wei [4–8] studied some exponential polynomials and used the results to investigate the stability and bifurcations for some systems. The well-known Jury criterion can be used to determine the moduli of the roots of a real coefficient polynomial less than one [9, 10], but the calculation is prolixly. The purpose of this paper is to provide a new algebraic criterion of zero for some exponential polynomials and a real coefficient polynomial. 2. Some Algebraic Results Let be a linear space over a number field and a subspace of . A complement space of in is a linear space of such that . A vector is said to be the projection of a vector along at if , where and . Let be a linear transformation on and a subspace of . A subspace is said to be a invariant subspace if for any . Let be a -complement space of for , if(1) is a invariant subspace;(2) is a complement space of in , that is, ;(3)for any , the projection of along at is always 0. Let , be two linear transformations on . We consider that and are concordant if there exist a nontrivial invariant subspaces of and , and a complement space of in , such that(1)at least one of constraints of on is a scalar transformation ;(2) is a -complement space of or for . Let . We say that and are concordant if linear transformations and of ？？ ,？？for all ,？？ ,？？for all are concordant. For example, and a reducible matrix are always in concordance. Theorem 2.1. Let be concordant. Then there exist an invertible matrix , such that or or or where .

Abstract:
In this paper, the sufficient conditions, in the sense of Liapunov, of delay-independent stability of the delay control system x(t)=Ax(t)+ATx(t-T) are obtained. The example presented shows that this stability criterion not only is concise and easy to use, but also can obtain some extension of the stable region.

Abstract:
The robust stability of neutral systems with time-varying discrete delay and norm-bounded uncertainties is studied. Firstly, a new variable is introduced to replace the uncertainties of the systems. Then, by constructing a new Lyapunov-Krasovskii functional, using integral inequality and introducing free weighting matrices, we derive a new asymptotic stability criterion in terms of the linear matrix inequality(LMI). The obtained criterion takes into account the neutral delay, discrete delay and the derivative of the discrete delay. This result is less conservative than some existing results. Finally, numerical examples demonstrate the improvements over existing results, and show the relationship among the neutral delay, discrete delay and the derivative of discrete delay.

Abstract:
We prove an existence result for stable vector bundles with arbitrary rank on an algebraic surface, and determine the birational structure of certain moduli space of stable bundles on a rational ruled surface.

Abstract:
Sufficient conditions for the Euclidean null controllability of non-linear delay systems with time varying multiple delays in the control and implicit derivative are derived. If the uncontrolled system is uniformly asymptotically stable and if the control system is controllable, then the non-linear infinite delay system is Euclidean null controllable. Journal of Applied Sciences and Environmental Management Vol. 9(3) 2005: 71-76

Abstract:
This paper analyzes the eigenvalue distribution of neutral differential systems and the corresponding difference systems, and establishes the relationship between the eigenvalue distribution and delay-independent stability of neutral differential systems. By using the ``Complete Discrimination System for Polynomials", easily testable necessary and sufficient algebraic criteria for delay-independent stability of a class of neutral differential systems are established. The algebraic criteria generalize and unify the relevant results in the literature. Moreover, the maximal delay bound guaranteeing stability can be determined if the systems are not delay-independent stable. Some numerical examples are provided to illustrate the effectiveness of our results.

Abstract:
This paper is concerned with the finite-time stability of Caputo fractional neural networks with distributed delay. The factors of such systems including Caputo’s fractional derivative and distributed delay are taken into account synchronously. For the Caputo fractional neural network model, a finite-time stability criterion is established by using the theory of fractional calculus and generalized Gronwall-Bellman inequality approach. Both the proposed criterion and an illustrative example show that the stability performance of Caputo fractional distributed delay neural networks is dependent on the time delay and the order of Caputo’s fractional derivative over a finite time. 1. Introduction It is well known that the fractional calculus is a generalization and extension of the traditional integer-order differential and integral calculus. The fractional calculus has gained importance in both theoretical and engineering applications of several branches of science and technology. It draws a great application in nonlinear oscillations of earthquakes and many physical phenomena such as seepage flow in porous media and in fluid dynamic traffic models. Many practical systems in interdisciplinary fields can be described through fractional derivative formulation. For more details on fractional calculus theory, one can see the monographs of Miller and Ross [1], Podlubny [2], Diethelm [3], and Kilbas et al. [4]. In the last few years, there has been a surge in the study of the theory of fractional dynamical systems. Some recent works the theory of fractional differential systems can be seen in [5–10] and references therein. In particular, for the first time, Lazarevi？ [7] investigated the finite-time stability of fractional time-delay systems. In [8], Lazarevi？ and Spasi？ further introduced the Gronwall’s approach to discuss the finite-time stability of fractional-order dynamic systems. Compared with the classical integer-order derivatives, fractional-order derivatives provide an excellent approach for the description of memory and hereditary properties of various processes. Therefore, it may be more accurate to model by fractional-order derivatives than integer-order ones. In [11–13], fractional operators were introduced into artificial neural network, and the fractional-order formulations of artificial neural network models were also proposed in research works about biological neurons. Recently, there has been an increasing interest in the investigation of the fractional-order neural networks, and some important and interesting results were obtained [13–19], due