[5] X. Yang, D. J. Evans, and G. M. Megson, “**Global** **asymptotic** **stability** in a class of Putnam-type equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 1, pp. 42–50, 2006.
[6] J. Kuang, Applied Inequalities, Shandong Science and Technology Press, Jinan, China, 2004.
[7] K. S. Berenhaut and S. Stevi´c, “The **global** attractivity of a higher order rational diﬀerence equa- tion,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 940–944, 2007.

, n = 0, 1, . . . ,
where all parameters α , β , a i , b i , a ij , b ij , i, j = 0, 1, . . . ,k, and the initial conditions x i , i ∈ {–k,. . . , 0}, are nonnegative. We investigate the **asymptotic** behavior of the solutions of the considered equation. We give simple explicit conditions for the **global** **stability** and **global** **asymptotic** **stability** of the zero or positive equilibrium of this equation.

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Received: August 28, 2018; Accepted: October 17, 2018; Published: November 9, 2018
Abstract: In the present world, due to the complicated dynamic properties of neural cells, many dynamic neural networks are described by neutral functional differential equations including neutral delay differential equations. These neural networks are called neutral neural networks or neural networks of neural-type. The differential expression not only defines the derivative term of the current state but also explains the derivative term of the past state. In this paper, **global** **asymptotic** **stability** of a neutral-type neural networks, with time-varying delays, are presented and analyzed. The neural network is made up of parts that include: linear, non-linear, non-linear delayed, time delays in time derivative states, as well as a part of activation function with the derivative. Different from prior references, as part of the considered networks, the last part involves an activation function with the derivative rather than multiple delays; that is a new class of neutral neural networks. This paper assumes that the activation functions satisfy the Lipschitz conditions so that the considered system has a unique equilibrium point. By constructing a Lyapunov-Krasovskii-type function and by using a linear matrix inequality analysis technique, a sufficient condition for **global** **asymptotic** **stability** of this neural network has been obtained. Finally, we present a numerical example to show the effectiveness and applicability of the proposed approach.

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On the other hand, stochastic phenomenon always appears in the electrical circuit design of neural networks [ 3 ]. Moreover, a neural network could be stabilized or destabilized by certain stochastic inputs [ 26 ]. Hence, stochastic effects should be taken into account in the design of delayed neural networks. Many interesting results on **stability** of stochastic neural networks with mixed delays have been reported via different approaches; see [ 27–35 ]. In particular, LMI-based technique is an important approach and has been successfully used to tackle various **stability** problems for stochastic neural networks with mixed delays [ 28–35 ]. The main advantages of the approach include firstly that it only needs tuning of parameters and matrices, and secondly it can be easily checked by resorting to the Matlab LMI toolbox. In [ 29 ], Balasubramaniam and Rakkiyappan investigated the **global** **asymptotic** **stability** of stochastic recurrent neural networks with discrete and distributed delays by utilizing the Lyapunov–Krasovskii functional and LMIs. Using the same approach, Balasubramaniam and Rakkiyappan [ 32 ] further investigated the **global** asymptotical **stability** for a class of Markovian jumping stochastic Cohen–Grossberg neural networks with discrete interval and distributed delays. In [ 33 , 34 ], Wang et al. studied the exponential **stability** of uncertain stochastic neural networks with discrete and distributed delays and robust **stability** for stochastic Hopfield neural networks with time delays. However, to the best of our knowledge, there are almost no LMI-based results on the problems of **global** **stability** of Cohen–Grossberg-type BAM neural networks with time-varying and distributed delays. This inspires our work.

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In this paper, based on Lyapunov-Krasovskii functionals and some inequality techniques, we have investigated the problem of **global** **asymptotic** **stability** for piecewise homoge- neous Markovian jump BAM neural networks with discrete and distributed time-varying delays. A linear matrix inequalities method has been developed to solve this problem. The suﬃcient condition has been established in terms of LMIs. A numerical example is given to demonstrate the usefulness of the derived LMI-based **stability** conditions.

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Then we apply the results to investigate the **stability** of equilibrium of T when it satisﬁes certain type of sublinear conditions with respect to the partial order deﬁned by a closed convex cone. The examples of application to nonlinear diﬀerence equations are also given.
2000 Mathematics Subject Classiﬁcation: 47H07, 47H09, 47H10.

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By using multiplier method, it is proven that the zero solution of the problem is globally asymptotically stable.
1. Introduction
The straight-line vertical position of marine risers has been investigated with respect to dynamic **stability** 1. It studies the following initial boundary value problem describing the dynamics of marine riser:

CUBIC STOCHASTIC DIFFERENCE EQUATIONS
ALEXANDRA RODKINA AND HENRI SCHURZ
Received 18 September 2003 and in revised form 22 December 2003
**Global** almost sure **asymptotic** **stability** of solutions of some nonlinear stochastic dif- ference equations with cubic-type main part in their drift and di ﬀ usive part driven by square-integrable martingale diﬀerences is proven under appropriate conditions in R 1 . As an application of this result, the **asymptotic** **stability** of stochastic numerical methods, such as partially drift-implicit θ-methods with variable step sizes for ordinary stochastic diﬀerential equations driven by standard Wiener processes, is discussed.

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x n n So，in what follows, we hypothesize that 0 b 1 .
when b (0,1) ,then Eq(1)can be described as a nonlinear equation instead of a rational difference equation one .Unfortunately, we have not found the effective solution to the **global** behavior of nonlinear difference equations of order greater than one. Therefore, to study the qualitative properties of nonlinear difference equations with higher order is theoretically meaningful.

Develop a black-box extremum-seeking method with an increased con- vergence rate compared to classical extremum-seeking methods that uses small-amplitude low-frequency perturbations.
Although to aim of applying extremum-seeking control is to optimize the steady- state performance of a plant, the optimal steady-state performance is commonly not obtained (not even in infinite time). This can be attributed to the use of performance-indicator measurements, often in combination with added perturba- tions, to find the optimal steady-state plant performance. While the steady-state values of the performance indicators are assumed to be measured, the measure- ments are different from the steady-state values due to measurement noise and the dynamic response of the plant to changing plant-parameter values. Nonethe- less, several extremum-seeking method have been proposed in the literature to obtain **asymptotic** convergence. These methods are based on regulating the am- plitude of the perturbations (Moura and Chang, 2013; Stankovi´c and Stipanovi´c, 2010; Wang et al., 2016) or omitting the perturbations entirely (Blackman, 1962; Frait and Eckman, 1962; Hunnekens et al., 2014). Commonly local convergence to the optimum is proved, often for a limited class of plants. A **global** **asymptotic** **stability** result for general nonlinear plants is missing. This brings us to the second research objective of this thesis.

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To appear in: Systems and Control Letters.
Abstract
In this short paper we deal with the **stability** analysis problem of non-autonomous non- linear systems, in cascade. In particular we give sucient conditions to guarantee that: (i) a globally uniformly stable (GUS) nonlinear time-varying (NLTV) system remains GUS when it is perturbed by the output of a globally uniformly asymptotically stable (GUAS) NLTV system, under the assumption that the perturbing signal is absolutely integrable; (ii) if in ad- dition the perturbed system is GUAS, it remains GUAS under the cascaded interconnection;

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Key Words: Asymptotic stability; Fractional Volterra type integral equation; Fractional diﬀerential equation with modification of the argument; Diﬀerential equations with fractional inte[r]

Received 9 July 2007; Accepted 19 November 2007 Recommended by Elena Braverman
We investigate the **global** **stability** character of the equilibrium points and the period-two solutions of y n+1 = (py n + y n − 1 )/(r + qy n + y n − 1 ), n = 0, 1, ..., with positive parameters and nonnegative initial conditions. We show that every solution of the equation in the title converges to either the zero equilibrium, the positive equilibrium, or the period-two solution, for all values of parameters outside of a specific set defined in the paper. In the case when the equilibrium points and period-two solution coexist, we give a precise description of the basins of attraction of all points. Our results give an aﬃrmative answer to Conjecture 9.5.6 and the complete answer to Open Problem 9.5.7 of Kulenovi´c and Ladas, 2002.

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In the first part of the paper, we consider the **stability** problem of a matrix polytope through common quadratic Lyapunov functions. We suggest a modified gradient algorithm. In the second part by using Bendixson’s theo- rem a sufficient condition for a stable member is given.
References

(December 23, 1998)
General **asymptotic** approach to the **stability** problem of multi-parameter solitons in Hamiltonian systems i∂E n /∂z = δH/δE n ∗ has been developed. It has been shown that **asymptotic** study of the soliton **stability** can be reduced to the calculation of a certain sequence of the determinants, where the famous determinant of the matrix consisting from the derivatives of the system invariants with respect to the soliton parameters is just the first in the series. The presented approach gives first analytical criterion for the oscillatory instability and also predicts novel stationary instability. Higher order approximations allow to calculate corresponding eigenvalues with arbitrary accuracy.

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DELAY DIFFERENCE SYSTEMS
HIDEAKI MATSUNAGA
Received 9 September 2003 and in revised form 22 December 2004
For the linear delay diﬀerence system x n+1 − x n = Ax n − k , where A is a 2 × 2 real constant matrix and k is a nonnegative integer, we present an explicit necessary and suﬃcient condition for the **asymptotic** **stability** of the zero solution of this system in terms of detA, tr A , and the delay k .

EQUATIONS ON TIME SCALES
GRO HOVHANNISYAN
Received 29 December 2005; Revised 5 April 2006; Accepted 7 April 2006
We examine the conditions of **asymptotic** **stability** of second-order linear dynamic equa- tions on time scales. To establish **asymptotic** **stability** we prove the **stability** estimates by using integral representations of the solutions via **asymptotic** solutions, error estimates, and calculus on time scales.

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stable, norm, linear systems, null solution, Schauder- Tychcoff Theorem, uniformly converges, equicontnu.. 7980 MATHEMATICS SUBJECT CLASSIFICATION CODES..[r]

Abstract In this article, we survey the **asymptotic** **stability** analysis of fractional diﬀerential systems with the Prabhakar fractional derivatives. We present the **stability** regions for these types of fractional diﬀerential systems. A brief comparison with the **stability** aspects of fractional diﬀerential systems in the sense of Riemann-Liouville fractional derivatives is also given.

Abstract In this article, we introduce the fractional diﬀerential systems in the sense of the Weber fractional derivatives and study the **asymptotic** **stability** of these systems. We present the **stability** regions and then compare the **stability** regions of fractional diﬀerential systems with the Riemann-Liouville and Weber fractional derivatives.

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