scholarly journals The Stability Cone for a Difference Matrix Equation with Two Delays

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
S. A. Ivanov ◽  
M. M. Kipnis ◽  
V. V. Malygina
Keyword(s):  
Neural Networks ◽  
Difference Equations ◽  
Matrix Equation ◽  
Geometric Algorithms ◽  
Difference Matrix ◽  
The Matrix ◽  
Two Delays ◽  
Matrix Difference Equations ◽  
The Stability

We provide geometric algorithms for checking the stability of matrix difference equations with two delays such that the matrix is nilpotent. We give examples of how our results can be applied to the study of the stability of neural networks.

New results for global stability of neutral-type delayed neural networks

2018 ◽  
Author(s):  
Neyir Ozcan
Keyword(s):  
Neural Networks ◽  
Time Delays ◽  
Matrix Theory ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Neural System ◽  
Delayed Neural Networks ◽  
The Matrix ◽  
The Stability ◽  
Stability Results

"This paper deals with the stability analysis of the class of neutral-type neural networks with constant time delay. By using a suitable Lyapunov functional, some delay independent sufficient conditions are derived, which ensure the global asymptotic stability of the equilibrium point for this this class of neutral-type neural networks with time delays with respect to the Lipschitz activation functions. The presented stability results rely on checking some certain properties of matrices. Therefore, it is easy to verify the validation of the constraint conditions on the network parameters of neural system by simply using some basic information of the matrix theory."

Local and global Hopf bifurcation analysis in a neutral-type neuron system with two delays

2018 ◽  
Vol 32 (08) ◽  
pp. 1850049
Author(s):  
Qiuyu Lv ◽  
Xiaofeng Liao
Keyword(s):  
Hopf Bifurcation ◽  
Difference Equations ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Global Hopf Bifurcation ◽  
Neuron System ◽  
Neural Network System ◽  
Two Delays ◽  
The Stability ◽  
Delay Differential

In recent years, neutral-type differential-difference equations have been applied extensively in the field of engineering, and their dynamical behaviors are more complex than that of the delay differential-difference equations. In this paper, the equations used to describe a neutral-type neural network system of differential difference equation with two delays are studied (i.e. neutral-type differential equations). Firstly, by selecting [Formula: see text], [Formula: see text] respectively as a parameter, we provide an analysis about the local stability of the zero equilibrium point of the equations, and sufficient conditions of asymptotic stability for the system are derived. Secondly, by using the theory of normal form and applying the theorem of center manifold introduced by Hassard et al., the Hopf bifurcation is found and some formulas for deciding the stability of periodic solutions and the direction of Hopf bifurcation are given. Moreover, by applying the theorem of global Hopf bifurcation, the existence of global periodic solution of the system is studied. Finally, an example is given, and some computer numerical simulations are taken to demonstrate and certify the correctness of the presented results.

scholarly journals Convergence analysis of gradient-based iterative algorithms for a class of rectangular Sylvester matrix equations based on Banach contraction principle

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Adisorn Kittisopaporn ◽  
Pattrawut Chansangiam ◽  
Wicharn Lewkeeratiyutkul
Keyword(s):  
Spectral Radius ◽  
Matrix Equation ◽  
Iterative Algorithms ◽  
Banach Contraction Principle ◽  
Contraction Principle ◽  
Convergence Factor ◽  
Sylvester Matrix ◽  
Iteration Matrix ◽  
Banach Contraction ◽  
The Matrix

AbstractWe derive an iterative procedure for solving a generalized Sylvester matrix equation $AXB+CXD = E$ A X B + C X D = E , where $A,B,C,D,E$ A , B , C , D , E are conforming rectangular matrices. Our algorithm is based on gradients and hierarchical identification principle. We convert the matrix iteration process to a first-order linear difference vector equation with matrix coefficient. The Banach contraction principle reveals that the sequence of approximated solutions converges to the exact solution for any initial matrix if and only if the convergence factor belongs to an open interval. The contraction principle also gives the convergence rate and the error analysis, governed by the spectral radius of the associated iteration matrix. We obtain the fastest convergence factor so that the spectral radius of the iteration matrix is minimized. In particular, we obtain iterative algorithms for the matrix equation $AXB=C$ A X B = C , the Sylvester equation, and the Kalman–Yakubovich equation. We give numerical experiments of the proposed algorithm to illustrate its applicability, effectiveness, and efficiency.

On the Necessary and Sufficient Conditions for the Stability of Linear Generalized Ordinary Differential, Linear Impulsive and Linear Difference Systems

2009 ◽  
Vol 16 (4) ◽  
pp. 597-616
Author(s):  
Shota Akhalaia ◽  
Malkhaz Ashordia ◽  
Nestan Kekelia
Keyword(s):  
Linear System ◽  
Difference Equations ◽  
Closed Interval ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Difference Systems ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Generalized Ordinary Differential Equations ◽  
Lyapunov Sense

Abstract Necessary and sufficient conditions are established for the stability in the Lyapunov sense of solutions of a linear system of generalized ordinary differential equations 𝑑𝑥(𝑡) = 𝑑𝐴(𝑡) · 𝑥(𝑡) + 𝑑𝑓(𝑡), where and are, respectively, matrix- and vector-functions with bounded total variation components on every closed interval from . The results are realized for the linear systems of impulsive, ordinary differential and difference equations.

Solution of the matrix equation AX + XB = C [F4]

1972 ◽  
Vol 15 (9) ◽  
pp. 820-826 ◽  
Author(s):  
R. H. Bartels ◽  
G. W. Stewart
Keyword(s):  
Matrix Equation ◽  
The Matrix

scholarly journals Improved Stability Criteria of Static Recurrent Neural Networks with a Time-Varying Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Lei Ding ◽  
Hong-Bing Zeng ◽  
Wei Wang ◽  
Fei Yu
Keyword(s):  
Neural Networks ◽  
Recurrent Neural Networks ◽  
Linear Matrix ◽  
Time Varying ◽  
Time Varying Delay ◽  
Improved Stability ◽  
The Stability ◽  
Delay Dependent ◽  
Delay Dependent Stability ◽  
Varying Delay

This paper investigates the stability of static recurrent neural networks (SRNNs) with a time-varying delay. Based on the complete delay-decomposing approach and quadratic separation framework, a novel Lyapunov-Krasovskii functional is constructed. By employing a reciprocally convex technique to consider the relationship between the time-varying delay and its varying interval, some improved delay-dependent stability conditions are presented in terms of linear matrix inequalities (LMIs). Finally, a numerical example is provided to show the merits and the effectiveness of the proposed methods.

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