scholarly journals CONVERGENCE OF SOLUTIONS OF TIME-VARYING LINEAR SYSTEMS WITH INTEGRABLE FORCING TERM

2008 ◽  
Vol 78 (3) ◽  
pp. 445-462 ◽  
Author(s):  
JITSURO SUGIE
Keyword(s):  
Sufficient Conditions ◽  
Zero Solution ◽  
Time Varying ◽  
Forcing Term ◽  
Varying Coefficients ◽  
Perturbed System ◽  
All Solutions ◽  
Time Varying Coefficients ◽  
Image Position ◽  
Homogeneous Linear

AbstractThe following system is considered in this paper: The primary goal is to establish conditions on time-varying coefficients e(t), f(t), g(t) and h(t) and a forcing term p(t) for all solutions to converge to the origin (0,0) as $t \to \infty $. Here, the zero solution of the corresponding homogeneous linear system is assumed to be neither uniformly stable nor uniformly attractive. Sufficient conditions are given for asymptotic stability of the zero solution of the nonlinear perturbed system under the assumption that q(t,0,0)=0.

Uniform global asymptotic stability for half-linear differential systems with time-varying coefficients

2011 ◽  
Vol 141 (5) ◽  
pp. 1083-1101 ◽  
Author(s):  
Masakazu Onitsuka ◽  
Jitsuro Sugie
Keyword(s):  
Linear System ◽  
Zero Solution ◽  
Integrable Case ◽  
Time Varying ◽  
Globally Asymptotically Stable ◽  
Varying Coefficients ◽  
Linear Differential Systems ◽  
Time Varying Coefficients ◽  
Linear Differential ◽  
Image Position

The present paper deals with the following system:where p and p* are positive numbers satisfying 1/p + 1/p* = 1, and ϕq(z) = |z|q−2z for q = p or q = p*. This system is referred to as a half-linear system. We herein establish conditions on time-varying coefficients e(t), f(t), g(t) and h(t) for the zero solution to be uniformly globally asymptotically stable. If (e(t), f(t)) ≡ (h(t), g(t)), then the half-linear system is integrable. We consider two cases: the integrable case (e(t), f(t)) ≡ (h(t), g(t)) and the non-integrable case (e(t), f(t)) ≢ (h(t), g(t)). Finally, some simple examples are presented to illustrate our results.

EXPONENTIAL CONVERGENCE BEHAVIOR OF FUZZY CELLULAR NEURAL NETWORKS WITH DISTRIBUTED DELAYS AND TIME-VARYING COEFFICIENTS

2009 ◽  
Vol 19 (07) ◽  
pp. 2455-2462 ◽  
Author(s):  
MAN-CHUN TAN
Keyword(s):  
Neural Networks ◽  
Sufficient Conditions ◽  
Cellular Neural Networks ◽  
Distributed Delays ◽  
Time Varying ◽  
Fuzzy Cellular Neural Networks ◽  
Lipschitz Continuous ◽  
Varying Coefficients ◽  
All Solutions ◽  
Time Varying Coefficients

In this paper, a class of fuzzy cellular neural networks (FCNNs) with distributed delays and time-varying coefficients is studied. By employing the mathematical analysis method, some sufficient conditions are derived for ensuring that all solutions of such FCNNs converge exponentially to zero. Both the Lipschitz continuous condition on activation functions and the differentiability of time-varying delays are not required in this study.

scholarly journals Qualitative Analyses of Differential Systems with Time-Varying Delays via Lyapunov–Krasovskiĭ Approach

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1196
Author(s):  
Cemil Tunç ◽  
Osman Tunç ◽  
Yuanheng Wang ◽  
Jen-Chih Yao
Keyword(s):  
Sufficient Conditions ◽  
Zero Solution ◽  
Unperturbed System ◽  
Time Varying ◽  
Time Varying Delay ◽  
Gronwall’S Inequality ◽  
Perturbed System ◽  
Linear Delay ◽  
Delay Differential ◽  
Varying Delay

In this paper, a class of systems of linear and non-linear delay differential equations (DDEs) of first order with time-varying delay is considered. We obtain new sufficient conditions for uniform asymptotic stability of zero solution, integrability of solutions of an unperturbed system and boundedness of solutions of a perturbed system. We construct two appropriate Lyapunov–Krasovskiĭ functionals (LKFs) as the main tools in proofs. The technique of the proofs depends upon the Lyapunov–Krasovskiĭ method. For illustration, two examples are provided in particular cases. An advantage of the new LKFs used here is that they allow to eliminate using Gronwall’s inequality. When we compare our results with recent results in the literature, the established conditions are more general, less restrictive and optimal for applications.

scholarly journals Model of time-varying linear systems and Kolmogorov equations

2015 ◽  
Vol 25 (2) ◽  
pp. 201-214
Author(s):  
Assen V. Krumov
Keyword(s):  
Analytical Solution ◽  
Linear Systems ◽  
Sufficient Conditions ◽  
Original System ◽  
Approximate Model ◽  
Time Varying ◽  
Kolmogorov Equations ◽  
Varying Coefficients ◽  
Method Of Solution ◽  
Time Varying Coefficients

Abstract In the paper an approximate model of time-varying linear systems using a sequence of time-invariant systems is suggested. The conditions for validity of the approximation are proven with a theorem. Examples comparing the numerical solution of the original system and the analytical solution of the model are given. For the system under the consideration a new criterion giving sufficient conditions for robust Lagrange stability is suggested. The criterion is proven with a theorem. Examples are given showing stable and non stable solutions of a time-varying system and the results are compared with the numerical Runge-Kutta solution of the system. In the paper an important application of the described method of solution of linear systems with time-varying coefficients, namely analytical solution of the Kolmogorov equations is shown.

scholarly journals The VIT Transform Approach to Discrete-Time Signals and Linear Time-Varying Systems

Eng ◽  
2021 ◽  
Vol 2 (1) ◽  
pp. 99-125
Author(s):  
Edward W. Kamen
Keyword(s):  
Discrete Time ◽  
Initial Conditions ◽  
Linear Time ◽  
Order Term ◽  
Initial Time ◽  
Time Varying ◽  
Varying Coefficients ◽  
Time Signals ◽  
Time Varying Systems ◽  
Time Varying Coefficients

A transform approach based on a variable initial time (VIT) formulation is developed for discrete-time signals and linear time-varying discrete-time systems or digital filters. The VIT transform is a formal power series in z−1, which converts functions given by linear time-varying difference equations into left polynomial fractions with variable coefficients, and with initial conditions incorporated into the framework. It is shown that the transform satisfies a number of properties that are analogous to those of the ordinary z-transform, and that it is possible to do scaling of z−i by time functions, which results in left-fraction forms for the transform of a large class of functions including sinusoids with general time-varying amplitudes and frequencies. Using the extended right Euclidean algorithm in a skew polynomial ring with time-varying coefficients, it is shown that a sum of left polynomial fractions can be written as a single fraction, which results in linear time-varying recursions for the inverse transform of the combined fraction. The extraction of a first-order term from a given polynomial fraction is carried out in terms of the evaluation of zi at time functions. In the application to linear time-varying systems, it is proved that the VIT transform of the system output is equal to the product of the VIT transform of the input and the VIT transform of the unit-pulse response function. For systems given by a time-varying moving average or an autoregressive model, the transform framework is used to determine the steady-state output response resulting from various signal inputs such as the step and cosine functions.

scholarly journals Estimating latent group structure in time-varying coefficient panel data models

2019 ◽  
Author(s):  
Jia Chen
Keyword(s):  
Panel Data ◽  
Data Models ◽  
Panel Data Models ◽  
Time Varying ◽  
Clustering Method ◽  
Varying Coefficient ◽  
Varying Coefficients ◽  
Time Varying Coefficients ◽  
Latent Group ◽  
Group Structures

Summary This paper studies the estimation of latent group structures in heterogeneous time-varying coefficient panel data models. While allowing the coefficient functions to vary over cross-sections provides a good way to model cross-sectional heterogeneity, it reduces the degree of freedom and leads to poor estimation accuracy when the time-series length is short. On the other hand, in a lot of empirical studies, it is not uncommon to find that heterogeneous coefficients exhibit group structures where coefficients belonging to the same group are similar or identical. This paper aims to provide an easy and straightforward approach for estimating the underlying latent groups. This approach is based on the hierarchical agglomerative clustering (HAC) of kernel estimates of the heterogeneous time-varying coefficients when the number of groups is known. We establish the consistency of this clustering method and also propose a generalised information criterion for estimating the number of groups when it is unknown. Simulation studies are carried out to examine the finite-sample properties of the proposed clustering method as well as the post-clustering estimation of the group-specific time-varying coefficients. The simulation results show that our methods give comparable performance to the penalised-sieve-estimation-based classifier-LASSO approach by Su et al. (2018), but are computationally easier. An application to a panel study of economic growth is also provided.

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