scholarly journals Review of Some Control Theory Results on Uniform Stability of Impulsive Systems

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1186 ◽  
Author(s):  
Bin Liu ◽  
Bo Xu ◽  
Guohua Zhang ◽  
Lisheng Tong
Keyword(s):  
Asymptotic Stability ◽  
Control Theory ◽  
Hybrid Systems ◽  
Discrete Time ◽  
Continuous Time ◽  
Uniform Stability ◽  
Impulsive Systems ◽  
Uniform Asymptotic Stability ◽  
Stability Results ◽  
Stability Concepts

This paper aims to review some uniform stability results for impulsive systems. For the review, we classify the models of impulsive systems into time-based impulsive systems and state-based ones, including continuous-time impulsive systems, discrete-time impulsive systems, stochastic impulsive systems, and impulsive hybrid systems. According to these models, we review, respectively, the related stability concepts and some representative results focused on uniform stability, including the results on uniform asymptotic stability, input-to-state stability (ISS), KLL -stability (uniform stability expressed by KLL -functions), event-stability, and event-ISS. And we formulate some questions for those not fully developed aspects on uniform stability at each subsection.

scholarly journals ON STABILITY CRITERIA OF FRACTAL DIFFERENTIAL SYSTEMS OF CONFORMABLE TYPE

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040009
Author(s):  
AWAIS YOUNUS ◽  
THABET ABDELJAWAD ◽  
TAZEEN GUL
Keyword(s):  
Asymptotic Stability ◽  
Control Theory ◽  
Stability Criteria ◽  
Differential Systems ◽  
Central Concern ◽  
Finite Dimensional ◽  
Linear And Nonlinear ◽  
Stability Results

In this paper, stability results of central concern for control theory are given for finite-dimensional linear and nonlinear local fractional or fractal differential systems. The main purpose of this paper is to provide some results on stability and asymptotic stability of conformable order systems, together with some illustrating examples.

scholarly journals Analysis and comparison of the stability of discrete-time and continuous-time linear systems

2016 ◽  
Vol 26 (4) ◽  
pp. 551-563
Author(s):  
Tadeusz Kaczorek
Keyword(s):  
Asymptotic Stability ◽  
Discrete Time ◽  
Continuous Time ◽  
Asymptotically Stable ◽  
Discrete Time Systems ◽  
Continuous Time Systems ◽  
The Matrix ◽  
Discretization Step ◽  
Time Systems ◽  
Time Linear

Abstract The asymptotic stability of discrete-time and continuous-time linear systems described by the equations xi+1 = Ākxi and x(t) = Akx(t) for k being integers and rational numbers is addressed. Necessary and sufficient conditions for the asymptotic stability of the systems are established. It is shown that: 1) the asymptotic stability of discrete-time systems depends only on the modules of the eigenvalues of matrix Āk and of the continuous-time systems depends only on phases of the eigenvalues of the matrix Ak, 2) the discrete-time systems are asymptotically stable for all admissible values of the discretization step if and only if the continuous-time systems are asymptotically stable, 3) the upper bound of the discretization step depends on the eigenvalues of the matrix A.

scholarly journals On the Uniform Stability of Impulsive Lotka-Volterra Type Systems with Supremums

2018 ◽  
Vol 4 (2) ◽  
Author(s):  
Ivanka Stamova ◽  
Gani Stamov
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Method ◽  
Sufficient Conditions ◽  
Type Systems ◽  
Uniform Stability ◽  
Volterra Model ◽  
Uniform Asymptotic Stability ◽  
Volterra Type

In this paper we propose an impulsive n- species Lotka-Volterra model with supremums. By using Lyapunov method we give sufficient conditions for uniform stability and uniform asymptotic stability of the positive states.

Asymptotic Stability for an Impulsive Model of Hematopoiesis with Time Delay

2012 ◽  
Vol 204-208 ◽  
pp. 4506-4512
Author(s):  
Rui Chen ◽  
Pei Jun Ju
Keyword(s):  
Time Delay ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Function Method ◽  
Uniform Stability ◽  
Uniform Asymptotic Stability ◽  
Lyapunov Function Method ◽  
Impulsive Model

A model of hematopoiesis with time delay and impulses is studied. Based on the Lyapunov function method, uniform stability and uniform asymptotic stability of the equilibria is discussed.

On Problem of Partial Stability for Functional Differential Systems with Holdover

2019 ◽  
Vol 20 (7) ◽  
pp. 398-404
Author(s):  
V. I. Vorotnikov
Keyword(s):  
Asymptotic Stability ◽  
Equilibrium Position ◽  
Functional Differential Equations ◽  
Uniform Stability ◽  
Uniform Asymptotic Stability ◽  
Partial Stability ◽  
Additional Group ◽  
Functional Differential Systems ◽  
Functional Differential ◽  
Zero Equilibrium

The theory of systems of functional differential equations is a significant and rapidly developing sphere of modern mathematics which finds extensive application in complex systems of automatic control and also in economic, modern technical, ecological, and biological models. Naturally, the problems arises of stability and partial stability of the processes described by the class of the equation. The article studies the problem of partial stability which arise in applications either from the requirement of proper performance of a system or in assessing system capability. Also very effective is the approach to the problem of stability with respect to all variables based on preliminary analysis of partial stability. We suppose that the system have the zero equilibrium position. A conditions are obtained under which the uniform stability (uniform asymptotic stability) of the zero equilibrium position with respect to the part of the variables implies the uniform stability (uniform asymptotic stability) of this equilibrium position with respect to the other, larger part of the variables, which include an additional group of coordinates of the phase vector. These conditions include: 1) the condition for uniform asymptotic stability of the zero equilibrium position of the "reduced" subsystem of the original system with respect to the additional group of variables; 2) the restriction on the coupling between the "reduced" subsystem and the rest parts of the system. Application of the obtained results to a problem of stabilization with respect to a part of the variables for nonlinear controlled systems is discussed.

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