Results on existence of smooth Lyapunov functions for (pre-)asymptotically stable hybrid systems with non-open basins of attraction

This paper is devoted to the study of the stability of a CD[Formula: see text] T cell viral infection model with diffusion. First, we discuss the well-posedness of the model and the existence of endemic equilibrium. Second, by analyzing the roots of the characteristic equation, we establish the local stability of the virus-free equilibrium. Furthermore, by constructing suitable Lyapunov functions, we show that the virus-free equilibrium is globally asymptotically stable if the threshold value [Formula: see text]; the endemic equilibrium is globally asymptotically stable if [Formula: see text] and [Formula: see text]. Finally, we give an application and numerical simulations to illustrate the main results.

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Stability Analysis for Autonomous Dynamical Switched Systems through Nonconventional Lyapunov Functions

Mathematical Problems in Engineering ◽

10.1155/2015/502475 ◽

2015 ◽

Vol 2015 ◽

pp. 1-12 ◽

Cited By ~ 5

Author(s):

V. Nosov ◽

J. A. Meda-Campaña ◽

J. C. Gomez-Mancilla ◽

J. O. Escobedo-Alva ◽

R. G. Hernández-García

Keyword(s):

Stability Analysis ◽

Finite Number ◽

Switched Systems ◽

Lyapunov Functions ◽

Switched System ◽

Linear Functions ◽

Asymptotically Stable ◽

Multiple Lyapunov Functions ◽

Stability Theorems ◽

The Stability

The stability of autonomous dynamical switched systems is analyzed by means of multiple Lyapunov functions. The stability theorems given in this paper have finite number of conditions to check. It is shown that linear functions can be used as Lyapunov functions. An example of an exponentially asymptotically stable switched system formed by four unstable systems is also given.

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Globally asymptotically stable hybrid observers design on SE (3)

2017 IEEE 56th Annual Conference on Decision and Control (CDC) ◽

10.1109/cdc.2017.8264101 ◽

2017 ◽

Cited By ~ 5

Author(s):

Miaomiao Wang ◽

Abdelhamid Tayebi

Keyword(s):

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Stable Hybrid

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Design of globally asymptotically stable nonlinear observers using Lyapunov functions

Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148) ◽

10.1109/acc.2001.946032 ◽

2001 ◽

Cited By ~ 1

Author(s):

B. Tibken

Keyword(s):

Lyapunov Functions ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Nonlinear Observers

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GLOBAL PROPERTIES OF A MODEL OF IMMUNE EFFECTOR RESPONSES TO VIRAL INFECTIONS

Advances in Complex Systems ◽

10.1142/s0219525907001252 ◽

2007 ◽

Vol 10 (04) ◽

pp. 495-503 ◽

Cited By ~ 5

Author(s):

XIA WANG ◽

XINYU SONG

Keyword(s):

Immune Response ◽

Immune Responses ◽

Lyapunov Functions ◽

Viral Infections ◽

Reproductive Number ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Immune Effector ◽

Global Properties ◽

Ctl Immune Response

This article proposes a mathematical model which has been used to investigate the importance of lytic and non-lytic immune responses for the control of viral infections. By means of Lyapunov functions, the global properties of the model are obtained. The virus is cleared if the basic reproduction number R0 ≤ 1 and the virus persists in the host if R0 > 1. Furthermore, if R0 > 1 and other conditions hold, the immune-free equilibrium E0 is globally asymptotically stable. The equilibrium E1 exists and is globally asymptotically stale if the CTL immune response reproductive number R1 < 1 and the antibody immune response reproductive number R2 > 1. The equilibrium E2 exists and is globally asymptotically stable if R1 > 1 and R2 < 1. Finally, the endemic equilibrium E3 exists and is globally asymptotically stable if R1 > 1 and R2 > 1.

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