Using an analog of the Lyapunov function with sign-alternating derivative in the study of global asymptotic stability of equilibria

2014 ◽  
Vol 96 (1-2) ◽  
pp. 204-207
Author(s):  
A. O. Ignat’ev
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Global Asymptotic Stability ◽  
Stability Of Equilibria

On Stability of Solutions of Certain Differential Equations of the Third Order

1967 ◽  
Vol 10 (5) ◽  
pp. 681-688 ◽  
Author(s):  
B.S. Lalli
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Stability Of Solutions ◽  
Third Order ◽  
The Third ◽  
Image Position

The purpose of this paper is to obtain a set of sufficient conditions for “global asymptotic stability” of the trivial solution x = 0 of the differential equation1.1using a Lyapunov function which is substantially different from similar functions used in [2], [3] and [4], for similar differential equations. The functions f1, f2 and f3 are real - valued and are smooth enough to ensure the existence of the solutions of (1.1) on [0, ∞). The dot indicates differentiation with respect to t. We are taking a and b to be some positive parameters.

STABILITY OF EQUILIBRIA OF NEURAL NETWORKS

1999 ◽  
Vol 09 (10) ◽  
pp. 1941-1955
Author(s):  
P. F. CURRAN ◽  
L. O. CHUA
Keyword(s):  
Neural Networks ◽  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Special Class ◽  
Sufficient Conditions ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Stability Of Equilibria

Sufficient conditions for local and global asymptotic stability of equilibria of some general classes of neural networks are presented. In the event that the interconnection matrix is block diagonally stable it is shown that the equilibrium is globally asymptotically stable if the cells are dissipative at the equilibrium. For a special class of networks the conditions of dissipativity are reduced to more readily-tested conditions of passivity. Equilibria are shown to be asymptotically stable essentially if the cells are locally passive.

Replacement of Lyapunov Function by Locally Convergent Robust Fixed Point Transformations in Model-Based Control a Brief Summary

2010 ◽  
Vol 14 (2) ◽  
pp. 224-236 ◽  
Author(s):  
József K. Tar ◽  
Keyword(s):  
Fixed Point ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Global Asymptotic Stability ◽  
External Disturbances ◽  
Model Based Control ◽  
Model Based ◽  
Point Transformations ◽  
Tuning Process ◽  
The Cost

Lyapunov’s 2ndMethod is a popular approach in the model-based control of nonlinear systems since normally it can guaranteeglobal asymptotic stability. However, the cost of the application of Lyapunov function frequently can be inefficient and complicated parameter tuning process containing unnecessary number of almost arbitrary control parameters, and vulnerability of the tuning process against not modeled, unknown external disturbances. Improved versions of the original, model-based tuning are especially sensitive to the external perturbations. All these disadvantages can simply be avoided by the application of robust fixed point transformations at the cost of giving up the guarantee of global asymptotic stability. Instead of that simple,stable iterative controlof local basin of attraction can be constructed on the basis of an approximate system model that can well compensate the effects of unknown external disturbances, too. Since the basin of convergence of the method in principle can be left, setting the three adaptive parameters of this controller needs preliminary simulation investigations. These statements are illustrated and substantiated via simulation results obtained for the adaptive control of various physical systems.

GLOBAL STABILITY FOR A PHYTOPLANKTON-NUTRIENT SYSTEM

2000 ◽  
Vol 08 (02) ◽  
pp. 195-209 ◽  
Author(s):  
OLIVIER PARDO
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Global Stability ◽  
Invariance Principle ◽  
Global Asymptotic Stability ◽  
Lasalle’S Invariance Principle ◽  
Bacterial Decomposition ◽  
Autonomous Differential Equations ◽  
Lasalle's Invariance Principle

The model proposed by A. H. Taylor et al. [18] is discussed, with a view to determining the global asymptotic stability of the equilibria. The system consists of two autonomous differential equations, modeling the couple Phytoplankton-Nutrient with no delay on the recycling efficiency of nutrient by bacterial decomposition. Two distinct cases, persistence and extinction of phytoplankton are considered. In each case we will state the local and then the global stability of the equilibria by constructing an appropriate Lyapunov function and using the LaSalle's invariance principle. Also, in the case of extinction of phytoplankton we have introduced a supply of nutrient in the system and we have revealed the bloom of phytoplankton, which appears biologically in upwelling conditions.

Conditions for the local and global asymptotic stability of the time–fractional Degn–Harrison system

2020 ◽  
Vol 21 (7-8) ◽  
pp. 749-759
Author(s):  
Rachida Mezhoud ◽  
Khaled Saoudi ◽  
Abderrahmane Zaraï ◽  
Salem Abdelmalek
Keyword(s):  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Lyapunov Method ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Linearization Method ◽  
Direct Lyapunov Method ◽  
Fractional Systems ◽  
Reaction Diffusion Model ◽  
Theoretical Results

AbstractFractional calculus has been shown to improve the dynamics of differential system models and provide a better understanding of their dynamics. This paper considers the time–fractional version of the Degn–Harrison reaction–diffusion model. Sufficient conditions are established for the local and global asymptotic stability of the model by means of invariant rectangles, the fundamental stability theory of fractional systems, the linearization method, and the direct Lyapunov method. Numerical simulation results are used to illustrate the theoretical results.

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