ON THE STABILITY OF DYNAMIC SYSTEMS WITH CERTAIN SWITCHINGS, WHICH CONSISTS OF LINEAR SUBSYSTEMS WITHOUT DELAY

2021 ◽  
Vol 3 ◽  
pp. 5-17
Author(s):  
Denis Khusainov ◽  
◽  
Alexey Bychkov ◽  
Andrey Sirenko ◽  
Jamshid Buranov ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Function ◽  
Dynamic Systems ◽  
Lyapunov Functions ◽  
Lyapunov Method ◽  
Positive Definite Function ◽  
Definite Function ◽  
The Stability ◽  
Further Development ◽  
Zero Equilibrium

This work is devoted to the further development of the study of the stability of dynamic systems with switchings. There are many different classes of dynamical systems described by switched equations. The authors of the work divide systems with switches into two classes. Namely, on systems with definite and indefinite switchings. In this paper, the system with certain switching, namely a system composed of differential and difference sub-systems with the condition of decreasing Lyapunov function. One of the most versatile methods of studying the stability of the zero equilibrium state is the second Lyapunov method, or the method of Lyapunov functions. When using it, a positive definite function is selected that satisfies certain properties on the solutions of the system. If a system of differential equations is considered, then the condition of non-positiveness (negative definiteness) of the total derivative due to the system is imposed. If a difference system of equations is considered, then the first difference is considered by virtue of the system. For more general dynamical systems (in particular, for systems with switchings), the condition is imposed that the Lyapunov function does not increase (decrease) along the solutions of the system. Since the paper considers a system consisting of differential and difference subsystems, the condition of non-increase (decrease of the Lyapunov function) is used.For a specific type of subsystems (linear), the conditions for not increasing (decreasing) are specified. The basic idea of using the second Lyapunov method for systems of this type is to construct a sequence of Lyapunov functions, in which the level surfaces of the next Lyapunov function at the switching points are either «stitched» or «contain the level surface of the previous function».

scholarly journals Stability of The Equilibrium of Nonlinear Dynamical Systems

2018 ◽  
Vol 71 (1) ◽  
pp. 71-80
Author(s):  
Irada A. Dzhalladova ◽  
Miroslava Růžičková
Keyword(s):  
Lyapunov Functions ◽  
Lyapunov Method ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Differential Equations ◽  
Fractional Part ◽  
Stability Domain ◽  
Nonlinear Dynamical ◽  
Quadratic Lyapunov Functions ◽  
The Stability ◽  
Zero Equilibrium

Abstract The algorithm for estimating the stability domain of zero equilibrium to the system of nonlinear differential equations with a quadratic part and a fractional part is proposed in the article. The second Lyapunov method with quadratic Lyapunov functions is used as a method for studying such systems.

scholarly journals ATTRACTOR–REPELLER PAIR, MORSE DECOMPOSITION AND LYAPUNOV FUNCTION FOR RANDOM DYNAMICAL SYSTEMS

2008 ◽  
Vol 08 (04) ◽  
pp. 625-641 ◽  
Author(s):  
ZHENXIN LIU ◽  
SHUGUAN JI ◽  
MENGLONG SU
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Function ◽  
Metric Space ◽  
Stability Theory ◽  
Lyapunov Functions ◽  
Random Dynamical Systems ◽  
Morse Decomposition ◽  
Compact Metric Space ◽  
Morse Decompositions ◽  
The Stability

In the stability theory of dynamical systems, Lyapunov functions play a fundamental role. In this paper, we study the attractor–repeller pair decomposition and Morse decomposition for compact metric space in the random setting. In contrast to [7,17], by introducing slightly stronger definitions of random attractor and repeller, we characterize attractor–repeller pair decompositions and Morse decompositions for random dynamical systems through the existence of Lyapunov functions. These characterizations, we think, deserve to be known widely.

scholarly journals Quantitative and Qualitative Methods in the Study of some Dynamic Systems

2015 ◽  
Vol 13 ◽  
pp. 168-171 ◽  
Author(s):  
Dumitru Bălă
Keyword(s):  
Dynamical Systems ◽  
Qualitative Methods ◽  
Dynamic Systems ◽  
Viscous Damping ◽  
Stability Of Dynamical Systems ◽  
The Stability ◽  
Quantitative And Qualitative Methods

In this paper we present several methods for the study of stability of dynamical systems. We analyze the stability of a hammer modeled by the free vibrator that collides with a sprung elastic mass taking into consideration the viscous damping too.

scholarly journals State Observation for Lipschitz Nonlinear Dynamical Systems Based on Lyapunov Functions and Functionals

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1424 ◽  
Author(s):  
Angelo Alessandri ◽  
Patrizia Bagnerini ◽  
Roberto Cianci
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Lyapunov Functions ◽  
Nonlinear Dynamical Systems ◽  
Estimation Error ◽  
Stability Conditions ◽  
Nonlinear Dynamical ◽  
State Observation ◽  
State Observers ◽  
The Stability

State observers for systems having Lipschitz nonlinearities are considered for what concerns the stability of the estimation error by means of a decomposition of the dynamics of the error into the cascade of two systems. First, conditions are established in order to guarantee the asymptotic stability of the estimation error in a noise-free setting. Second, under the effect of system and measurement disturbances regarded as unknown inputs affecting the dynamics of the error, the proposed observers provide an estimation error that is input-to-state stable with respect to these disturbances. Lyapunov functions and functionals are adopted to prove such results. Third, simulations are shown to confirm the theoretical achievements and the effectiveness of the stability conditions we have established.

LYAPUNOV FUNCTION AND GLOBAL PROPERTIES OF VIRUS DYNAMICS WITH CTL IMMUNE RESPONSE

2008 ◽  
Vol 01 (04) ◽  
pp. 443-448 ◽  
Author(s):  
XIA WANG ◽  
YOUDE TAO
Keyword(s):  
Immune Response ◽  
Lyapunov Function ◽  
Lyapunov Functions ◽  
Disease Model ◽  
Virus Dynamics ◽  
Dynamics Model ◽  
Global Properties ◽  
The Stability ◽  
Ctl Immune Response

The stability of infections disease model with CTL immune response in vivo is considered in this paper. Explicit Lyapunov functions for our dynamics model with CTL immune response with nonlinear incidence of the form βVqTpfor the case q ≤ 1 are introduced, and global properties of the model are thereby established.

Control Vector Lyapunov Functions for Large-Scale Impulsive Systems

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lyapunov Function ◽  
Lyapunov Functions ◽  
Large Scale ◽  
Lyapunov Theory ◽  
System Stability ◽  
Dynamical System Theory ◽  
Control Vector ◽  
Vector Lyapunov Functions

This chapter extends the notion of control vector Lyapunov functions to impulsive dynamical systems. Vector Lyapunov theory has been developed to weaken the hypothesis of standard Lyapunov theory to enlarge the class of Lyapunov functions that can be used for analyzing system stability. In particular, the use of vector Lyapunov functions in dynamical system theory offers a very flexible framework since each component of the vector Lyapunov function can satisfy less rigid requirements as compared to a single scalar Lyapunov function. Using control vector Lyapunov functions, the chapter develops a universal hybrid decentralized feedback stabilizer for a decentralized affine in the control nonlinear impulsive dynamical system that possesses guaranteed gain and sector margins in each decentralized input channel. These results are used to develop hybrid decentralized controllers for large-scale impulsive dynamical systems with robustness guarantees against full modeling and input uncertainty.

scholarly journals A Generalized Nonuniform Contraction and Lyapunov Function

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Fang-fang Liao ◽  
Yongxin Jiang ◽  
Zhiting Xie
Keyword(s):  
Lyapunov Function ◽  
Lyapunov Functions ◽  
Linear Equations ◽  
Complete Characterization ◽  
Nonlinear Perturbations ◽  
Linear Perturbations ◽  
The Stability

For nonautonomous linear equationsx′=A(t)x, we give a complete characterization of general nonuniform contractions in terms of Lyapunov functions. We consider the general case of nonuniform contractions, which corresponds to the existence of what we call nonuniform(D,μ)-contractions. As an application, we establish the robustness of the nonuniform contraction under sufficiently small linear perturbations. Moreover, we show that the stability of a nonuniform contraction persists under sufficiently small nonlinear perturbations.

scholarly journals ENERGY METHODS IN THE STABILITY PROBLEM FOR THE 𝔰𝔬(4) FREE RIGID BODY

2013 ◽  
Vol 23 (02) ◽  
pp. 1350032 ◽  
Author(s):  
PETRE BIRTEA ◽  
IOAN CAŞU
Keyword(s):  
Lyapunov Function ◽  
Rigid Body ◽  
Nonlinear Stability ◽  
Lyapunov Functions ◽  
Stability Problem ◽  
Energy Methods ◽  
Linear Combinations ◽  
Free Rigid Body ◽  
Constants Of Motion ◽  
The Stability

For the 𝔰𝔬(4) free rigid body the stability problem for the isolated equilibria has been completely solved using Lie-theoretical and topological arguments. For each case of nonlinear stability previously found, we construct a Lyapunov function. These Lyapunov functions are linear combinations of Mishchenko's constants of motion.

Stability of Dynamical Systems With Multiple Nonlinearities

1987 ◽  
Vol 109 (4) ◽  
pp. 410-413 ◽  
Author(s):  
Norio Miyagi ◽  
Hayao Miyagi
Keyword(s):  
Dynamical System ◽  
Stability Analysis ◽  
Dynamical Systems ◽  
Lyapunov Function ◽  
Stability Region ◽  
Direct Method ◽  
Essential Feature ◽  
Point Of View ◽  
Stability Of Dynamical Systems ◽  
The Stability

This note applies the direct method of Lyapunov to stability analysis of a dynamical system with multiple nonlinearities. The essential feature of the Lyapunov function used in this note is a non-Lure´ type Lyapunov function which surpasses the Lure´-type Lyapunov function from the point of view of the stability region guaranteed. A modified version of the multivariable Popov criterion is used to construct non-Lure´ type Lyapunov function, which allow for the dynamical sytems with multiple nonlinearities.

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