scholarly journals Remarks on a variant of Lyapunov-LaSalle theorem

2019 ◽  
Vol 16 (2) ◽  
pp. 1056-1066 ◽  
Author(s):  
Songbai Guo ◽  
◽  
Wanbiao Ma ◽  
Keyword(s):  
Lasalle Theorem

Modeling and Impedance Control of a Two-Manipulator System Handling a Flexible Beam

1997 ◽  
Vol 119 (4) ◽  
pp. 736-742 ◽  
Author(s):  
Dong Sun ◽  
Yunhui Liu
Keyword(s):  
Asymptotic Stability ◽  
Force Feedback ◽  
Impedance Control ◽  
Interaction Force ◽  
Flexible Beam ◽  
New Approach ◽  
Vibration Dynamics ◽  
Lasalle Theorem ◽  
Internal Forces ◽  
Proper Response

This paper presents a new approach of transporting a flexible beam handled by two manipulators to a desired position/orientation while suppressing its vibration, and simultaneously controlling the internal forces between the manipulators and the beam to avoid any damage on the system. The algorithm combines impedance control and an I-type force feedback into one scheme by designing a proper response of the interaction force. No information about the vibration is used in the controller. The asymptotic stability is investigated by using LaSalle theorem, based on the vibration dynamics of the beam approximated by m assumed modes (m → ∞ ). Simulations demonstrate the validity of the proposed method.

On Singular Orbits and Global Exponential Attractive Set of a Lorenz-Type System

2019 ◽  
Vol 29 (06) ◽  
pp. 1950082
Author(s):  
Haijun Wang
Keyword(s):  
Lyapunov Functions ◽  
Three Dimensional ◽  
Type System ◽  
Homoclinic Orbits ◽  
Global Dynamics ◽  
Heteroclinic Cycles ◽  
Mathematical Expressions ◽  
Lasalle Theorem ◽  
Singular Orbits ◽  
Attractive Sets

This paper deals with some unsolved problems of the global dynamics of a three-dimensional (3D) Lorenz-type system: [Formula: see text], [Formula: see text], [Formula: see text] by constructing a series of Lyapunov functions. The main contribution of the present work is that one not only proves the existence of singularly degenerate heteroclinic cycles, existence and nonexistence of homoclinic orbits for a certain range of the parameters according to some known results and LaSalle theorem but also gives a family of mathematical expressions of global exponential attractive sets for that system with respect to its parameters, which is available only in very few papers as far as one knows. In addition, numerical simulations illustrate the consistence with the theoretical conclusions. The results together not only improve and complement the known ones, but also provide support in some future applications.

scholarly journals Adaptive Output Feedback Control for a Class of Stochastic Nonlinear Systems with SiISS Inverse Dynamics

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Na Duan ◽  
Hai-Kuan Liu
Keyword(s):  
Nonlinear Systems ◽  
Output Feedback ◽  
Closed Loop ◽  
Inverse Dynamics ◽  
Output Feedback Control ◽  
Stochastic Nonlinear Systems ◽  
Adaptive Stabilization ◽  
Adaptive Output Feedback Control ◽  
Lasalle Theorem ◽  
Small Gain

The adaptive stabilization scheme based on tuning function for stochastic nonlinear systems with stochastic integral input-to-state stability (SiISS) inverse dynamics is investigated. By combining the stochastic LaSalle theorem and small-gain type conditions on SiISS, an adaptive output feedback controller is constructively designed. It is shown that all the closed-loop signals are bounded almost surely and the stochastic closed-loop system is globally stable in probability.

scholarly journals Traveling waves for a nonlocal dispersal SIR epidemic model with the mass action infection mechanism

2021 ◽  
Author(s):  
Xin Wu ◽  
Zhaohai Ma
Keyword(s):  
Equilibrium State ◽  
Traveling Waves ◽  
Epidemic Model ◽  
Mass Action ◽  
Sir Epidemic Model ◽  
Nonlocal Dispersal ◽  
New Model ◽  
Infection Mechanism ◽  
Sir Epidemic ◽  
Lasalle Theorem

This paper is concerned with a nonlocal dispersal susceptible–infected–recovered (SIR) epidemic model adopted with the mass action infection mechanism. We mainly study the existence and non-existence of traveling waves connecting the infection-free equilibrium state and the endemic equilibrium state. The main difficulties lie in the fact that the semiflow generated here does not admit the order-preserving property. Meanwhile, this new model brings some new challenges due to the unboundedness of the nonlinear term. We overcome these difficulties to obtain the boundedness of traveling waves with the speed $c>c_{\min}$ by some analysis techniques firstly and then prove the existence of traveling waves by employing Lyapunov–LaSalle theorem and Lebesgue dominated convergence theorem. By utilizing a approximating method, we study the existence of traveling waves with the critical wave speed $c_{\min}$. Our results on this new model may provide some implications on disease modelling and controls.

A Michaelis–Menten Predator–Prey Model with Strong Allee Effect and Disease in Prey Incorporating Prey Refuge

2018 ◽  
Vol 28 (06) ◽  
pp. 1850073 ◽  
Author(s):  
Sangeeta Saha ◽  
Alakes Maiti ◽  
G. P. Samanta
Keyword(s):  
Allee Effect ◽  
Equilibrium Points ◽  
Prey Refuge ◽  
Susceptible Population ◽  
Bifurcation Theorem ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Lasalle Theorem ◽  
Transcritical Bifurcations

Here, we have proposed a predator–prey model with Michaelis–Menten functional response and divided the prey population in two subpopulations: susceptible and infected prey. Refuge has been incorporated in infected preys, i.e. not the whole but only a fraction of the infected is available to the predator for consumption. Moreover, multiplicative Allee effect has been introduced only in susceptible population to make our model more realistic to environment. Boundedness and positivity have been checked to ensure that the eco-epidemiological model is well-behaved. Stability has been analyzed for all the equilibrium points. Routh–Hurwitz criterion provides the conditions for local stability while on the other hand, Bendixson–Dulac theorem and Lyapunov LaSalle theorem guarantee the global stability of the equilibrium points. Also, the analytical results have been verified numerically by using MATLAB. We have obtained the conditions for the existence of limit cycle in the system through Hopf Bifurcation theorem making the refuge parameter as the bifurcating parameter. In addition, the existence of transcritical bifurcations and saddle-node bifurcation have also been observed by making different parameters as bifurcating parameters around the critical points.

Stability Theory via Vector Lyapunov Functions

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical Systems ◽  
Stability Theory ◽  
Lyapunov Functions ◽  
Nonlinear Dynamical Systems ◽  
Comparison System ◽  
Nonlinear Dynamical ◽  
Vector Lyapunov Functions ◽  
Lasalle Theorem ◽  
System States ◽  
The Stability

This chapter describes a fundamental stability theory for nonlinear dynamical systems using vector Lyapunov functions. It first introduces the notation and definitions before developing stability theorems via vector Lyapunov functions for continuous-time and discrete-time nonlinear dynamical systems. It then extends the theory of vector Lyapunov functions by constructing a generalized comparison system whose vector field can be a function of the comparison system states as well as the nonlinear dynamical system states. It also presents a generalized convergence result which, in the case of a scalar comparison system, specializes to the classical Krasovskii–LaSalle theorem. In the analysis of large-scale nonlinear interconnected dynamical systems, several Lyapunov functions arise naturally from the stability properties of each individual subsystem.

scholarly journals Stochastic Versions of the LaSalle Theorem

1999 ◽  
Vol 153 (1) ◽  
pp. 175-195 ◽  
Author(s):  
Xuerong Mao
Keyword(s):  
Lasalle Theorem
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