Investigation of invariant sets with random perturbations with the use of the Lyapunov function

1998 ◽  
Vol 50 (2) ◽  
pp. 355-359 ◽  
Author(s):  
O. M. Stanzhitskii
Keyword(s):  
Lyapunov Function ◽  
Invariant Sets ◽  
Random Perturbations

scholarly journals Invariant Sets of Hybrid Autonomous Systems with Disturbance

2010 ◽  
Vol 2010 ◽  
pp. 1-12
Author(s):  
Li Jian-Qiang ◽  
Zhu Zexuan ◽  
Ji Zhen ◽  
Pei Hai-Long
Keyword(s):  
Lyapunov Function ◽  
Hybrid Systems ◽  
Autonomous Systems ◽  
Invariant Set ◽  
Invariant Sets ◽  
Common Lyapunov Function ◽  
Transition Mode ◽  
Convex Problem

The concept and model of hybrid systems are introduced. Invariant sets introduced by LaSalle are proposed, and the concept is extended to invariant sets in hybrid systems which include disturbance. It is shown that the existence of invariant sets by arbitrary transition in hybrid systems is determined by the existence of common Lyapunov function in the systems. Based on the Lyapunov function, an efficient transition method is proposed to ensure the existence of invariant sets. An algorithm is concluded to compute the transition mode, and the invariant set can also be computed as a convex problem. The efficiency and correctness of the transition algorithm are demonstrated by an example of hybrid systems.

Random perturbations of recursive sequences with an application to an epidemic model

1995 ◽  
Vol 32 (3) ◽  
pp. 559-578 ◽  
Author(s):  
Daniel Pierre Loti Viaud
Keyword(s):  
Dynamical System ◽  
Lyapunov Function ◽  
Finite Number ◽  
Epidemic Model ◽  
Sample Path ◽  
Random Perturbations ◽  
Main Condition ◽  
Limit Sets ◽  
Discrete Time Dynamical System ◽  
Recursive Sequences

We investigate the asymptotic sample path behaviour of a randomly perturbed discrete-time dynamical system. We consider the case where the trajectories of the non-perturbed dynamical system are attracted by a finite number of limit sets and characterize a case where this property remains valid for the perturbed dynamical system when the perturbation converges to zero. For this purpose, no further assumptions on the perturbation are needed and our main condition applies to the limit sets of the non-perturbed dynamical system. When the limit sets reduce to limit points we show that this main condition is more general than the usual assumption of the existence of a Lyapunov function for the non-perturbed dynamical system. An application to an epidemic model is given to illustrate our results.

Random perturbations of recursive sequences with an application to an epidemic model

1995 ◽  
Vol 32 (03) ◽  
pp. 559-578 ◽  
Author(s):  
Daniel Pierre Loti Viaud
Keyword(s):  
Dynamical System ◽  
Lyapunov Function ◽  
Finite Number ◽  
Epidemic Model ◽  
Sample Path ◽  
Random Perturbations ◽  
Main Condition ◽  
Limit Sets ◽  
Discrete Time Dynamical System ◽  
Recursive Sequences

We investigate the asymptotic sample path behaviour of a randomly perturbed discrete-time dynamical system. We consider the case where the trajectories of the non-perturbed dynamical system are attracted by a finite number of limit sets and characterize a case where this property remains valid for the perturbed dynamical system when the perturbation converges to zero. For this purpose, no further assumptions on the perturbation are needed and our main condition applies to the limit sets of the non-perturbed dynamical system. When the limit sets reduce to limit points we show that this main condition is more general than the usual assumption of the existence of a Lyapunov function for the non-perturbed dynamical system. An application to an epidemic model is given to illustrate our results.

scholarly journals Stabilization of fuzzy systems with constrained controls by using positively invariant sets

2006 ◽  
Vol 2006 ◽  
pp. 1-17 ◽  
Author(s):  
A. El Hajjaji ◽  
A. Benzaouia ◽  
M. Naib
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Fuzzy Systems ◽  
Sufficient Conditions ◽  
Invariant Sets ◽  
Design Phase ◽  
Common Lyapunov Function ◽  
Polyhedral Set ◽  
Takagi Sugeno ◽  
Positively Invariant Sets

We deal with the extension of the positive invariance approach to nonlinear systems modeled by Takagi-Sugeno fuzzy systems. The saturations on the control are taken into account during the design phase. Sufficient conditions of asymptotic stability are given ensuring at the same time that the control is always admissible inside the corresponding polyhedral set. Both a common Lyapunov function and piecewise Lyapunov function are used.

scholarly journals STATIONARY DISTRIBUTION AND EXTINCTION OF STOCHASTIC CORONAVIRUS (COVID-19) EPIDEMIC MODEL

Fractals ◽  
2021 ◽  
Author(s):  
AMIR KHAN ◽  
HEDAYAT ULLAH ◽  
MOSTAFA ZAHRI ◽  
USA WANNASINGHA HUMPHRIES ◽  
TOURIA KARITE ◽  
...  
Keyword(s):  
Brownian Motion ◽  
Lyapunov Function ◽  
Unique Solution ◽  
Epidemic Model ◽  
Feasible Region ◽  
Random Perturbations ◽  
Suggested Model ◽  
Susceptible Population ◽  
Ergodic Distribution ◽  
Very High

The aim of this paper is to model corona-virus (COVID-19) taking into account random perturbations. The suggested model is composed of four different classes i.e. the susceptible population, the smart lockdown class, the infectious population, and the recovered population. We investigate the proposed problem for the derivation of at least one unique solution in the positive feasible region of nonlocal solution. For one stationary ergodic distribution, the necessary result of existence is developed by applying the Lyapunov function and the condition for the extinction of the disease is also established. The obtained results show that the effect of Brownian motion and noise terms on the transmission of the epidemic is very high. If the noise is large the infection may decrease or vanish. For validation of our obtained scheme, the results for all the classes of the problem have been simulated numerically.

Action-Minimizing Curves for Tonelli Lagrangians

2017 ◽  
Author(s):  
Alfonso Sorrentino
Keyword(s):  
Critical Value ◽  
The Other ◽  
Invariant Sets ◽  
Simple Pendulum ◽  
New Concepts ◽  
Peierls Barrier ◽  
Average Action

This chapter discusses the notion of action-minimizing orbits. In particular, it defines the other two families of invariant sets, the so-called Aubry and Mañé sets. It explains their main dynamical and symplectic properties, comparing them with the results obtained in the preceding chapter for the Mather sets. The relation between these new invariant sets and the Mather sets is described. As a by-product, the chapter introduces the Mañé's potential, Peierls' barrier, and Mañé's critical value. It discusses their properties thoroughly. In particular, it highlights how this critical value is related to the minimal average action and describes these new concepts in the case of the simple pendulum.

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