Stability of a Nonlinear Langevin System of ML-Type Fractional Derivative Affected by Time-Varying Delays and Differential Feedback Control

The Langevin system is an important mathematical model to describe Brownian motion. The research shows that fractional differential equations have more advantages in viscoelasticity. The exploration of fractional Langevin system dynamics is novel and valuable. Compared with the fractional system of...

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Bibliographic Details
Published in:Fractal and fractional 2022-12, Vol.6 (12), p.725
Main Author: Zhao, Kaihong
Format: Article
Language:English
Subjects:
Banach spaces
Brownian motion
Comparative analysis
Control systems
Differential equations
differential feedback control
existence and stability
Existence theorems
Feedback control
Fixed points (mathematics)
Fractional calculus
Langevin system
Mathematical models
ML-fractional derivative
Nonlinear analysis
Numerical analysis
Simulation methods
Stability analysis
System dynamics
Systems stability
Time varying control
time-varying delay
Viscoelasticity
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Summary:The Langevin system is an important mathematical model to describe Brownian motion. The research shows that fractional differential equations have more advantages in viscoelasticity. The exploration of fractional Langevin system dynamics is novel and valuable. Compared with the fractional system of Caputo or Riemann–Liouville (RL) derivatives, the system with Mittag–Leffler (ML)-type fractional derivatives can eliminate singularity such that the solution of the system has better analytical properties. Therefore, we concentrate on a nonlinear Langevin system of ML-type fractional derivatives affected by time-varying delays and differential feedback control in the manuscript. We first utilize two fixed-point theorems proposed by Krasnoselskii and Schauder to investigate the existence of a solution. Next, we employ the contraction mapping principle and nonlinear analysis to establish the stability of types such as Ulam–Hyers (UH) and Ulam–Hyers–Rassias (UHR) as well as generalized UH and UHR. Lastly, the theoretical analysis and numerical simulation of some interesting examples are carried out by using our main results and the DDESD toolbox of MATLAB.
ISSN:2504-3110
2504-3110
DOI:10.3390/fractalfract6120725