Controllability and Observability Results of an Implicit Type Fractional Order Delay Dynamical System

Recently, several research articles have investigated the existence of solutions for dynamical systems with fractional order and their controllability. Nevertheless, very little attention has been given to the observability of such dynamical systems. In the present work, we explore the outcomes of c...

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Bibliographic Details
Published in:Mathematics (Basel) 2022-12, Vol.10 (23), p.4466
Main Authors: Ahmad, Irshad, Ahmad, Saeed, ur Rahman, Ghaus, Ahmad, Shabir, De la Sen, Manuel
Format: Article
Language:English
Subjects:
Analysis
Banach spaces
Control
Control theory
Controllability
Coronaviruses
COVID-19
Delay lines
Disease transmission
Dynamical systems
fixed point theorem
Fixed points (mathematics)
fractional differential equations
Functions, Implicit
grammian matrix
Integral equations
Investigations
observability
Observability (systems)
Sparsity
System theory
Theorems
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Summary:Recently, several research articles have investigated the existence of solutions for dynamical systems with fractional order and their controllability. Nevertheless, very little attention has been given to the observability of such dynamical systems. In the present work, we explore the outcomes of controllability and observability regarding a differential system of fractional order with input delay. Laplace and inverse Laplace transforms, along with the Mittage–Leffler matrix function, are applied to the proposed dynamical system in Caputo’s sense, and a general solution is obtained in the form of an integral equation. Then, we set out conditions for the controllability of the underlying model, regarding the linear case. We then expound controllability conditions for the nonlinear case by utilizing the fixed point result of Schaefer and the Arzola–Ascoli theorem. Using the fixed point concept, we prove the observability of the linear case using the observability Grammian matrix. The necessary and sufficient conditions for the nonlinear case are investigated with the help of the Banach contraction mapping theorem. Finally, we add some examples to elaborate on our work.
ISSN:2227-7390
2227-7390
DOI:10.3390/math10234466