Analysis of Caputo linear fractional dynamic systems with time delays through fixed point theory

TL;DR

This paper analyzes the stability of linear time-varying Caputo fractional systems with delays using fixed point theory, establishing conditions for global stability and asymptotic behavior.

Contribution

It introduces a fixed point approach to assess stability of fractional systems with delays, including controlled and uncontrolled cases, under new sufficiency conditions.

Findings

Established conditions for global stability independent of delay sizes

Proved existence of unique fixed points leading to asymptotic stability

Extended analysis to systems with state feedback control

Abstract

This paper investigates the global stability and the global asymptotic stability independent of the sizes of the delays of linear time-varying Caputo fractional dynamic systems of real fractional order possessing internal point delays. The investigation is performed via fixed point theory in a complete metric space by defining appropriate non-expansive or contractive self- mappings from initial conditions to points of the state- trajectory solution. The existence of a unique fixed point leading to a globally asymptotically stable equilibrium point is investigated in particular under easily testable sufficiency-type stability conditions. The study is performed for both the uncontrolled case and the controlled case under a wide class of state feedback laws.