Investigating the controllability of differential systems with nonlinear fractional delays, characterized by the order 0 &lt; <i>η</i> ≤ 1 &lt; <i>ζ</i> ≤ 2

Abstract In this study, we investigate systems known as nonlinear fractional delay differential (nLFDD) systems, characterized by finite state delays and fractional orders within the range of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>0</m:mn> <m:mo>&lt;</m:mo> <m:mi>η</m:mi> <m:mo>≤</m:mo> <m:mn>1</m:mn> <m:mo>&lt;</m:mo> <m:mi>ζ</m:mi> <m:mo>≤</m:mo> <m:mn>2</m:mn> </m:math> 0\lt \eta \le 1\lt \zeta \le 2 , situated infinite-dimensional settings. We utilize the controllability Gramian matrix to establish both necessary and sufficient conditions for the controllability of linear fractional delay differential systems that fall within the order range of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>0</m:mn> <m:mo>&lt;</m:mo> <m:mi>η</m:mi> <m:mo>≤</m:mo> <m:mn>1</m:mn> <m:mo>&lt;</m:mo> <m:mi>ζ</m:mi> <m:mo>≤</m:mo> <m:mn>2</m:mn> </m:math> 0\lt \eta \le 1\lt \zeta \le 2 . Moreover, the Schauder fixed point theorem is employed to delineate the sufficient conditions required for the controllability of nLFDD systems, which are defined by finite state delays and fractional orders in the specified range. To substantiate the theoretical constructs put forth, we provide two illustrative examples.