Investigation of controllability criteria for Caputo fractional dynamical systems with delays in both state and control

Abstract

This study examines the controllability criteria for linear and semilinear fractional dynamical systems with delays in both state and control variables in the framework of the Caputo fractional derivative. To establish the controllability criteria for linear fractional dynamical systems, the study derives necessary and sufficient conditions by employing the positive definiteness of the Grammian matrix. Extending this analysis to semilinear fractional dynamical systems, Krasnoselskii’s fixed point theorem is employed to derive sufficient conditions for the existence of a solution. Furthermore, in addressing semilinear fractional dynamical systems with delays in both state and control, Banach’s fixed point theorem is employed to derive sufficient conditions for the existence of a solution. In order to enhance the comprehension of the theoretical results, the study presents three specific examples along with appropriate graphical representations.