Pantographic formulation of a nonlinear system of fractional order with delays and examination of its controllability

Controllability is a significant qualitative aspect of dynamical systems where control theory refers to steering from a random initial state to any desirable state with the application of an admissible control in a finite time interval. Controllability is also important in finite and infinite dimensional spaces like systems described by ordinary and partial differential equations respectively since many phenomena in applied sciences can be modeled successfully through the related equations. From this point of view, this paper explores the existence and controllability of linear as well as nonlinear pantograph type systems with state delay. The proposed system is initially transformed into a fixed-point problem, and the solution to the system is expressed in terms of the Mittag–Leffler function with delay. In that way, the controllability of the linear system with state delay has been shown, besides establishing the sufficient conditions required for the controllability of the nonlinear fractional Pantograph type systems with state delay with the help of the Arzelà–Ascoli theorem and Schauder’s fixed-point techniques. A numerical example has also been provided for the clarification of the derived results, demonstrating the controllability of the linear system. The fractional order delay systems with solutions of linear and nonlinear pantograph types, as proposed in this study, have verified the applicability, adaptation and optimal harmonization in order that the relevant mathematical models can be fit to the systems for the sake of coming up with solution-oriented schemes concerning the real-world problems.