Controllability of pantograph-type nonlinear non-integer order differential system with input delay

One of the most famous classes of differential equations is the pantograph equation, which is a unique kind of functional differential equation with proportional delay. The pantograph equation which can be addressed both numerically and analytically owing to their different practical applications for modeling natural systems and nonlinearity. In linear state equations with time-varying behavior, the relation of the state variables to input signal may vary over time, and delay differential equations are those where a quantity's rate of change is dependent upon its value at a prior time point. Accordingly, a pantograph-type fractional order differential model with input delay is formulated and investigated in this study where the system's solution is needed to examine the intended outcome, which is posed as a fixed-point problem employing the Mittag-Leffler function and the Laplace transform. The existence of solution and other dynamical aspects of implicit differential equations have also been conducted in-depth. The underlying model's controllability is subsequently assessed while taking into account a number of auxiliary conditions on the independent variable and the nonlinear function under consideration. Using various fixed-point theorems, the necessary, as well as sufficient requirements for the newly developed nonlinear fractional-order pantograph-type differential system equipped with input delay, have been investigated. An example is further provided to establish authenticity and validation as an integral feature of different real life related processes where more exact control, precise predictions and robust performance are of pivotal significance in the mathematical modeling of nonlinear problems in natural systems.