Controllability and Observability Analysis of a Fractional-Order Neutral Pantograph System

In the recent past, a number of research articles have explored the stability, existence, and uniqueness of the solutions and controllability of dynamical systems with a fractional order (FO). Nevertheless, aside from the controllability and other dynamical aspects, very little attention has been given to the observability of FO dynamical systems. This paper formulates a novel type of FO delay system of the Pantograph type in the Caputo sense and explores its controllability and observability results. This research endeavor begins with the conversion of the proposed dynamical system into a fixed-point problem by utilizing Laplace transforms, the convolution of Laplace functions, and the Mittag-Leffler function (MLF). We then set out Gramian matrices for both the controllability and observability of the linear parts of our proposed dynamical system and prove that both the Gramian matrices are invertible, thus confirming the controllability and observability in a given domain. Considering the controllability and observability results of the linear part along with some other assumptions, we investigate the controllability and observability results related to the nonlinear system. The Banach contraction result, the fixed-point result of Schaefer, the MLF, and the Caputo FO derivative are used as the main tools for establishing these results. To establish the authenticity of the established results, we add two examples at the end of the manuscript.