Uncertain Asymptotic Stability Analysis of a Fractional-Order System with Numerical Aspects

We apply known special functions from the literature (and these include the Fox (Formula presented.) function, the exponential function, the Mittag-Leffler function, the Gauss Hypergeometric function, the Wright function, the (Formula presented.) function, the Fox–Wright function and the Meijer (Formula presented.) function) and fuzzy sets and distributions to construct a new class of control functions to consider a novel notion of stability to a fractional-order system and the qualified approximation of its solution. This new concept of stability facilitates the obtention of diverse approximations based on the various special functions that are initially chosen and also allows us to investigate maximal stability, so, as a result, enables us to obtain an optimal solution. In particular, in this paper, we use different tools and methods like the Gronwall inequality, the Laplace transform, the approximations of the Mittag-Leffler functions, delayed trigonometric matrices, the alternative fixed point method, and the variation of constants method to establish our results and theory.