The behavior of the analytical solutions of the fractional differential equation described\nby the fractional order derivative operators is the main subject in many stability problems. In this\npaper, we present a new stability notion of the fractional differential equations with exogenous\ninput. Motivated by the success of the applications of the Mittag-Leffler functions in many areas\nof science and engineering, we present our work here. Applications of Mittag-Leffler functions in\ncertain areas of physical and applied sciences are also very common. During the last two decades, this\nclass of functions has come into prominence after about nine decades of its discovery by a Swedish\nMathematician Mittag-Leffler, due to the vast potential of its applications in solving the problems\nof physical, biological, engineering, and earth sciences, to name just a few. Moreover, we propose\nthe generalized Mittag-Leffler input stability conditions. The left generalized fractional differential\nequation has been used to help create this new notion. We investigate in depth here the Lyapunov\ncharacterizations of the generalized Mittag-Leffler input stability of the fractional differential equation\nwith input.
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