DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2020049 DYNAMICAL SYSTEMS SERIES S Volume 13, Number 3, March 2020 pp. 853–865 FRACTIONAL INPUT STABILITY AND ITS APPLICATION TO NEURAL NETWORK Ndolane Sene ∗ D´ epartement de Math´ ematiques de la D´ ecision Universit´ e Cheikh Anta Diop de Dakar Laboratoire Lmdan BP 5683 Dakar Fann, S´ en´ egal Abstract. This paper deals with fractional input stability, and contributes to introducing a new stability notion in the stability analysis of fractional dif- ferential equations (FDEs) with exogenous inputs using the Caputo fractional derivative. In particular, we study the fractional input stability of FDEs with exogenous inputs. A Lyapunov characterization of this notion is proposed and several examples are provided to explain the fractional input stability of FDEs with exogenous inputs. The applicability and simulation of this method are illustrated by studying the particular class of fractional neutral networks. 1. Introduction. Numerous works on stability notions exist in the literature, and deal with the stability of fractional differential equations (FDEs) without inputs. The stability notions in fractional calculus are: Mittag-Leffler [20], asymptotic [13, 20], fractional exponential [30], practical, and conditional (with respect to small input) [29] stability, among others. Numerous works [4, 12, 14, 33, 34, 35] exist on the stability notions of differential equations with exogenous inputs, and various results have been provided, see also [31]. All of these results were obtained with ordinary derivative. In the context of FDEs with exogenous inputs, many works have investigated the problem of stabilization or synchronization. The conditional asymptotic stability of FDEs with exogenous inputs has recently been introduced in the literature (see[29]). Conditional stability simply refers to the stability with respect to small input. In this paper, we focus on studying the input-to-state sta- bility (ISS) in the context of the non-integer order. Traditionally, the ISS property of nonlinear systems requires that the norm of solutions be upper-bounded by a vanishing transient term, depending on the initial state, plus a term that is some- what proportional to the magnitude of the input signal applied to the given system [8]. The ISS property was introduced by Sontag in 1989 [33]. In our opinion, this property can be studied in the context of FDEs with exogenous inputs, which can be demonstrated by obtaining the solution of the FDE defined by D c α x = Ax + Bu. 2010 Mathematics Subject Classification. Primary: 26A33, 93D05; Secondary: 93D25. Key words and phrases. Caputo fractional derivative, fractional differential equations, frac- tional input stability. * Corresponding author: Ndolane Sene. 853