Stability Properties of Neutral Delay Fractional Systems with Caputo Derivatives Due to the Perron Condition

In this article, we consider a class of nonhomogeneous neutral linear systems with Caputo-type fractional derivatives, having incommensurate orders of differentiation and distributed delays. The main goal is to investigate the influence of the Perron condition on the stability properties of the corresponding homogeneous neutral linear system, when the nonhomogeneous system satisfies this condition. We first prove that, for any partially absolutely continuous initial functions, the investigated nonhomogeneous system has a unique global absolutely continuous solution. Furthermore, if the nonhomogeneous system satisfies the Perron condition, we establish that the fundamental and the extended fundamental matrices of the corresponding homogeneous system are uniformly bounded under certain boundedness conditions, which are also used in the classical case for systems with first-order derivatives. This uniform boundedness implies that the zero solution of the homogeneous system is uniformly stable. Finally, it is proved that the extended fundamental matrix Q(t,s) tends to zero as t → ∞, thereby demonstrating that the zero solution of the investigated homogeneous system is uniformly asymptotically stable.