Delay-dependent stability switches in fractional differential equations

Abstract

This paper discusses stability properties of a linear fractional delay differential system involving both delayed as well as non-delayed terms. As a main result, the explicit stability dependence on a changing time delay is described, including conditions for the appearance, number and exact calculations of stability switches for this system when its stability property turns into instability and vice versa in view of a monotonically increasing lag. Some supporting asymptotic results are stated as well. The proof technique is based on analysis of the generalized delay exponential function of the Mittag-Leffler type combined with D-decomposition method. The obtained results are illustrated via a fractional Lotka-Volterra population model and applied to a stabilization problem of the control theory.