A Qualitative Theory of Time Delay Nonlinear Fractional-Order Systems

This paper analyses fractional-order systems with time-varying delays. First, we present some results on the existence, uniqueness, exponential boundedness, and convergence rate of solutions to an equilibrium point of mixed fractional-order systems with time-varying delays. In particular, we show that in general the convergence rate of solutions to an equilibrium point is subpolynomial. This is a significant difference from ordinary differential equations. Then, we investigate the Mittag--Leffler stability of time delay fractional-order systems. To do this, we use the linearization method combined with a new weighted type norm which is compatible with the dependence on history and the hereditary property of these models. Based on an integral presentation of solutions and some special properties of Mittag--Leffler functions, we also obtain a criterion on the asymptotic stability of fractional-order systems with unbounded time-varying delays. Finally, some examples with simulations are given to illustrate the effectiveness of the theoretical results.