Malkin Lyapunov asymptotic stability of Caputo fractional-order time delay nonlinear system

The idea of Malkin stability provides an important processing technique for the Lyapunov stability of the equilibrium point of the system. That is the first derivative of the Lyapunov function along the system trajectory is no longer required to be negative semi-definite, but the equilibrium point of the system can still be stable. In this paper, inspired by Malkin stability theorem of integer-order system, a Malkin stability theorem for Caputo fractional-order time delay nonlinear systems is given by using Razumikhin technique. Moreover, our result also guarantees the attractiveness of the equilibrium point of fractional-order time delay systems besides stability. Finally, the result of this paper is applied to Caputo fractional-order systems with both discrete and distributed delays, and the asymptotically stable criterion of the system equilibrium point is given by establishing an augmented Lyapunov functional.