Abstract
The main purpose of this paper is to investigate the finite-time stability of Hadamard fractional differential equations (HFDEs). Firstly, the standard definitions of finite-time stability of HFDEs in compatible Banach spaces are proposed. In light of the method of successive approximation and Beesack inequality with weakly singular kernel, the criteria of finite-time stability for linear and nonlinear HFDEs are established, respectively. Then with regard to linear HFDEs with pure delay, a novel fractional delayed matrix function (also called delayed Mittag-Leffler matrix function) is given. Specific to nonlinear HFDEs with constant time delay, both Beesack inequality and Hölder inequality are utilized in the framework of the generalized Lipschitz condition. Finally, several indispensable simulations are implemented to verify the effectiveness and practicability of the main results.
Highlights
Fractional calculus, an important branch of mathematics, was born in 1695 and emerged almost simultaneously with classical calculus
The main purpose of this paper is to investigate the finite-time stability of Hadamard fractional differential equations (HFDEs)
To cope with the linear and nonlinear HFDEs with and without delay, the method of successive approximation and a new delayed Mittag-Leffler matrix function are employed in the presence of generalized Lipschitz conditions
Summary
Fractional calculus, an important branch of mathematics, was born in 1695 and emerged almost simultaneously with classical calculus. In [13], the authors apply the Gronwall’s inequality approach to derive the sufficient conditions on finite-time stability for linear nonhomogeneous FDEs. In addition, the finite-time stability of nonlinear FDEs with time-varying delay is investigated based on the Laplace transform and “inf-sup” method [26]. On the foundation of a new fractional Gronwall’s inequality with time delay, the authors in [7] provide a sufficient condition for finite-time stability of Caputo FDEs. Delayed Mittag-Leffler matrix method could be deemed as a novel approach to explore finite-time stability, more details refer to [17].
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