Nonlinear Dyn DOI 10.1007/s11071-016-2676-6 ORIGINAL PAPER Asymptotic stability and stabilization of a class of nonautonomous fractional order systems Bichitra Kumar Lenka · Soumitro Banerjee Received: 20 August 2015 / Accepted: 6 February 2016 © Springer Science+Business Media Dordrecht 2016 Abstract Many physical systems from diverse fields of science and engineering are known to give rise to fractional order differential equations. In order to con- trol such systems at an equilibrium point, one needs to know the conditions for stability. In this paper, the conditions for asymptotic stability of a class of nonautonomous fractional order systems with Caputo derivative are discussed. We use the Laplace trans- form, Mittag–Leffler function and generalized Gron- wall inequality to derive the stability conditions. At first, new sufficient conditions for the local and global asymptotic stability of a class of nonautonomous frac- tional order systems of order α where 1 <α< 2 are derived. Then, sufficient conditions for the local and global stabilization of such systems are proposed. Using the results of these theorems, we demonstrate the stabilization of some fractional order nonautonomous systems which illustrate the validity and effectiveness of the proposed method. B. K. Lenka (B) Department of Mathematics and Statistics, Indian Institute of Science Education and Research Kolkata, Mohanpur Campus 741246, West Bengal, India e-mail: bkl12rs001@iiserkol.ac.in S. Banerjee Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur Campus 741246, West Bengal, India e-mail: soumitro@iiserkol.ac.in Keywords Nonautonomous fractional order systems · Asymptotic stability · Mittag–Leffler function · Generalized Gronwall inequality · Stabilization · Linear feedback control 1 Introduction Fractional calculus is the generalization of ordinary derivative and/or integral to noninteger-order deriva- tive and/or integral. Due to the presence of memory and its nonlocal nature, fractional derivatives can be fruitfully used to model many real problems arising in various fields of science and engineering [6, 12, 13, 26]. For example, viscoelastic systems, dielectric polariza- tion, electrode–electrolyte polarization, and electro- magnetic waves were well described by fractional dif- ferential equations [2, 11, 29, 30]. Investigating the stability of fractional order sys- tems is one of the important problems in the theory of fractional calculus and its application in fractional control theory [21, 23, 32]. Many criteria for the sta- bility of fractional order systems have been proposed by researchers and can be found in the survey [15]. For example, the stability results for linear fractional order systems can be found in [1, 9, 15, 19, 20, 22– 25, 27, 31, 36]. For nonlinear fractional order systems, Li et al. [16, 17] proposed the Mittag–Leffler stability and Lyapunov direct method to investigate stability for fractional order nonlinear systems. Sadati et al. [28] studied the Mittag–Leffler stability for fractional non- 123