Asymptotic stability and stabilization of a class of nonautonomous fractional order systems

Abstract

Many physical systems from diverse fields of science and engineering are known to give rise to fractional order differential equations. In order to control 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 transform, Mittag–Leffler function and generalized Gronwall inequality to derive the stability conditions. At first, new sufficient conditions for the local and global asymptotic stability of a class of nonautonomous fractional order systems of order $$\alpha $$ where $$1<\alpha <2$$ 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.