Asymptotic stability and stabilization of a class of nonautonomous fractional order systems
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.
KeywordsNonautonomous fractional order systems Asymptotic stability Mittag–Leffler function Generalized Gronwall inequality Stabilization Linear feedback control
The author Bichitra Kumar Lenka sincerely acknowledges the University Grants Commission, India, under Grant No. F. 2-6/2008 (SA-I) for financial support.
- 22.Malti, R., Cois, O., Aoun, M., Levron, F., Oustaloup, A.: Computing impluse response energy of fractional transfer function. In: Proceedings 15th IFAC World Congress, Barcelona, Spain (2002)Google Scholar
- 23.Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: Proceeding IMACS-IEEE CESA, vol. 2, pp. 963–968 (1996)Google Scholar
- 28.Sadati, S.J., Baleanu, D., Ranjbar, A., Ghaderi, R., Abdeljawad, T.: Mittag–Leffler stability theorem for fractional nonlinear systems with delay. Abstr. Appl. Anal. 108651 (2010). doi: 10.1155/2010/108651