Non-existence of finite-time stable equilibria in fractional-order nonlinear systems

• Published in: Automatica (Oxford) Vol. 50; no. 2; pp. 547 - 551
• Main Authors: Shen, Jun, Lam, James
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01.02.2014; Elsevier
• Subjects: Applied sciences; Asymptotic properties; Computer science; control theory; systems; Control system analysis; Control theory. Systems; Delay; Dynamical systems; Equilibria; Exact sciences and technology; Finite-time stability; Fractional-order nonlinear system; Initial value problems; Linear Lyapunov function; Lyapunov functions; Nonlinear dynamics; Stability; System theory; Trajectories; Finite-time stability; Fractional-order nonlinear system; Equilibria; Linear Lyapunov function; Initial value problem; Non linear control; Delay system; Non linear system; Positive system; Modeling; Fractional derivative; Asymptotic stability; Time interval; Lyapunov function; Non existence
• ISSN: 0005-1098; 1873-2836
• DOI: 10.1016/j.automatica.2013.11.018

We note that in the literature it is often taken for granted that for fractional-order system without delays, whenever the system trajectory reaches the equilibrium, it will stay there. In fact, this is the well-known phenomenon of finite-time stability. However, in this paper, we will prove that for fractional-order nonlinear system described by Caputo’s or Riemann–Liouville’s definition, any equilibrium cannot be finite-time stable as long as the continuous solution corresponding to the initial value problem globally exists. In addition, some examples of stability analysis are revisited and linear Lyapunov function is used to prove the asymptotic stability of positive fractional-order nonlinear systems.