Advances in Lyapunov theory of Caputo fractional-order systems

摘要

A lemma widely used for Lyapunov stability analysis of Caputo fractional-order systems (CFOSs): let x(t)is an element of R-n be a vector of differentiable functions, then for any time instant t >= t(0), 1/2(t0)(C)D(t)(alpha)[x(T)(t)Px(t)] <= x(T) (t) (Pt0Dt alpha)-D-C x(t), for any alpha is an element of(0,1], where P is an element of R-nxn is a positive definite matrix, is pointed out not applicable, due to the fact that the solution of a CFOS may be not differentiable, even if the vector field function is analytic. To make up for this blank, we apply the most recent results on the continuation and smoothness of solutions to prove the following estimation for the Caputo fractional derivative of any quadratic Lyapunov function: D-C(0)t(alpha)[x(T)(t)Px(t)] <= x(T)(t)P(0)(C)D(t)(alpha)x(t)+[(C)(0)D(t)(alpha)x(T)(t)]Px(t), alpha is an element of(0,1), where x(t) is a real solution of the CFOS (C)(0)D(t)(alpha)x = f(t,x), x(0) = x(0), with some certain hypotheses. Moreover, a few other unclear concerns about existing results on the Lyapunov theory of CFOSs are eliminated. Finally, numerical examples are provided to illustrate these results.