Advances in Lyapunov theory of Caputo fractional-order systems

Abstract

A lemma widely used for Lyapunov stability analysis of Caputo fractional-order systems (CFOSs): let $x(t)\in {\mathbb {R}}^n$ be a vector of differentiable functions, then for any time instant $t\ge t_0$, $1/2\,_{t_0}^CD_t^\alpha [x^T(t)Px(t)]\le x^T(t)P\,{_{t_0}^CD}_t^\alpha x(t)$, for any $\alpha \in (0,1]$, where $P\in {\mathbb {R}}^{n\times n}$ 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: $_{0}^CD_t^\alpha [x^{T}(t)Px(t)]\le x^T(t)P{_{0}^CD}_t^\alpha x(t)+ [{_{0}^CD}_t^\alpha x^T(t)]Px(t)$, $\alpha \in (0,1)$, where x(t) is a real solution of the CFOS ${_{0}^C}D_{t}^{\alpha }x=f(t,x)$, $x(0)=x_0$, 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.