In this work, we analyze various stability (from stability to global uniform asymptotic stability) of Caputo fractional-order nonautonomous systems (CFONSs). With f(t,x) being only continuous, it is proven that the Caputo fractional derivative (CFD) of a general Lyapunov function V (t,x(t)) along solutions is continuous and moreover, t0CD tαV (t,x(t))≤ ∂V (t,x) ∂t(t,x(t))t0CD tαt + ∂V (t,x) ∂x ⋅ f(t,x)(t,x(t)), on the maximal interval of existence (MIE) of x(t). This together with the continuation of solutions suffices to prove the various stability theorems for CFONSs that are as general as those for integer-order systems, and make them practically applicable. The work reduces the assumption on vector field functions f for stability analysis from continuously differentiable (CD) to only continuous, which advances existing results to a large extent. Finally, some derived results are applied to real examples with numerical simulations.

中文翻译：

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Caputo分数阶非自治系统分析的进展：从稳定性到全局一致渐近稳定性

在这项工作中，我们分析了 Caputo 分数阶非自治系统 (CFONS) 的各种稳定性（从稳定性到全局一致渐近稳定性）。和F(吨,X)仅是连续的，证明了一般 Lyapunov 函数的 Caputo 分数导数 (CFD)五 (吨,X(吨))沿着解决方案是连续的，而且， 吨0CD 吨α五 (吨,X(吨))≤ ∂五 (吨,X) ∂吨(吨,X(吨))吨0CD 吨α吨 + ∂五 (吨,X) ∂X ⋅ F(吨,X)(吨,X(吨)), 在最大存在间隔 (MIE) 上X(吨). 这与解的连续性足以证明 CFONS 的各种稳定性定理与整数阶系统的稳定性定理一样普遍，并使它们实际适用。这项工作减少了对向量场函数的假设F用于从连续可微（CD）到仅连续的稳定性分析，这在很大程度上推进了现有结果。最后，将一些推导出的结果应用于具有数值模拟的实际示例。