Converse Lyapunov Theorem for Nabla Asymptotic Stability Without Conservativeness
针对线性时不变Nabla分数阶系统的Lyapunov方法保守性问题，提出逆Lyapunov定理，证明系统渐近稳定等价于存在正定Lyapunov函数且其一阶差分为负定，并推广到非线性系统。
This article focuses on the conservativeness issue of the existing Lyapunov method for linear time-invariant (LTI) nabla fractional-order systems and proposes a converse Lyapunov theorem to overcome the conservative problem. It is shown that the LTI nabla fractional-order system is asymptotically stable if and only if there exist a positive-definite Lyapunov function whose first-order difference is negative definite. After developing a systematic scheme to construct such Lyapunov candidates, the Lyapunov indirect method is derived for the nonlinear system. Finally, the effectiveness and practicability of the proposed methods are substantiated with four examples.