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Converse Lyapunov Theorem for Nabla Asymptotic Stability Without Conservativeness

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.