Admissible Convexity for Time-Varying Fractional Lyapunov Inequalities

Fractional Lyapunov inequalities provide an effective tool for stability analysis of fractional-order systems, since classical chain and product rules are not directly applicable to fractional operators. This paper investigates admissible convexity conditions for time-varying Lyapunov functions in continuous Caputo and discrete nabla fractional settings. It is shown that state convexity alone is insufficient: a historical ordering condition is needed to ensure the proper sign of the memory terms. The continuous inequality is characterized by a weighted integral of historical residuals, whereas the discrete nabla counterpart is characterized by a weighted sum over historical grid points. Reverse inequalities for concave functions are also derived under reversed ordering conditions. Product-form inequalities with nonnegative convex state factors and nonnegative nonincreasing time factors are recovered as natural admissible cases. Numerical examples and Lyapunov applications are presented to verify the results.