The Fractional Lyapunov Direct Method for ψ-Caputo Fractional-Order Systems

We establish new Lyapunov stability theory for ψ-Caputo fractional-order systems by strengthening Lyapunov functions under reasonable guiding wings of Class-K_∞ functions and their fractional derivative inequalities. The new generalized ψ-Gronwall inequalities and conceptual definitions of stability that are linked with the ψ-Mittag-Leffler function were introduced. Our main results are Lyapunov stability theorems whenever one finds a potential Lyapunov function that has upper and lower bounds and obeys typical Lyapunov fractional differential inequalities along imagined real trajectories of such systems. This theory works with some typical worked-out dynamic models, in which the stability dynamics are discussed.