Time-varying Lyapunov functions for nonautonomous nabla fractional order systems. (July 2022) - Wei, Yiheng
Abstract: The classical Leibniz rule for the integer difference cannot be easily applicable for the fractional counterpart. It further leads to a great difficulty in the calculation of the Lyapunov functions with product form. To overcome such a challenge, several fractional difference inequalities are developed for Lyapunov functions which are the product of a time sequence and a function regarding to system state. To further enrich the design of time-varying Lyapunov function, the differentiable convex condition is introduced and then three elegant inequalities are derived. Those inequalities hold for the Caputo/Riemann–Liouville/Grünwald–Letnikov definitions which bring the possibility of the Lyapunov stability analysis for nonautonomous nabla fractional order systems. Finally, illustrative examples serve to illustrate the applicability and practicability of the theoretical results. Highlights: Two fractional difference inequalities on time-varying function are derived. A fractional difference inequality is developed for the differentiable convex function. All the mentioned inequalities are applicable to the Caputo/Riemann–Liouville/ Grünwald–Letnikov definitions. On this basis, the Lyapunov stability of nonautonomous nonlinear nabla fractional order systems can be judged.