Global asymptotic stability for general nonautonomous ψ-Caputo fractional systems and applications

Abstract

Extending beyond asymptotic stability of the zero solution of integer-order differential systems remains a very difficult and challenging problem to date, and the question of how to extend for $\psi$-Caputo fractional order systems remains unknown. We develop a new theoretical framework by introducing the $\psi$-comparison principle and adopting the use of the generalized $\psi$-Laplace transform. First, we formulate fundamental linear comparison theory by identifying a potential Metzler matrix that gives order-dependent conditions to asymptotic stability of their zero solution. Then, we give new extensions to nonlinear systems by identifying an extra non-negative constant matrix that forms several new criteria for asymptotic stability. We also develop a general comparison theorem that looks for the possibility to identify a relatively asymptotic stability stable system, which further allows predicting the asymptotic stability of the zero solution of the original system. We demonstrate our novel theory by incorporating new results into some advanced nonlinear systems to demonstrate the novel significance of applicable results for an effective asymptotic analysis.