Lyapunov stability theorems for $$\psi $$-Caputo derivative systems

Abstract

We introduce the stability concepts for \(\psi \)-Caputo derivative systems in the sense of Lyapunov’s idea. Two new Lyapunov theorems are formulated for the stability analysis of such systems. Many new inequalities that involve \(\psi \)-Caputo derivative are developed for the application of the proposed Lyapunov theorems. Examples are provided and it is shown that the proposed theorems are very useful for predicting the limiting behavior of the trajectory of such systems starting from any initial condition imposed at any initial time on the real axis.