Invariant set theorems for non-autonomous time-fractional systems

Though the commonly used fractional Lyapunov stability and LaSalle theorems have been beneficial in the development of the theory and applications of fractional derivatives, their proofs hold certain flaws which can make their applicability questionable. From this point, we established new invariant set-based stability theorems for fractional order Caputo systems. As a consequence, we have obtained the fractional version of the Lyapunov stability theorem. In addition, sufficient conditions for the uniform asymptotic stability of Caputo systems are derived. Finally, two illustrative applications from population dynamics are presented to validate the effectiveness of the theoretical results.