Lyapunov-Based Analysis of Partial Practical Stability in Tempered Fractional Calculus

This study presents a comprehensive Lyapunov-based framework for analyzing partial practical stability in nonlinear tempered fractional-order systems (TFOS). We develop novel stability concepts including β*-practical uniform generalized Mittag–Leffler stability (β*-PUGMLS) and β*-practical uniform exponential stability (β*-PUES) with respect to system substates. Through carefully constructed Lyapunov functions, we establish sufficient conditions under which the system’s states converge to a predefined neighborhood of the origin. The theoretical framework provides Mittag–Leffler and exponential stability criteria for tempered fractional-order systems, extending classical stability theory to this important class of systems. Furthermore, we apply these stability results to design stabilizing feedback controllers for a specific class of triangular TFOS, demonstrating the practical utility of our theoretical developments. The efficacy of the proposed stability criteria and control strategy is validated through several illustrative examples, showing that system states converge appropriately under the derived conditions. This work contributes significantly to the stability theory of fractional-order systems and provides practical tools for controlling complex nonlinear systems in the tempered fractional calculus framework.