Practical Exponential Stability of Tempered ϖ-Fractional Systems: Lyapunov Criteria and Applications to Perturbed and Controlled Systems

In this paper, we investigate the practical exponential stability of a class of nonlinear systems governed by the tempered ϖ-Caputo fractional derivative. A new Lyapunov-based criterion is established to derive sufficient conditions ensuring ϖ-practical exponential stability. The obtained result is formulated in a general framework involving suitable growth bounds on the Lyapunov function together with a tempered fractional derivative inequality and a boundedness condition on a weighted integral term. The proposed theorem provides an explicit practical exponential estimate for the system trajectories and extends existing stability results that are available for standard fractional and tempered fractional systems. To demonstrate the applicability of the developed theory, two applications are presented. First, the general criterion is applied to a class of perturbed tempered ϖ-fractional systems, for which verifiable sufficient conditions are derived in terms of quadratic Lyapunov functions and perturbation bounds. Second, a state-feedback stabilization result is established for a class of nonlinear tempered fractional control systems, showing that the proposed theorem can be used as an effective tool for closed-loop practical exponential stabilization. Finally, numerical examples are provided to validate the theoretical developments and to illustrate the effectiveness of the proposed approach. An additional test case with η3>0 is included to demonstrate the nontrivial range of Theorem 1. Furthermore, a socio-economic tempered fractional cobweb model is incorporated to show how the proposed criterion applies to price-adjustment dynamics with memory and persistent market perturbations.