Robust Hyperexponential Stabilization via Nested Exponential Conformable Derivatives

This paper presents a novel class of conformable integro-differential operators designed to model systems with rapid and ultra-rapid dynamics. This class of local operators enables the design of controllers and observers that induce hyperexponential convergence and provide robustness against bounded disturbances and dynamic uncertainties. The proposed method leverages Nested Exponential Functions (NEFs) and Nested Exponential Factorial Functions (NEFFs) to capture fast dynamics effectively. Additionally, the proposed study examines the Fundamental Theorem of Calculus in the context of Nested Exponential Conformable (NEC) operators, unveiling structural properties, such as stability and robustness, that produce dynamical systems with enhanced hyperexponential convergence and faster dynamics compared to existing approaches. Stability results for NEC systems are established, and some illustrative examples based on numerical simulations are presented to demonstrate the reliability of the proposed approach.