Novel Stability Criteria for Discrete and Hybrid Systems via Ramanujan Inner Products

摘要

This paper introduces a Ramanujan inner product and its corresponding norm, establishing a novel framework for the stability analysis of hybrid and discrete-time systems as an alternative to traditional Euclidean metrics. We establish new $ε$-$δ$ stability conditions that utilize the unique properties of Ramanujan summations and their relationship with number-theoretic concepts. The proposed approach provides enhanced robustness guarantees and reveals fundamental connections between system stability and arithmetic properties of the system dynamics. Theoretical results are rigorously proven, and simulation results on numerical examples are presented to validate the efficacy of the proposed approach.