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

TL;DR

This paper proposes a new stability analysis framework for hybrid and discrete systems using Ramanujan inner products, offering improved robustness and novel insights into the connection between system stability and number theory.

Contribution

It introduces a Ramanujan inner product-based stability criterion, providing an alternative to Euclidean metrics and linking system stability with number-theoretic properties.

Findings

Enhanced robustness guarantees for hybrid and discrete systems.

Fundamental connections established between stability and arithmetic properties.

Theoretical proofs and numerical simulations validate the approach.

Abstract

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 $\epsilon$-$\delta$ 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.