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

Shyam Kamal, Sunidhi Pandey, Thach Ngoc Dinh

TL;DR

The paper addresses stability analysis for discrete-time and hybrid systems where dynamics exhibit arithmetic structure that Euclidean metrics may smooth over. It introduces a Ramanujan inner product and associated norm to form Ramanujan-stabilizing conditions, including a discrete-time Ramanujan stability criterion and Ramanujan-Lyapunov results for hybrid systems. The key contributions are the formal Ramanujan stability theorems, robustness results against arithmetic disturbances, and numerical examples demonstrating less conservative, structure-aware guarantees. This framework connects stability to number-theoretic properties of the dynamics, with practical implications for systems with sampling, scheduling, or arithmetic-pattern disturbances.

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 $ε$-$δ$ 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.

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

TL;DR

The paper addresses stability analysis for discrete-time and hybrid systems where dynamics exhibit arithmetic structure that Euclidean metrics may smooth over. It introduces a Ramanujan inner product and associated norm to form Ramanujan-stabilizing conditions, including a discrete-time Ramanujan stability criterion and Ramanujan-Lyapunov results for hybrid systems. The key contributions are the formal Ramanujan stability theorems, robustness results against arithmetic disturbances, and numerical examples demonstrating less conservative, structure-aware guarantees. This framework connects stability to number-theoretic properties of the dynamics, with practical implications for systems with sampling, scheduling, or arithmetic-pattern disturbances.

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 - 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.

Paper Structure

This paper contains 28 sections, 5 theorems, 19 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Classical Stability Notions
  4. Stability of Discrete-Time Nonlinear Systems
  5. Stability of Nonlinear Hybrid Systems
  6. Ramanujan Inner Product
  7. Properties of the Ramanujan Norm
  8. Comparison with the Euclidean Norm
  9. Ramanujan Gain Function
  10. Main Stability Theorem
  11. Ramanujan Stability Criterion
  12. Robustness under Arithmetic Disturbances
  13. Control-Theoretic Interpretation.
  14. Hybrid System Framework and Stability Definitions
  15. Hybrid System Model
  16. ...and 13 more sections

Key Result

Theorem 1

Let $\mathcal{S}$ be the class of complex sequences $a=(a_n)_{n\in\mathbb Z}$ that admit a Ramanujan expansion $a(n) = \sum_{d\ge 1} \alpha_d\,c_d(n)$, with coefficient sequence $(\alpha_d)_{d\ge1}$ satisfying the weighted square--summability condition $\sum_{d\ge1} \varphi(d)\,|\alpha_d|^2 < \infty

Figures (2)

  • Figure 1: Norm of state of system \ref{['dts']} showing that the Euclidean norm has persistent spikes due to prime-time disturbances, while the Ramanujan norm decays smoothly to zero.
  • Figure 2: Norm of state trajectories of system \ref{['hbd']} showing evolution of Euclidean and Ramanujan norm.

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Definition 3: Ramanujan Sum
  • Definition 4
  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Definition 5: Ramanujan Gain Function
  • Theorem 2
  • ...and 8 more