Best Ulam constants for damped linear oscillators with variable coefficients

TL;DR

This paper derives the best Ulam stability constants for non-autonomous damped linear oscillators using Riccati equations, providing robust bounds applicable to solutions with finite-time blow-up or global existence.

Contribution

It introduces a method to compute minimal Ulam constants for non-autonomous linear differential equations modeling damped oscillators, expanding stability analysis tools.

Findings

Derived explicit formulas for Ulam constants.

Applicable to equations with finite-time blow-up solutions.

Validated results with illustrative examples.

Abstract

This study uses an associated Riccati equation to study the Ulam stability of non-autonomous linear differential vector equations that model the damped linear oscillator. In particular, the best (minimal) Ulam constants for these non-autonomous linear differential vector equations are derived. These robust results apply to vector equations with solutions that blow up in finite time, as well as to vector equations with solutions that exist globally on $(-\infty,\infty)$. Illustrative, non-trivial examples are presented, highlighting the main results.