Hyers-Ulam stability for differential systems with $2\times 2$ constant coefficient matrix

TL;DR

This paper establishes new necessary and sufficient conditions for Hyers-Ulam stability in 2x2 linear differential systems with constant coefficients and determines the minimal stability constants, supported by illustrative examples.

Contribution

It introduces the first explicit calculation of the best Hyers-Ulam constant for such systems and provides comprehensive criteria for stability.

Findings

New necessary and sufficient conditions for stability.

Explicit formulas for minimal Hyers-Ulam constants.

Application to second-order differential equations.

Abstract

We explore the Hyers-Ulam stability of perturbations for a homogeneous linear differential system with $2\times 2$ constant coefficient matrix. New necessary and sufficient conditions for the linear system to be Hyers-Ulam stable are proven, and for the first time, the best (minimal) Hyers-Ulam constant for systems is found in some cases. Several examples are provided. Obtaining the best Hyers-Ulam constant for second-order constant coefficient differential equations illustrates the applicability of the strong results.