Conditions for asymptotic stability of first order scalar differential-difference equation with complex coefficients

TL;DR

This paper derives explicit necessary and sufficient conditions based on complex coefficients for the asymptotic stability of first order scalar differential-difference equations, facilitating practical stability analysis.

Contribution

It provides a complete characterization of stability conditions directly in terms of polynomial coefficients, including applications to retarded PDEs.

Findings

Conditions for root placement in the complex plane are explicitly derived.

The stability criteria are applicable to differential-difference equations with complex coefficients.

Examples demonstrate the practical use of the conditions in various applications.

Abstract

We investigate a scalar characteristic exponential polynomial with complex coefficients associated with a first order scalar differential-difference equation. Our analysis provides necessary and sufficient conditions for allocation of the roots in the complex open left half-plane what guarantees asymptotic stability of the differential-difference equation. The conditions are expressed explicitly in terms of complex coefficients of the characteristic exponential polynomial, what makes them easy to use in applications. We show examples including those for retarded PDEs in an abstract formulation.