Solution estimates and stability tests for linear neutral differential equations

TL;DR

This paper derives explicit exponential stability criteria and solution estimates for scalar neutral differential equations with measurable coefficients and delays, covering both asymptotic and transient behaviors.

Contribution

It provides the first exponential estimates for solutions of neutral differential equations with measurable coefficients and delays, including stability tests and transient behavior analysis.

Findings

Explicit exponential stability tests derived.

Solution estimates valid on finite segments.

First-time exponential estimates for such equations.

Abstract

Explicit exponential stability tests are obtained for the scalar neutral differential equation $$ \dot{x}(t)-a(t)\dot{x}(g(t))=-\sum_{k=1}^m b_k(t)x(h_k(t)), $$ together with exponential estimates for its solutions. Estimates for solutions of a non-homogeneous neutral equation are also obtained, they are valid on every finite segment, thus describing both asymptotic and transient behavior. For neutral differential equations, exponential estimates are obtained here for the first time. Both the coefficients and the delays are assumed to be measurable, not necessarily continuous functions.