Characterisation of Stability and Decay Rates in a Weakly Damped Second Order Linear Differential Equation

TL;DR

This paper establishes precise conditions for the stability and decay rates of solutions to a weakly damped second order linear differential equation with external forcing, extending understanding of asymptotic behavior in such systems.

Contribution

It provides necessary and sufficient conditions for solution convergence and decay, and characterizes when forced solutions mimic unforced asymptotic behavior.

Findings

Necessary and sufficient conditions for convergence

Characterization of decay to zero

Sharp conditions for asymptotic equivalence

Abstract

This paper gives necessary and sufficient conditions for the convergence of the solution of a weakly damped second order linear differential equation that is subjected to outside forcing, for which solutions of the unforced equation are asymptotically stable. Conditions are also given which characterise when the solution and its derivative tend to zero. Finally, we give sharp sufficient conditions under which the solution of the forced equation has the same asymptotic behaviour as the unforced equation, to leading order.