Exponential stability of second order delay differential equations through Floquet theory

TL;DR

This paper develops a Floquet theory-based approach for analyzing exponential stability in second order delay differential equations, enabling stabilization strategies previously deemed impossible.

Contribution

It introduces a novel Floquet theory for second order delay differential equations that preserves the order and extends classical stability results.

Findings

New Floquet theory for second order delay equations

Conditions for exponential stabilization using period and delay choices

Demonstration of stabilization where standard methods fail

Abstract

In this paper, we obtain results on exponential stability of second order delay differential equations, which are based on a version of the Floquet theory for delay differential equations of the second order we proposed. Our version allows researchers to preserve the order of equation and to obtain analogues of the classical results of the Floquet theory known for ordinary differential equations. On the basis of our version of the Floquet theory, new original unexpected results on the exponential stability are proposed. We demonstrate that choosing period of coefficients and delays of the gain in corresponding intervals allows to achieve the exponential stabilization in the cases considered as impossible when the standard technique was applied.