Exponential stability for a system of second and first order delay differential equations

TL;DR

This paper investigates the exponential stability of a coupled second-order delay differential system with an indirect feedback control, providing explicit conditions under which solutions decay exponentially.

Contribution

It introduces explicit sufficient conditions ensuring exponential stability for a coupled delay differential system with indirect control, extending existing stability results.

Findings

Derived explicit stability conditions for the system

Proved exponential decay of solutions under these conditions

Extended stability analysis to coupled delay systems with control

Abstract

Exponential stability of the second order linear delay differential equation in $x$ and $u$-control $$ \ddot{x}(t)+a_1(t)\dot{x}(h_1(t))+a_2(t)x(h_2(t))+a_3(t)u(h_3(t))=0 $$ is studied, where indirect feedback control $\dot{u}(t)+b_1(t)u(g_1(t))+b_2(t)x(g_2(t))=0$ connects $u$ with the solution. Explicit sufficient conditions guarantee that both $x$ and $u$ decay exponentially.