Exponential stability of systems of vector delay differential equations with applications to second order equations

TL;DR

This paper develops new explicit exponential stability conditions for systems of vector delay differential equations using various mathematical techniques, and applies these results to second order equations.

Contribution

It introduces novel explicit stability criteria for vector delay differential systems and applies them to second order equations, enhancing stability analysis methods.

Findings

Derived new exponential stability conditions for vector systems.

Established explicit stability tests for second order delay equations.

Applied classical techniques to modern vector differential systems.

Abstract

Various results and techniques, such as Bohl-Perron theorem, a priori solution estimates, M-matrices and the matrix measure, are applied to obtain new explicit exponential stability conditions for the system of vector functional differential equations $$ \dot{x_i}(t)=A_i(t)x_i(h_i(t)) +\sum_{j=1}^n \sum_{k=1}^{m_{ij}} B_{ij}^k(t)x_j(h_{ij}^k(t)) + \sum_{j=1}^n\int\limits_{g_{ij}(t)}^t K_{ij}(t,s)x_j(s)ds,~i=1,\dots,n. $$ Here $x_i$ are unknown vector functions, $A_i, B_{ij}^k, K_{ij}$ are matrix functions, $h_i,h_{ij}^k, g_{ij}$ are delayed arguments. Using these results, we deduce explicit exponential stability tests for second order vector delay differential equations.