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

Abstract

Various results and techniques, such as the Bohl-Perron theorem, a priori estimates of solutions, M-matrices and the matrix measure, are applied to obtain new explicit exponential stability conditions for the system of vector functional differential equationsx˙i(t)=Ai(t)xi(hi(t))+∑j=1n∑k=1mijBijk(t)xj(hijk(t))+∑j=1n∫gij(t)tKij(t,s)xj(s)ds,i=1,…,n. Here xi are unknown vector-functions, Ai,Bijk,Kij are matrix functions, hi,hijk,gij are delayed arguments. Using these results, we deduce explicit exponential stability tests for second order vector delay differential equations.