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

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 equations x˙_i(t)=A_i(t)x_i(h_i(t))+∑j=1n∑k=1m_ijB_ij^k(t)x_j(h_ij^k(t))+∑j=1n∫g_ij(t)tK_ij(t,s)x_j(s)ds,i=1,…,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.