Uniform exponential stability of linear delayed integro-differential vector equations

Uniform exponential stability of a linear delayed integro-differential vector equation x˙(t)=∑k=1mA_k(t)x(h_k(t))+∑k=1l∫g_k(t)tP_k(t,s)x(s)ds,t∈[0,∞), where x=(x₁,…,x_n)^T is an unknown vector-function, is considered. It is assumed that m, l are positive integers, matrices A_k, P_k and delays h_k, g_k are Lebesgue measurable. The main result is of an explicit type, depending on all delays, and its proof is based on an a priori estimation of solutions, a Bohl-Perron type result, and utilization of the matrix measure. As particular cases, it includes (2^m+l−1) mutually different sufficient conditions. Some of them are formulated separately as corollaries. Advantages of derived explicit results over the existing ones are demonstrated on examples and open problems are proposed as well.