On necessary and sufficient conditions for exponential stability in linear Volterra integro–differential equations

Abstract. Suppose K is a continuous matrix-valued function such that ∫ ∞ 0 t2|K(t) | dt <∞, and let X be the matrix-valued solution of the resolvent problem X′(t) = AX(t) + ∫ t 0 K(t − s)X(s) ds, t> 0; X(0) = I. If the solution X is in L1(0,∞), then the following are equivalent: (a) There are β> 0, c> 0 such that |K(t) | ≤ ce−βt, t ≥ 0; (b) There are α> 0, c0> 0 such that |X(t) | ≤ c0e−αt, |X′(t) | ≤ c0e−αt, |X′′(t) | ≤ c0e−αt, t ≥ 0. 1