Integrally small perturbations of linear nonautonomous systems

Let A(t) and B(t) (t ≥ 0) be variable n × n-matrices. Assuming that the system ẋ = A(t)x is exponentially stable and the matrix norm of the integral ∫ ^t ₀ (B(s) - A(s)) ds is sufficiently small, for the system ẋ = B(t)x we derive explicit stability conditions, which improve the well-known ones in appropriate situations. The results are illustrated by a numerical example.