Integrally Small Perturbations of Linear Nonautonomous Systems

Let A(t) and B(t) (t≥0) be variable n×n-matrices. Assuming that the system (x)\dot]=A(t)x\dot{x}=A(t)x is exponentially stable and the matrix norm of the integral ò₀^t (B(s)-A(s)) ds\int_{0}^{t} (B(s)-A(s))\,ds is sufficiently small, for the system (x)\dot]=B(t)x\dot{x}=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.