Slowly varying system ẋ = A(t)x

A limiting case of great importance in engineering is that of slowly varying parameters. For systems described by
\dot{x} = A(t)x
, one would intuitively expect that if, for each
t
, the frozen system is stable, then the time-varying system should also be stable. Provided
A(t)
is small enough, Rosenbrock has shown that this is the case [1]. Rosenbrock used a continuity argument [1, p. 75]. In this correspondence explicit bounds and slightly sharper results are obtained. Finally, it is pointed out that these results are useful in the study of the exact behavior of non-linear lumped systems with slowly varying operating points.