On the global controllability of perturbed controllable systems

By using a recent theorem of Davison and Kunze [1], it is shown that, if certain conditions hold such that the system
\dot{x} = A(x,t)x + B(x,t)u
is globally controllable, then the perturbed system
\dot{x} = [A(x,t) + \epsilon\tilde{A}(x,t)]x + [B(x,t) + \epsilon\tilde{B}(x,t)u
, where
\tilde{A}
and
\tilde{B}
are bounded, is also globally controllable, provided ε is small enough. In particular, if
\dot{x} = A(t)x + B(t)u
is controllable, then so is the perturbed system
\dot{x} = [A(t) + \epsilon\tilde{A}(x,t)]x + [B(t) + \epsilon\tilde{B}(x,t)]u
.