A controllability condition for nonlinear systems

It is shown that a control system characterized by a differential equation of the formdot{x}(t) = A(t)x(t) + B(t)u(t) + f(t,x(t)), whose linear part is controllable, is itself controllable provided the nonlinear functionfsatisfies a condition amounting to stating that its "dc gain" approaches zero asparallelxparallel rightarrow infty. The proof uses the Schauder fixed-point theorem