  We address consecutively two problems. First we introduce a class of so
called Fre'chet generalized controls for a multi-input control-affine system
with non-commuting controlled vector fields. For each control of the class one
is able to define a unique generalized trajectory,and the input-to-trajectory
map turns out to be continuous with respect to the Fre'chet metric. On the
other side, the class of generalized controls is broad enough to settle the
second problem, which is proving existence of generalized minimizers of
Lagrange variational problem with functionals of low (in particular linear)
growth. Besides we study possibility of Lavrentiev-type gap between the infima
of the functionals in the spaces of ordinary and generalized controls.
