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

Fre'chet Generalized Trajectories and Minimizers for Variational Problems of Low Coercivity