Fréchet Generalized Trajectories and Minimizers for Variational Problems of Low Coercivity

We address consecutively two problems. First, we introduce a class of so-called Fr´echet 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 Fr´echet metric. On the other side, the class of generalized controls is broad enough to settle the second problem, which is to prove existence of generalized minimizers of Lagrange variational problem with functionals of low (in particular linear) growth. Besides, we study the possibility of Lavrentiev-type gap between the infima of the functionals in the spaces of ordinary and generalized controls.