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

TL;DR

This paper introduces Fre'chet generalized controls for complex control systems, establishing continuity of the input-to-trajectory map and proving the existence of generalized minimizers for variational problems with low-growth functionals, while exploring Lavrentiev gaps.

Contribution

It develops a new class of generalized controls for non-commuting systems and proves existence results for variational problems with low coercivity.

Findings

Continuity of the input-to-trajectory map under Fre'chet controls

Existence of generalized minimizers for low-growth functionals

Analysis of Lavrentiev gaps between control spaces

Abstract

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.