Lipschitz regularity of controls and inversion mapping for a class of smooth extremization problems

TL;DR

This paper establishes Lipschitz regularity of controls and constructs an inversion mapping for constrained extremal problems in smooth optimal control systems, even under weaker conditions than classical assumptions.

Contribution

It provides a novel regularity result for controls in constrained extremization problems with weaker Lagrangian conditions and constructs a Lipschitz inversion mapping for extremals.

Findings

Controls are Lipschitz continuous under weaker conditions.

A locally Lipschitz inversion mapping for extremals is constructed.

Regularity results apply to affine control systems with smooth dynamics.

Abstract

In the contest of optimal control problems, regularity results for optima are known when addressing fiber-strictly convex Lagrangian. For infinite time horizons, or for settings with infinite dimensional dynamics, the equivalence between minima/maxima and extremals could break down. Commonly, this is due to a loss of convexity/concavity of the cost functional or to a presence of state constraints, in which further controllability assumptions are needed. For many science applications, this a trend is not required, as in energy saving problems. In the present paper, we deal with the set of a functional's extremals subject to end-point restrictions. We consider an affine control system and a cost functional associated to an autonomous Lagrangian. The dynamics is smooth, satisfying the Lie bracket condition, and the functional is assumed merely Fr\'echet differentiable. Here we provide a regularity result for controls in the context of constrained extremization problems, under weaker conditions on Lagrangian than the not met classical ones. More precisely, we show a characterization for the Lipschitz regularity of controls associated with the extremal trajectories steering two fixed points, assuming the absence of singular controls. As main application, we construct a locally Lipschitz inversion mapping from the ambient space to the set of constrained extremals.