A geometrically based criterion to avoid infimum-gaps in Optimal Control

TL;DR

This paper introduces a geometric criterion based on control family abundance and normality to prevent infimum gaps in optimal control problems, extending previous results beyond convex dynamics.

Contribution

It establishes a general, non-convex condition involving control family abundance and normality that guarantees the absence of infimum gaps in optimal control.

Findings

Normality implies no infimum gap under control family abundance.

Introduces a non-convex, dynamical-topological notion of abundance.

Proves an open mapping theorem for Quasi-Differential-Quotient cones.

Abstract

In optimal control theory the expression infimum gap means a strictly negative difference between the infimum value of a given minimum problem and the infimum value of a new problem obtained by the former by extending the original family V of controls to a larger family W. Now, for some classes of domain-extensions -- like convex relaxation or impulsive embedding of unbounded control problems -- the normality of an extended minimizer has been shown to be sufficient for the avoidance of an infimum gaps. A natural issue is then the search of a general hypothesis under which the criterium 'normality implies no gap' holds true. We prove that, far from being a peculiarity of those specific extensions and from requiring the convexity of the extended dynamics, this criterium is valid provided the original family V of controls is abundant in the extended family W. Abundance, which is stronger than the mere C^0-density of the original trajectories in the set of extended trajectories, is a dynamical-topological notion introduced by J. Warga, and is here utilized in a 'non-convex' version which, moreover, is adapted to differential manifolds. To get the main result, which is based on set separation arguments, we prove an open mapping result valid for Quasi-Differential-Quotient (QDQ) approximating cones, a notion of 'tangent cone' resulted as a peculiar specification of H. Sussmann's Approximate-Generalized-Differential-Quotients (AGDQ) approximating cone.