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Minimizers that are not Impulsive Minimizers and Higher Order Abnormality

Monica Motta, Michele Palladino, Franco Rampazzo

Abstract

This paper addresses two related problems in optimal control. The first investigation consists of compatibility issues between two classical approaches to deriving necessary conditions for optimal control problems with a final target: the set-separation approach and penalization techniques. These methods generally lead to non-equivalent conditions, mainly due to their reliance on different notions of tangency at the target. We address this issue by considering Quasi Differential Quotient (QDQ) approximating cones (which are fit for the set-separation approach) and identifying conditions under which the Clarke tangent cone (which is a typical tool within penalization techniques) is also a QDQ approximating cone. In particular, we show that this property holds under suitable local invariance assumptions or when the target coincides locally with an $r$-prox regular set. In the second part of the paper we apply this compatibility result to the study of infimum-gap phenomena in optimal control problems with unbounded controls and impulsive extensions. In particular, we establish a connection between the occurrence of infimum gaps for strict-sense minimizers and abnormality in a higher-order Maximum Principle involving Lie brackets. While the abnormality-gap correspondence beyond first-order conditions has been already established for extended-sense --i.e. impulsive-- minimizers, a topological argument involving the former and the utilization of the above compatibility issues allow us to extend this correspondence to strict-sense minimizers.

Minimizers that are not Impulsive Minimizers and Higher Order Abnormality

Abstract

This paper addresses two related problems in optimal control. The first investigation consists of compatibility issues between two classical approaches to deriving necessary conditions for optimal control problems with a final target: the set-separation approach and penalization techniques. These methods generally lead to non-equivalent conditions, mainly due to their reliance on different notions of tangency at the target. We address this issue by considering Quasi Differential Quotient (QDQ) approximating cones (which are fit for the set-separation approach) and identifying conditions under which the Clarke tangent cone (which is a typical tool within penalization techniques) is also a QDQ approximating cone. In particular, we show that this property holds under suitable local invariance assumptions or when the target coincides locally with an -prox regular set. In the second part of the paper we apply this compatibility result to the study of infimum-gap phenomena in optimal control problems with unbounded controls and impulsive extensions. In particular, we establish a connection between the occurrence of infimum gaps for strict-sense minimizers and abnormality in a higher-order Maximum Principle involving Lie brackets. While the abnormality-gap correspondence beyond first-order conditions has been already established for extended-sense --i.e. impulsive-- minimizers, a topological argument involving the former and the utilization of the above compatibility issues allow us to extend this correspondence to strict-sense minimizers.
Paper Structure (20 sections, 10 theorems, 86 equations, 1 figure)

This paper contains 20 sections, 10 theorems, 86 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A theoretical issue
  3. An application to the question of infimum gaps
  4. Preliminaries
  5. Basic notation
  6. Cone transversality
  7. Some generalized normal and tangent cones
  8. Prox-regular sets
  9. QDQ approximating cones and Quasi Prox Regularity
  10. Quasi Differential Quotients (QDQ) and conic approximation
  11. Quasi Prox-Regularity and Clarke tangent cones
  12. Impulsive extension of a minimum problem and infimum gap phenomena
  13. An unbounded control problem and its impulsive extension
  14. Extended processes with local infimum gap are isolated processes
  15. Embedded strict-sense minimizers with local infimum gap are limits of isolated processes
  16. ...and 5 more sections

Key Result

Proposition 2.1

\newlabelteo20 Two convex cones ${\mathcal{K}}_1$, ${\mathcal{K}}_2\subseteq \mathbb{R}^n$ are transversal if and only if either they are strongly transversal or they are complementary linear subspaces, namely ${\mathcal{K}}_1\oplus{{\mathcal{K}}}_2=\mathbb{R}^n$ (i.e., ${\mathcal{K}}_1 +{\mathcal

Figures (1)

  • Figure 1: Tangent and normal cones

Theorems & Definitions (33)

  • Definition 1
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.1
  • Proof 1
  • Definition 2
  • Remark 2.1
  • Proposition 2.3
  • Definition 1
  • Remark 3.1
  • ...and 23 more