Measuring Singularity of Generalized Minimizers for Control-Affine Problems

TL;DR

This paper investigates the regularization of control-affine optimal control problems, establishing conditions for convergence of regularized solutions and introducing a measure called the degree of singularity to quantify the problem's complexity.

Contribution

It provides a functional-theoretic proof that regularized infima converge to the original infimum under general conditions and introduces the degree of singularity as a new measure for analyzing singularity.

Findings

Regularized infima converge to the original infimum under broad conditions.

The degree of singularity relates to the order of singularity in linear-quadratic problems.

Partial results are established for nonlinear control-affine problems.

Abstract

An open question contributed by Yu. Orlov to a recently published volume "Unsolved Problems in Mathematical Systems and Control Theory", V.D. Blondel, A. Megretski (eds), Princeton Univ. Press, 2004, concerns regularization of optimal control-affine problems. These noncoercive problems in general admit 'cheap (generalized) controls' as minimizers; it has been questioned whether and under what conditions infima of the regularized problems converge to the infimum of the original problem. Starting with a study of this question we show by simple functional-theoretic reasoning that it admits, in general, positive answer. This answer does not depend on commutativity/noncommtativity of controlled vector fields. It depends instead on presence or absence of a Lavrentiev gap. We set an alternative question of measuring "singularity" of minimizing sequences for control-affine optimal control problems by so-called degree of singularity. It is shown that, in the particular case of singular linear-quadratic problems, this degree is tightly related to the "order of singularity" of the problem. We formulate a similar question for nonlinear control-affine problem and establish partial results. Some conjectures and open questions are formulated.