Measuring singularity of generalized minimizers for control-affine problems

Abstract

An open question contributed by Y. Orlov to a recently published volume Unsolved Problems in Mathematical Systems and Control Theory (V. D. Blondel and A. Megretski, eds.), Princeton Univ. Press (2004), concerns regularization of optimal control-affine problems. These noncoercive problems admit, in general, “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 from a study of this question, we show by simple functional-theoretic reasoning that it admits, in general, a positive answer. This answer does not depend on commutativity/noncommutativity of controlled vector fields. Instead, it depends on the 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 socalled 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.