  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.
