SUB-RIEMANNIAN METRICS

3

though his result failed to include the abnormal extremals.) When the

Maximum Principle is in fact applied to this case, the possibility that

some minimizers might be "abnormal" emerges directly by mechani-

cally writing the necessary conditions. This, of course, does not yet

answer the question whether abnormal minimizers actually exist, but

it provides a radically different perspective on the problem: whereas a

differential geometer's natural inclination is to look for the true charac-

terization of minimizers in the form of a modified version of the geodesic

equation, based on making variations as in the classical derivation of

the Euler-Lagrange equations, the control theorist's predisposition is

to start from the opposite direction, applying the Maximum Principle

to conclude that a minimizer is either a "normal extremal" or an "ab-

normal" one, inferring from this that abnormal extremal minimizers

probably exist, and then setting out to prove that they do.

The problem of the existence of strictly abnormal minimizers was

finally settled in 1991, when R. Montgomery, in [17], produced an ex-

ample of such a minimizer, thereby showing that the intuition derived

from the control-theoretic point of view was in fact the right one, even

though Montgomery himself was led to his discovery by physical rather

than control-theoretic considerations. Montgomery's very long and in-

genious optimality proof was somewhat simplified in 1992 by I. Kupka

(cf. [14]). However, neither of these proofs makes it possible to go

beyond individual examples and prove, for instance, that large classes

of abnormal extremals are optimal. In [16], we studied an example for

which the optimality proof was much simpler. Moreover, this example

had the extra virtue of being such that, after suitable changes of coordi-

nates, one can transform fairly general situations into normal forms to

which the proof of [16] applies. This was made precise in the preprint

[25], where a general "optimality lemma" was announced, which basi-

cally describes the most general situation where a method similar to

that of [16] can be applied to prove optimality. Using this optimality