  When applying methods of optimal control to motion planning or stabilization
problems, some theoretical or numerical difficulties may arise, due to the
presence of specific trajectories, namely, singular minimizing trajectories of
the underlying optimal control problem. In this article, we provide
characterizations for singular trajectories of control-affine systems. We prove
that, under generic assumptions, such trajectories share nice properties,
related to computational aspects; more precisely, we show that, for a generic
system -- with respect to the Whitney topology --, all nontrivial singular
trajectories are of minimal order and of corank one. These results, established
both for driftless and for control-affine systems, extend previous results. As
a consequence, for generic systems having more than two vector fields, and for
a fixed cost, there do not exist minimizing singular trajectories. We also
prove that, given a control system satisfying the LARC, singular trajectories
are strictly abnormal, generically with respect to the cost. We then show how
these results can be used to derive regularity results for the value function
and in the theory of Hamilton-Jacobi equations, which in turn have applications
for stabilization and motion planning, both from the theoretical and
implementation issues.

Optimization and Control