The stability notions expressed here are usually introduced in the
time-varying setting
. Since the vast majority of
planning problems in this book are time-invariant, the presentation
was confined to time-invariant vector fields. There is, however, one
fascinating peculiarity in the topic of finding a feedback plan that
stabilizes a system. *Brockett's condition* implies that for some
time-invariant systems for which continuous, time-varying feedback
plans exist, there does not exist a continuous time-invariant feedback
plan [143,156,996]. This includes the class of
driftless control systems, such as the simple car and the unicycle.
This implies that to maintain continuity of the vector field, a time
dependency must be introduced to allow the vector field to vary as
is approached! If continuity of the vector field is not
important, then this concern vanishes.

Steven M LaValle 2012-04-20