Suppose a velocity field
is given along with an
equilibrium point . Let denote a *candidate
Lyapunov function*, which will be used as an auxiliary device for
establishing the stability of . An appropriate must be
determined for the particular vector field . This may be quite
challenging in itself, and the details are not covered here. In a
sense, the procedure can be characterized as ``guess and verify,''
which is the way that many solution techniques for differential
equations are described. If succeeds in establishing
stability, then it is promoted to being called a *Lyapunov
function* for .

It will be important to characterize how varies in the
direction of flow induced by . This is measured by the *Lie
derivative*,

This results in a new function , which indicates for each the change in along the direction of .

Several concepts are needed to determine stability. Let a function
be called a *hill* if it is continuous, strictly increasing,
and . This can be considered as a one-dimensional
navigation function, which has a single local minimum at the goal,
0. A function
is called
*locally positive definite* if there exists some open set
and a hill function such that
and
for all . If can be chosen as , and if is bounded, then is called *globally
positive definite* or just *positive
definite*. In some spaces this may
not be possible due to the topology of ; such issues arose when
constructing navigation functions in Section 8.4.4. If
is unbounded, then must additionally approach infinity as
approaches infinity to yield a positive definite
[846]. For
, a quadratic form , for which
is a positive definite matrix, is a globally positive definite
function. This motivates the use of quadratic forms in Lyapunov
stability analysis.

The Lyapunov theorems can now be stated [156,846]. Suppose that is locally positive definite at . If there exists an open set for which , and on all , then is Lyapunov stable. If is also locally positive definite on , then is asymptotically stable. If and are both globally positive definite, then is globally asymptotically stable.

Steven M LaValle 2012-04-20