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15.1.2 Lyapunov Functions

Suppose a velocity field
is given along with an
equilibrium point, . Can the various forms of stability be
easily determined? One of the most powerful methods to prove
stability is to construct a Lyapunov function. This will be
introduced shortly, but first some alternatives are briefly mentioned.

If is linear, which means that for some constant
matrix and
, then stability questions with
respect to the origin,
, are answered by finding the
eigenvalues of [192]. The state is asymptotically
stable if and only if all eigenvalues of have negative real parts.
Consider the scalar case,
, for which
and
is a constant. The solution to this differential equation is
, which converges to 0 only if . This can be
easily extended to the case in which
and is an
diagonal matrix for which each diagonal entry (or
eigenvalue) is negative. For a general matrix, real or complex
eigenvalues determine the stability (complex eigenvalues cause
oscillations). Conditions also exist for Lyapunov stability. Every
equilibrium state of
is Lyapunov stable if the
eigenvalues of all have nonpositive real parts, and the
eigenvalues with zero real parts are distinct roots of the
characteristic polynomial of .

If is nonlinear, then stability can sometimes be inferred by
linearizing about and performing linear stability
analysis. In many cases, however, this procedure is inconclusive (see
Chapter 6 of [156]). Proving the stability of a vector
field is a challenging task for most nonlinear systems. One approach
is based on LaSalle's invariance principle
[39,156,585] and is particularly useful for showing
convergence to any of multiple goal states (see Section 5.4 of
[846]). The other major approach is to construct a *Lyapunov function*, which is used as an intermediate tool to
indirectly establish stability. If this method fails, then it still
may be possible to show stability using other means. Therefore, it is
a sufficient condition for stability, but not a necessary one.

**Subsections**
Steven M LaValle
2020-08-14