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2.7 Lyapunov Stability Theorems 75

Note that LaSalle’s theorem is actually more general than we have
described. In fact, it can be used to ascertain the stability of sets rather than
just an equilibrium point. The basic idea is that since V(x) is lower bounded
V(x)>c, then the derivative V has to gradually vanish, and that the trajectory
is eventually confined to the set where V=0. The following definitions is
useful in explaining the more general LaSalle’s theorem.

DEFINITION 2.7–5 A set G is said to be an invariant set of a dynamical
system if every trajectory which starts in G remains in G.

As examples of invariant sets, we can give the whole state-space, an
equilibrium point and a limit cycle. By using this concept, LaSalle was able
to describe the convergence to sets rather than to just equilibrium points.
For example, we can use this result to show that the limit cycle is a stable
attracting set of the Van der Pol oscillator.
The Linear Time-Invariant Case

In the case where the system under consideration is linear and time-invariant,
Lyapunov theory is well developed and the choice of a Lyapunov function is
simple. In fact, in this case, the various stability concepts in definitions 2.6.1
are identical. Lyapunov theory then provides necessary as well as sufficient
conditions for stability as discussed in this section. For the proofs consult
[Khalil 2001].

THEOREM 2.7–3: Given a linear time-invariant system

The system is stable if and only if there exists a positive definite solution P to
the equation

T
A P+PA=-Q
where Q is an arbitrary positive-definite matrix.

Note that the stability of the whole system was obtained in the last theorem
since in this case, the origin is the unique equilibrium point and its stability is
equivalent to the system being stable. In addition, no reference was made to
what kind of stability is implied since all stability concepts are equivalent in
the very special case of linear, time-invariant systems [Khalil 2001]. Also
note that this result is equivalent to testing that all eigenvalues of A have
negative real parts [Kailath 1980]. In the following, we include a table
summarizing Lyapunov Stability theorems.

Copyright © 2004 by Marcel Dekker, Inc.