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78 Introduction to Control Theory

then, note

(2.7.3)

Now, we can show by separation of variables and integrating that

T
x min (P)x V(t) V(0)e - t (2.7.4)
or that

(2.7.5)

Therefore, x(t) is approaching the origin at rate faster than /2. In fact, it
can be shown that the fastest convergence rate estimate is obtained when
Q=I.

Krasovskii Theorem

There are some cases when a Lyapunov function for autonomous nonlinear
systems is easily obtained using Krasovkii’s theorem stated below.

THEOREM 2.7–4: Consider the autonomous nonlinear system x=f(x) with
the origin being an equilibrium point. Let A(x)=∂f/∂x. Then, a sufficient
condition for the origin to be AS is that there exists 2 symmetric positive-
definite matrices, P and Q such that for all x≠0, the matrix

F(x)=A(x) P+PA(x)+Q (1)
T
T
is 0 in some ball B about the origin. The function V(x)=f(x) Pf(x) is then a
Lyapunov function for the system. If B= and if V(x) is radially unbounded
then the system is GAS.

Copyright © 2004 by Marcel Dekker, Inc.