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74 Introduction to Control Theory

THEOREM 2.7–2: LaSalle
Given the autonomous nonlinear system

and let the origin be an equilibrium point. Then,

1. Asymptotic Stability: Suppose a Lyapunov function V(x) has been found
such that for , V(x)>0 and Then the origin is
asymptotically stable if and only if only at x=0.
2. Global Asymptotic Stability: The origin is GAS if above and
V(x) is radially unbounded.

Unfortunately, in many applications with time-varying trajectories, the open-
loop systems are not autonomous, and more advanced results such as the
ones described later will be called upon to show global asymptotic stability.

EXAMPLE 2.7–8: LaSalle Theorem
Consider the autonomous system

The origin is an equilibrium point. Moreover, consider a Lyapunov function
candidate

Leading to

Since for all x 1=x 2, we need to check whether the origin is the only
point where It can be seen from the state equation that x 1=x 2 can
only happen at the origin, therefore the origin is GAS.

Copyright © 2004 by Marcel Dekker, Inc.