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76 Introduction to Control Theory

Lyapunov Stability Theorems

1. Autonomous Systems: , such that
• The origin is an equilibrium point.
• The set .
• There exists V(x) continuous, continuously differentiate, V(0)=0

2. Non-autonomous Systems: x=f(t, x), such that
• The origin is an equilibrium point at t=t 0 .
• The set .
• There exists V(t, x) continuous, continuously differentiate in t and
x, V(t, 0)=0
• α(||x||), ß(||x||), and γ(||x||) class K functions.

Lyapunov Theorems may be used to design controllers that can stabilize
linear time-invariant systems as described in the next example.

Copyright © 2004 by Marcel Dekker, Inc.