文档介绍：Lecture Note #1 on
Stability and Fundamental of Lyapunov Function Theory
Prof. Hste, we will be focusing
on the situations that the limit set is an equilibrium point. One popular approach to the analysis of
equilibrium point is the linearization technique. Under a generic condition, it can be shown that
the equilibrium point of an autonomous nonlinear system is asymptotically stable (or unstable) if
and only if the equilibrium point x = 0 of the corresponding linearized system is asymptotically
stable (or unstable).
Consider an autonomous vector field
x· = fx(), x ∈ ℜn ()
Let a point x ∈ ℜn be an equilibrium solution of Eq. that
f()x = 0
., a solution which does not change in time.
In order to determine the stability of xt() we must understand the nature of solutions near xt() .
Let
xt()= xt()+ yt(). ()
Substituting Eq. () into Eq. () and Taylor expanding about xt() gives
· 2
x·()t = x()t + y· = fxt()()++Df() x() t y O()y ()
n ·
where Df is the derivative of f and . denotes a norm on ℜ .