Chapter 4Chapter 4Lyapunov Stabilityy p y

The Invariance PrincipleThe Invariance PrincipleTime Varying Systems

Ω

E

ExampleExample

3: ( ) 0 0E V x b x x : ( ) 0 0: ( , ) (0,0)

E V x b x xM x x

• Assume M contains a point with a nonzero

iti th th l ti t th t

( , ) ( , )

position x, then the acceleration at that point is 30 1 0k k

This implies that the trajector ill mo e

30 1 0x x xm m

• This implies that the trajectory will move out of E. Not invariant. Contradiction!

Derivative of VDerivative of V

( ), ( ),lim

V x t t t t V x t tdVd

( ), ( ), ( ), ( ),lim

t

t

dt tV x t t t t V x t t t V x t t t V x t t

t

( ), ( ), ( ), ( ),lim lim

t

t t

tV x t t t t V x t t t V x t t t V x t t

t t

( , )

t tV V V Vx f x tt x t x