Stability Criterion 2: Lyapunov Stability Theorem

Example

XTX

murmur

Example I X 2 2

iz 2 2

Oe xi t xi

dVf Vix 2x I 12 2 2

2XC X 12 2 12 Xz 2 27

2 5 4 1 2 4 3Z X Xz 2 22

Thus Vey D iff x 2 0

Vex 0 for all x f to

Example EERO

Hx

vexonyOeo

This shows x o is asymp stable

Lyapunov stability theorem

ITTI.ES

TppT7efI fx O xTPx O

O

Lyapunov stability theorem

etiO

Lyapunov stability theorem

von

f

It

Time-derivative of Lyapunov function

O

U xTPx xtPxCAxJpxtxTPcAxxTATpx xTPAXxT AT A x

Let Q CAMPAThen Vex xTQx 7

Time-derivative of Lyapunov function

IVey co

asyrup stable

Vex so

Lyapunov stable

Recap

OO

O