Lyapunov Function: Show 0 is Stable

[rjcarril]

Assume that f(0) = 0 and Df(0) has eigenvalues with negative real parts. Con-
struct a Lyapunov function to show that 0 is asymptotically stable.

 

[rjcarril]

Assume that f(0) = 0 and Df(0) has eigenvalues with negative real parts. Con-
struct a Lyapunov function to show that 0 is asymptotically stable.

I know it is a strict lyapunov function, but i cannot figure out how to solve it for the general case.

 

[pasmith]

Consider $V(x) = \|x\|^2 = x \cdot x$. Then $\nabla V = 2x$ and
$$\dot V = \nabla V \cdot \dot x = \nabla V \cdot f(x) = 2x \cdot (Df(0) \cdot x) + O(\|x\|^3).$$
What does the condition on the eigenvalues of Df(0) imply about the sign of $x \cdot (Df(0) \cdot x)$? What does that imply about $\dot V$ for $\|x\|$ sufficiently small?

Can you prove that if $\dot V < 0$ on a neighbourhood of 0 then 0 is asymptotically stable?