# Lyapunov function help!

by rjcarril
Tags: function, lyapunov
 P: 2 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.
P: 2
 Quote by 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.
 P: 683 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?

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