# Lyapunov function - second derivative

I have a question concerning Lyapunov function, let's say that I have the first time derivative of Lyapunov function (V) and it is as follow: $\dot{V}(S)=-kS^2-\bar{k}|S|$. I need to derive a second time derivative and I am not sure if my solution is correct. Can someone confirm it or give me a hint how to get to the correct solution. $\ddot{V}(S)=-2kS\dot{S}-\frac{S}{|S|}\bar{k}\frac{S\dot{S}}{|S|}=-2kS\dot{S}-\bar{k}\frac{S^2}{|S|^2}\dot{S}=-2kS\dot{S}-\bar{k}\dot{S}$. Is it correct?

• Why do you need the second time derivative of the Lyapunov function? Normally the first derivative should be enough. If it is negative definite (and the Lyapunov function itself is positive definite) then you have asymptotic stability. When $\dot{V}\leq -\alpha\, V$ with $\alpha > 0$ then you have exponential stability. – Kwin van der Veen Aug 19 '17 at 21:11
• I have to apply Barbalat's lemma, that's why I need 2nd time derivative of V. – krzesniak1 Aug 20 '17 at 16:02
• I was not aware of that lemma. Good to know. – Kwin van der Veen Aug 20 '17 at 16:09

The derivative of |x| with respect to x is 1 if x> 0, -1 if x< 0, undefined at x= 0. That can, of course, be written as $\frac{x}{|x|}$. So the derivative of $\overline{k} |S|$ with respect to time is $\frac{S}{|S|}\dot{S}$. I don't know why you have $\frac{S}{|S|}$ twice. That is not correct.