Question on derivative of a function

Question

I have this exercise :

We consider the system :

$$\begin{aligned}x_1'=&x_2 \\ x_2'=&-h_1(x_1)-x_2-h_2(x_3)\\ x_3'=&x_2-x_3\end{aligned}$$ where $h_1$ et $h_2$ are locally lipschitz, $h_i(0)=0$ and $yh_i(y)>0$ for all $y\neq0, \; (i=1,2).$

(a) Show that the origin is the unique equilibrium point of the system .

(b) Show that the functional $$V(x)=\int_0^{x_1} h_1(y) dy +\frac{x_2^2}{2}+\int_0^{x_3}h_2(y) dy$$ is positive definite for all $x=(x_1,x_2,x_3)\in \mathbb{R}^3$

(c) Show that the origin is asymptotically stable

For (a) and (b) that's ok , but to answer (c) i must calculate $V'$.

And i have a probleme with $V'$.

Can someone help me to find $V'$?

Please help me

Thank you .

Answer 1

First look at: http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign if you are not sure of how to differentiate V(x) w.r.t $x_1$ or $x_2$ etc.

Then,

$V'(x) = \nabla V(x).f(x)$, where $f(x)$ is the r.h.s of your system. i.e. $f(x)=[x_1' \: x_2' \: x_3']^T$

And $\nabla V(x)= [\frac{\partial(V)}{\partial x_1} \: \frac{\partial(V)}{\partial x_2} \: \frac{\partial(V)}{\partial x_3}]^T $