<s>[INST] Stability Function system Hi I have this question

For the system
$$\begin{align}
\dot p_1 &= -p_1 + 6p_2 \\
\dot p_2 &= -7p_2+(p_1+p_2)\cos p_1 
\end{align}
$$
Use the function  $L(p)=\frac{1}{2}(p_1^2 + p_2^2)$ ,and show that the origin is locally asymptotically stable.

What I have done so far:

*

*Differentiated $L(p)$ to get $\dot L(p)$


*Substituted in the values of $\dot p_1$ and $\dot p_2$
 [/INST] Since
$$x_1^2 + x_2^2 \ge 2x_1x_2 \implies x_1x_2 \le \frac{1}{2}x_1^2+\frac{1}{2}x_2^2$$
for all real numbers $x_1,x_2$, we have
\begin{align}\dot V(x)&=-x_1^2+x_1x_2+(x_1x_2+x_2^2)\sin x_1-3x_2^2\\&\le
-x_1^2+\frac{1}{2}x_1^2+\frac{1}{2}x_2^2+\left(\frac{1}{2}x_1^2+\frac{1}{2}x_2^2+x_2^2\right)|\sin x_1|-3x_2^2\\&=
-\frac{1}{2}x_1^2-\frac{5}{2}x_2^2+\left(\frac{1}{2}x_1^2+\frac{3}{2}x_2^2\right)|\sin x_1|\\&\le-x_2^2\\&
\le 0 
\end{align}
as $|\sin x_1|\le 1$. Global asymptotic stability follows from $\dot V(x) < 0$ when $(x_1,x_2)\neq 0$.
</s>