Stability Function system

Question

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:

1. Differentiated $L(p)$ to get $\dot L(p)$

2. Substituted in the values of $\dot p_1$ and $\dot p_2$

Answer 1

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 $(x_1,x_2)\neq 0$.

Answer 2

As $ |\sin x_1|\le1$ you get $$\begin{align} \dot V&\le -x_1^2+x_2x_1+|x_2x_1|\,|\sin x_1|+x_2^2|\sin x_1|-3x_2^2 \\ &\le -x_1^2+x_2x_1+|x_2x_1| +x_2^2 -3x_2^2 \\ &\le -x_1^2+2|x_1x_2|-2x_2^2 \\ &=-(|x_1|-|x_2|)^2-|x_2|^2 \\ &<0 \end{align}$$ for $(x_1,x_2)\ne(0,0)$.