I have the system that I want to show the global asymptotic stability of the origin
$$\dot{x_1} = x_2 \\ \dot{x_2} = -g(k_1 x_1 + k_2 x _2) $$ where k1 and k2 are positive numbers.
Also, $$g(y)y > 0 $$ for all $y\neq 0$ and $$\lim_{|y|\rightarrow \infty } \int_{0}^{y} g(\xi)d\xi = +\infty $$
With this information, I can see that I want to bring the derivative of my Lyapunov function to take form of
$$\dot{V}=-(k_1 x_1+k_2 x_2)g(k_1 x_1 + k_2 x_2) \\$$ so that I can say the derivative is negative definite, and that my Lyapunov function should take the combination of integral and quadratic form to have radially unboundedness. I tried to back track from V_dot, just to realize it's not so feasible to get the original function from its partial derivatives. Second approach I tried is with
$$V = \int_{0}^{x_1}g(\xi)d\xi + 0.5x_2^2$$ but it seems that its derivatives wouldn't tell me much info either. What would be a good next step to do to the candidate function?
