Consider the system:
$$\dot{x}_1 = x_2 $$ $$\dot{x}_2 = -g(k_1 x_1 + k_2 x_2), k_1>0, k_2>0$$
where the nonlinearity g(·) is such that
$$g(y)y > 0, \forall y \neq 0 $$
$$lim_{|y| \to \infty} \int_{0}^{y} g(\alpha) d(\alpha) = +\infty $$
Using an appropriate Lyapunov function, show that the equilibrium x = 0 is globally asymptotically stable.
ATTEMPT:
I know if:
$$\dot{x}_2 = -f(x_1) - g(x_2)$$ then I can define Lyapunov function as:
$$V(x_1, x_2) = (\int_{0}^{x_1}f(x)dx) + 1/2x_2^2$$.
Then,
$$\dot{V(x_1, x_2)}= -x_2 g(x_2) <0$$
But here, the function $$g(k_1 x_1 + k_2 x_2)$$ cannot be split into these two function of that depend on only $x_1$ and $x_2$ respectively. What should the choice of Lyapunov function be for this case?