Consider a system $\dot{x}=F(x)$ with phase flow denoted by $g^t$. Suppose $0$ is an asymptotically stable equilibrium point and for all $t\geq 0$ $$ |g^t(x)|\leq \phi(t)|x| $$ where $\phi\in C([0,\infty),\mathbb{R})$ satisfies$\int_0^\infty \phi^2\leq \infty$. Define $$ L(x):=\int_0^\infty |g^t(x)|^2\mathrm{d}t $$ Show that $L$ is a strict Lyapunov function w.r.t. $x^e=0$.
For previous preparation, I have proved that $L$ is well-defined (i.e., the integral is finite). And I noticed that $\forall x\in\mathbb{R}^n,\ \forall t\geq 0,$ $$ L(g^t(x))=\int_t^\infty |g^s(x)|^2\mathrm{d}s $$ But I stuck at calculate $\nabla L$, and I have no idea how to use the asymptotically stable condition.
Appreciate any help!