Consider the nonlinear system $$\dot{x}(t)=f(x,u)$$ where $f:\mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^n$ is the vector field governing the evolution of the systems. Now consider the $C^1$ function $V:\mathbb{R}^n \to \mathbb{R}$ with $V(0)=0$. Verify that the functional $$H\left(x(\cdot);u(\cdot) \right):=-\int_0^{\infty}\frac{\partial V}{\partial x}\left[x(t)\right]f(x(t),u(t))\,\mathrm{d}t \tag{1}$$ is a feedback invariant as long as $\displaystyle\lim_{t \to \infty} x(t)=0 \tag{2}$
Attemp: (1) can be written as $$\begin{align}H(x,u)&=-\int_0^{\infty}\nabla V(x(t))\cdot f(x(t),u(t))\,\mathrm{d}t \\&=-\int_0^{\infty}\nabla V(x(t)) \cdot \dot{x}(t) \, \mathrm{d}t=\int_0^{\infty}- \frac{\mathrm{d}}{\mathrm{d}t} \left(V(x(t) \right)\, \mathrm{d}t\\&=V(x(0))-\lim_{t \to \infty}V(x(t))=V(x(0)) ~~\text{\{using (2)\}}\end{align}$$ So $H$ is independent of control input $u(t)$ which makes it feedback invariant. Is my approach correct ?
( The question is from linear system theory by Joao P. Hespanha Chapter 20, exercise 20.1. )