My question is: it is possible to proof the bounded real lemma for $H_\infty$ performance with the following procedure?
The $H_\infty$ performance is defined as: \begin{align} \parallel G_{ru}\parallel_{\infty} & <\gamma \end{align} where $G_{ru}$ represent the transfer from the input $u$ to the output $y$. It is commonly mentioned that this performance is achieved with the following criteria performance
\begin{align} J_{ru}=\int_{0}^{\infty}y^{T}(t)y(t)d\tau & \leq & \gamma^{2}\int_{0}^{\infty}u^{T}(t)u(t)d\tau\ \end{align}
Then, I suppose that the proof of the lemma can done by deriving a Lyapunov equation through $J_{ru}$ \begin{align} V(x)=X^TPX \end{align}
such that
\begin{align} J_{ru}=\int_{0}^{\infty}\left(y^{T}(t)y(t)- \gamma^{2}u^{T}(t)u(t)+\frac{d\dot{V}(x(t))}{dt}\right)d\tau-V(x)<0 \end{align}
Then by deriving the Lyapunov equation the typical representation of bounded real-lemma can be achieved. My question is about if this proof is correct?. Before hands thanks for your answer.