I don't see why the following three forms of the LQR optimal control problem are equivalent:
For $\begin{cases} x'=Ax+Bu \\ x(t_0)=x_0\end{cases}$ find
$$\min_{u\in L^2(t_0,T; \mathbb{R}^m)} J(u)=\frac{1}{2}\left\{\int_{t_0}^T ||Cx||^2+||u||^2 dt + ||Rx(T)||^2\right \}$$
$$\min_{u\in L^2(t_0,T; \mathbb{R}^m)} J(u)=\frac{1}{2}\left\{\int_{t_0}^T x^TQx+u^TRu+2x^TNu\ dt + x(T)^TPx(T)\right\}$$
where $Q,P$ are positive semi-definite matrices and $R$ is positive definite.
- $$\min_{u\in L^2(t_0,T; \mathbb{R}^m)} J(u)=\frac{1}{2}\left\{\int_{t_0}^T x^TQx+u^TRu\ dt + x(T)^TPx(T)\right\}$$
where $Q,P$ are positive semi-definite matrices and $R$ is positive definite.
I found them posed in all 3 versions but I don't see how to pass from one form to another.
The square of the 2-norm of a vector $v$ can be expressed in many different ways, such as $\|v\|^2$ but also as $v^\top v$. Equation 1. in your question uses the first notation, where as the other two the second. Namely equation 1. can also be written as following when using the second notation
\begin{align} \min_{u\in L^2(t_0,T; \mathbb{R}^m)} J(u) &= \frac{1}{2}\left\{\int_{t_0}^T (C\,x)^\top (C\,x) + u^\top u\,dt + (R\,x)^\top (R\,x)\right \} \\ &= \frac{1}{2}\left\{\int_{t_0}^T x^\top C^\top C\,x + u^\top I\,u\,dt + x^\top R^\top R\,x\right \} \end{align}
So when comparing this to equation 2. and 3. it can be noted that $Q$, $R$, $N$ and $P$ should be equal to $C^\top C$, $I$, $0$ and $R^\top R$ respectively.
It can be noted that 2. is a slightly more general form of 3. and that integral can also be written as
$$ \min_{u\in L^2(t_0,T; \mathbb{R}^m)} J(u) = \frac{1}{2}\left\{\int_{t_0}^T\begin{bmatrix}x \\ u\end{bmatrix}^\top \begin{bmatrix}Q & N \\ N^\top & R\end{bmatrix} \begin{bmatrix}x \\ u\end{bmatrix}\,dt + x(T)^\top P\,x(T)\right \} $$