For $x: \mathbb R\rightarrow \mathbb R^n$ and $u:\mathbb R \rightarrow \mathbb R^m$ we have the constrained ordinary differential equation (ODE) \begin{align} \dot x(t) &= f(x(t),u(t)) \tag 1\\ 0&\geq h(x(t),u(t)) \tag 2 \end{align} where $f:\mathbb R^{n\times m} \rightarrow\mathbb R^n$ and $h: \mathbb R^{n\times m}\rightarrow \mathbb R^n$.
From the notation I guess the domain of $f$ and $h$ are matrices, i.e. $\mathbb R^{n\times m}$. But is that not a typo? Should it not be $+$ instead of $\times$? I mean the following:
$f$ and $h$ are a compositions of $x$ with the range $\mathbb R^n$ and $u$ with the range $\mathbb R^m$. Therefore \begin{align} f&:\mathbb R^n \times \mathbb R^m = \mathbb R^{n+m} \rightarrow \mathbb R^n \tag 3 \\ g&:\mathbb R^n \times \mathbb R^m = \mathbb R^{n+m} \rightarrow \mathbb R^n \tag 4 \end{align} I.e. the cartesian product of $\mathbb R^n$ and $\mathbb R^m$.
Is this correct or am I wrong?