Notation of inf In this paper (equation 4.1) the following formula is listed:
$\inf_{u \in R} \left \{ \frac{\partial V}{\partial \boldsymbol{x}}f(\boldsymbol{x},u)  \right \} < 0, \quad \forall \boldsymbol{x} \neq \boldsymbol{0} $
Now I don't understand what the term $\inf_{u \in R}$ indicates. I know that inf stands for infimum, but I can not make any sense out of this notation.

The subscript gives context to the infimum.  You could also write it as
$$
\inf\left\{\frac{\partial V}{\partial x} f(x,u)\ \bigg| \ u \in R\right\}
$$
