Notation of inf

Question

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.

Answer 1

The notation $$\inf_{u \in \Bbb R} \left \{ \frac{\partial V}{\partial {\bf x}}f({\bf x},u) \right \} < 0, \quad \forall \ {\bf x}\neq {\bf 0}$$

is the same as: $$\inf \left \{ \frac{\partial V}{\partial {\bf x}}f({\bf x},u)\mid u \in \Bbb R \right \} < 0, \quad \forall \ {\bf x}\neq {\bf 0}$$

That is, the infimum is being taken on $u$.

Answer 2

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