Here I do not understand the sentence
Note that even if $L$ does not depend on $u$, the cost $J$ depends on the control $u(\cdot)$ through $x(\cdot)$ which is the trajectory that this control generates.
How can an input $u$ generate a trajectory when it is not present in $L$? It seems that $u$ generates $x$ just by the fact that it is not present somewhere else!