Control function for rough paths ($\omega (s,u)+\omega (u,t)\le \omega (s,t)$)

A control is a continuous nonnegative function on ordered pairs of times, zero on the diagonal and superadditive as displayed. It measures the size of increments of a rough path, typically through $|X_{s,t}^{(k)}|\le \omega (s,t{)}^{k/p}$. Continuous finite variation gives suitable controls, which let local estimates be stated without tying them to a particular time parametrization.