The simplest method of control of metabolism is by substrate concentration. So if the concentration of NADH and private is low, lactate dehydrogenase cannot function. When they build up it can and will.
The situation is more complicated than this, because pyruvate will only build up if it is not converted to acetyl CoA by the pyruvate dehydrogenase complex. As I recall this is under allosteric control by the ATP and ADP concentrations which will change In these circumstances. (Links to standard text description to follow.)