Hi Natasha,
Reversibility is generally implied in thermodynamic vs kinetic control in this way:
Usually for a reaction to be kinetically controlled, it should be irreversible.
Usually for a reaction to be thermodynamically controlled, it should be reversible.
But normally I don't relate those two things to determine the kinetic/thermodynamic control of a reaction, I just look at what type of species is being formed, which I find is the most reliable way of determining the reaction control.
I hope this helps,
Katt