What should a pilot of any large commercial passenger airplane do if they feel sleepy?
What is the maximum time per day during which the pilot may control the airplane ?
If a pilot is not rested enough to safely operate the aircraft, they should not fly. If a pilot is flying and falling asleep, they should switch out with another pilot or land.
Current FAA regulations for domestic flights generally limit pilots to eight hours of flight time during a 24-hour period.
For the Air Force, actual flight duty periods vary depending on the aircraft type and mission but we are required to get 12 hours of crew rest with the opportunity for 8 hours of uninterrupted rest prior to flying.
You can read the FAA's Fact Sheet on Pilot Flight Time, Rest, and Fatigue here.
The first thing a pilot should do if they feel sleepy is Check the CO sensor. Drowsiness is the first sign of Carbon Monoxide Poisoning. Checking for CO poisoning only takes a second, and can be easily mitigated and remedied.
NOTE: Apparently a Pulse Oximeter will not indicate a CO poisoning problem.
Hopefully, it is not CO poisoning, and they just partied to long the night before.