AN INTL B747 PILOT REPORTS A PILOTAGE ERROR AFTER 20 HOURS ON DUTY FOLLOWING A MIN REST PERIOD.

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WHILE LEVELING OFF DURING A DSCNT TO 11000 FT FOR AN APCH INTO ZZZ1; THE ACFT BEGAN TO SHUDDER. I APPLIED PWR; BUT COULD NOT ARREST THE SPD DECAY WITHOUT LOWERING THE NOSE; WHICH CAUSED ME TO DSND 500 FT BELOW THE ASSIGNED ALT. WE WERE TOLD TO CONTINUE THE DSCNT TO 10000 FT AT WHICH TIME I REALIZED THAT THE SPD BRAKES WERE STILL OUT; CAUSING THE PROB. THE ENTIRE CREW WAS SO FATIGUED FROM OUR ALMOST 20 HR DAY THAT NO ONE RECOGNIZED THE PROB. I FEEL THAT CREW FATIGUE PLAYED A MAJOR ROLE IN THIS INCIDENT. THE TRIP FOR ME STARTED ON NOV/XA/07. THIS LEG STARTED 17 DAYS LATER WHEN MY AUGMENTED CREW (3 PLTS AND 2 FE'S) WERE TO OPERATE ZZZZ1-ZZZZ2. THE CREW STARTED OUT RESTED FOR AN XA30L DEP BUT DUE TO MAINT PROBS THAT WERE KNOWN PRIOR TO ALERTING THE CREW; THE CREW RAN OUT OF CREW DUTY DAY BEFORE THE MAINT PROB WAS FIXED. THE CREW RETURNED FOR MINIMUM CREW REST (10 HRS). WE WERE THEN ALERTED TO OPERATE THE 2 LEGS SCHEDULED FOR A 19 HR DUTY DAY. WHEN WE RETURNED TO THE AIRPLANE; NO ONE WAS REALLY RESTED. I MYSELF ONLY WAS ABLE TO GET 1-2 HRS OF SLEEP. UNFORTUNATELY; PLTS ARE HUMAN BEINGS AND CANNOT SLEEP ON COMMAND. WE WERE JUST GETTING SLEEPY WHEN IT WAS TIME TO WAKE UP AGAIN. THERE MUST BE A BETTER SYS TO ACCOUNT FOR HUMAN FACTORS.