For most commercial flight director, autopilot and auto-throttle systems (e.g. the A330-300, B737-800), the flight director/autopilot flies the altitude hold (ALT) or vertical speed (V/S) mode by closing the flight path/altitude loop with pitch control, while the auto-throttle maintains airspeed. It's only in flight level change (FLC/CLB/DES) or go-around (GA) mode that the autopilot controls airspeed via pitch control and the auto-throttle controls a thrust rating.
Closing the loop around airspeed via throttle and closing the loop around flight path via pitch control work fine when the aircraft is on the front-side of the thrust curve. However, this control strategy leads to instability when flying on the back-side. While this scenario is not likely to be encountered in normal flight operations, the best climb speed for maximum weight would be pretty close to the boundary (with some margin built in by the manufacturers).
How do the automatic flight controls handle flying on the back-side? Or do manufacturers ensure that the floor speed/AOA for system engagement always have margin to the minimum drag?