One common scheme for controlling a register booster fan is to detect a temperature difference in the ductwork. If the temperature drops while in "AC mode", the HVAC is blowing cold air. If the temperature goes up while in "heating mode", the HVAC is blowing warm air.

I wanted to see how simple this idea would be to implement. I put a temperature sensor inside the ductwork in one of my upstairs rooms. I also have a smart thermostat from which I can find out exactly when the AC or heat is on. (It's a warm summer here, so it's AC.) Both of those devices are integrated into my Home Assistant server. Here are graphs of a few days or so of both.

First, the temperature inside the ductwork:

Next, the smart thermostat activity.

The solid blue areas are the times when the AC is active. It lines up nicely with the temperature drop of 15-20 degrees (F) in the ductwork. The smart thermostat also has a control to let me run the fan some configured percentage of the time for circulation and filtering purposes, but that is not reflected in the graph.