The way these systems are controlled (in my terrestial experience) is to have the system be capable of cooling below the desired set point. Then you can PID the temperature using a heater or a thermal bridge/heat switch that will cause the system to warm up. It's complicated, and a lot of very interesting thermodynamics involved.

It also depends if you just need "as cold as possible" or if you need to know the temperature and can calibrate out fluctuation, or if you genuinely need stability.

https://bluefors.com/stories/differences-between-pulse-tube-...