The processes involved in solar fusion have been well understood since the 1930s [1,2]. Hans Bethe won a Nobel Prize for this in 1967. The problem is that one cannot produce stellar densities and pressures in any kind of apparatus.

[1] https://en.wikipedia.org/wiki/CNO_cycle

[2] https://en.wikipedia.org/wiki/Proton%E2%80%93proton_chain