My Thermador PDR304ZS electric oven won't stay on.
(It is part of a dual fuel range.)

It will heat up to the desired (set) temperature
with no problem. But as the oven cools it will never
start up again to maintain the heat at the desired
temperature.

The indicator lights may be of interest:

The "oven on" indicator light is on while the oven heats
up, but it goes off when the oven reaches the desired
temperature. It stays off. (the operation manual says
it should stay on, while the oven is on.)

The "heating" indicator light is on while the oven heats
up. It goes off when the oven reaches the desired
temperature. After the oven cools for a bit, it goes back
on, but the oven does not heat. The "heating" light then
stays on, but the oven never does heat up.

I replaced the control board, but that did not help.
Any help would be appreciated, thanks in advance !

Yes, it that was the part number. control board part 676192.

I did not hear any relay clicking sounds.
But will restart the oven and listen carefully.
The thermador oven fan is pretty loud.

If I do not hear the relay, what does that mean ?

After running the oven again, I do not hear any relay
clicking sound(s) when the oven is supposed to turn
back on. (we do hear relay clicking sounds only when
we first turn the oven on.)

We don't hear any relay clicking sounds.

On some earlier threads I had read about faulty relay
connections, and one of the relay connections was
loose and very carbonized/burnt, so I tried to clean it as
best I could with sand paper. Also on the previous control
board this relay connector was corroded and burnt looking.

No, the relay connector on the new board looks fine.
On the old board I replaced, it looked burnt.