I have a 2002 ST4S and I had big problem with it's temper that the engine can start sometimes but sometimes not. I have to walk away having a coffee and most of the time when I come back , the engine starts untill even this could not work after the battery is empty out , recharged, and empty out.
I tool the bike to Ducati dealer and test it on computer, nothing abnormal was found.
They changed the relay and the sensors for me. Now, it's temper is a lot better. It does not start sometimes but after a few teimes of trun the key on and off, it starts. Seems like the problem is still there but just not as bad as before.