Is it obligatory for one who doubts whether he performed the prayer after falling asleep and forgetting whether he prayed it or not, especially if he suffers from obsessive whispers (waswas), to repeat it?

The treatment for obsessive thoughts is to disregard them and not pay attention to them. If you are certain that you slept during the prayer before completing it, then you must make it up. However, if you are unsure whether you prayed it or not after the time has elapsed, you may take the easier opinion as long as you are afflicted with obsessive thoughts. Nevertheless, making it up as a precaution is more preferable.