To fall asleep употребляется в прошедшем времени.
I fell asleep last night/yesterday/last month, где fell это Past Simple от глагола to fall.
Если мы говорим в настоящем времени, ссылаемся на свой режим сна или на то, что мы делаем каждую ночь, то используется форма fall asleep где fall это Present Simple от глагола to fall.
I always fall asleep after reading a chapter of my book. (это то, что происходит каждую ночь и является частью повседневной жизни).
Примеры:
I went to sleep after midnight last night, I was very tired this morning.
He goes to sleep at 10 PM every night and wakes up at 7 AM every morning.
“Did you go to sleep late last night?” asked his mother.
I always fall asleep while watching the television.
When he does a late shift at work, it takes him a long time to fall asleep when he gets home.
The two boys were so tired after the birthday party that they fell asleep on the couch without having their dinner.
“I fell asleep on the train and missed my stop last night.
I had to walk an extra 20 minutes to get home” she complained to her colleague the next day.
