Slang or idiom for submissive [closed]

Try doormat

one that submits without protest to abuse or indignities.

An even better word is kowtower

 someone who agrees too easily or eagerly to do what someone else wants you to do : to obey someone with power in a way that seems weak.

But the best one is he/she is my bitch

A submissive person (often female), who does what others want.

Pushover

a person who is easily persuaded, influenced, or seduced.

You could also just refer to puppet for your "controlled by" sense:

a person, group, state, etc, that appears independent but is in fact controlled by another

• someone's bitch (noun) could be used (where someone is a proper noun), though I don't like this phrasing
• One might say a person is whipped (adj.), particularly if referring to a romantic relationship
• pushover is probably the best noun