Rent control definitely works, because owners lock in the original price of the building with a fixed-rate mortgage. If you buy a building in 1960 to rent out, you can rent it out at 1960s prices pretty much forever, with only a tiny fraction added to cover increasing property taxes (subsidized for owners of rent-stabilized apartments) and the increasing costs of routine maintenance.
If I were to become Dictator of New York City tomorrow, I think I'd freeze rents as a factor on the acquisition cost of the property + prevailing interest rates on 30 year fixed mortgages. If you want to make today's market rates on your property, you have to build a brand new building. This incentivizes the creation of new housing, and provides stability to existing tenants through rent stabilization even for newcomers to the city.