For those who just want a summary, the conclusion is... it's complicated. From the paper:

"Rent controls appear to be quite effective in terms of slowing the growth of rents paid for dwellings subject to control. However, this policy also leads to a wide range of adverse effects affecting the whole society."

So you get the first order effect you wanted "rent stays low in the area" but you get all kinds of adverse second order effects (higher rents outside controlled areas, poorer maintenance, lower new construction rates, etc.)