Some rent control in this country has been going on since the beginning of World War II, and the legal basis for the rent control is that it is to protect from shortages in housing caused by the war.
The best way to analyze the effect of rent control on housing would be to plot the housing shortage vs the time since controls were instituted. The problems will always start, not when the controls are imposed, but when society (a.k.a. “the market”) realizes that the controls are likely to be imposed. And they will get worse with the passage of time.