As an econ major, I can't stress how terrible this idea is. Rent controls do not make escalating rent problems better, they make them worse. Sometimes far worse.
Long story short, rent control is a manipulation of the market where demand exceeds supply (see image below). This isn't a guess or assumption, it's a science. With rent controls, there will be fewer rental options available and a greater amount of demand for those remaining units. The stronger the controls, the greater this effect. If you have ever wondered why 50-100 people are all applying for one rent-controlled apartment in SF, this is it.
You may think that if you are already in an apartment, that this will not effect you in a negative way. That couldn't be further from the truth. The stronger the gap between the market and the current rental price, the more likely your landlord will be doing everything possible to get you out of the building. Likely this will include include minimizing maintenance, refusing to do any improvements, trying to convert multi-family rentals to condos, and a variety of other items that I won't even go into. As soon as your are out of the apartment, you will be hit with far higher rents for lower quality units that more people are competing to get (again, SF). Many people will resort to "black market" apartments or tenement housing. Some will become homeless.
The damage does not end there. Rent controls will reduce the market value of all apartments affected. That means lower property taxes for the City of San Jose, and less revenue for services--including the already strapped SJPD and SJFD departments. Landlords will have less incentive to maintain their properties (in fact the opposite when trying to kick people out), leading to substandard housing and blight in both urban and suburban areas.
Thankfully, it seems like Mayor Sam Liccardo does not share the same sentiment on rent control as Raul Peralez. It's a very difficult topic to stand up to, because most voters will not understand economics and the topic of rent control is completely counter-intuitive. Sorry for the soapbox moment, but I really believe this will cause substantial damage to the rental market as can be seen in every single city around the world with strict rent controls (especially SF and New York).