Government price controls have a 4000 year history of miserable failure. See "Forty Centuries of Wage and Price Controls."
Except this still will not help due to the economic forces driving urbanization. Silicon Valley is a self-fulfilling prophecy and creating more supply will only help to drive more demand.
Every new person newly moving into a heavily urbanized area will drive 0.5 more persons wanting to go there due to new network and auxiliary jobs created (groceries, haircutter, ...). As more than 70% of our jobs are by now defined by interactions with other people, urban centers of excellence are doomed to grow.
This is why building new housing is only ever a short-term solution as well. The only long-term solution to urbanization and gentrification is making these areas less attractive by offering less jobs in a major area.
This is empirically false:
https://dwtd9qkskt5ds.cloudfront.net/blog/wp-content/uploads...
http://blogs-images.forbes.com/trulia/files/2014/02/TruliaPr...
QED.
Home prices aren't being driven up by developer greed or predatory landlords, they are stabilized at a rate acceptable to the market. In this case, since the availability is limited you are allowing a small percentage of the total population (those with outsized buying power) to define what is acceptable.
Rent control (alone) just pushes the problem down the road. Build more and everyone will be happy...eventually.