One scheme used in some European cities is to let the prices float freely, but kick in a subsidy for existing residents when rent in an area increases by more than a certain % of their salary, softening the impact of market shifts on neighborhood turnover. This of course has the impact of increasing prices for newcomers relative to a situation where such a subsidy didn't exist, because fewer existing residents will be forced out due to rent increases, thereby freeing up fewer empty apartments for newcomers to rent. But that's the goal of the program, to reduce the number of existing residents unwillingly forced out of their neighborhood by rent increases, by shifting the margins a bit. With some kind of actual quantitative assignment of utilities it'd be possible to rationally design a subsidy program that buffers the impact of large changes without neutralizing them or setting up a long-term two-tier market (for example, subsidies can cover less than 100% of increases, and phase out over time).
I can't see any problems with this plan.