How can rent controls cause a decrease in supply of housing? Do the landlords break down their buildings all because of rent control?
All economists agree that rent control does in fact decrease the supply of housing. As I understand it, you're asking how this happens because you don't see how the supply will go down from what it already is.
The first thing to understand is that this analysis is talking in part about the long term/the future. If rent controls are imposed, people who were thinking about building apartment buildings won't because the profits won't be high enough.
The other thing that can happen is not that landlords tear down their buildings, but that they just stop keeping them up and maintaining them like they would have before. This reduces the supply of decent housing because eventually the apartments deteriorate to being low-quality and eventually to being unlivable.
Rent control refers to legal restriction placed on the maximum rent the land lords are allowed to charge on for the properties rented out by them. A restriction of this kind restricts the profit that landlords earn on their properties rented. Because of this there will be less people interested in buying new properties for renting out to others or even in continuing to let out on rent the properties owned by them. They may find that the trouble involved in renting out is not worth the trouble and may decide to sell off the properties or just keep it vacant.
In this way, most certainly rent control tends to reduce the availability of houses on rent. But this does not mean that total availability of house will decrease. People still need houses, and because of this the demand for houses for self occupation may increase, as many people will be unable to get suitable house on rent.
On the question of breaking down buildings, I am not sure if land lords will breakdown buildings owned by them just because of rent control. Just breaking down the buildings will mean end of the rental income altogether rather than higher income. Thus breaking down of houses should be attributed to the ability of landlords to create better facilities in place of older ones rather than just the low rents of the old buildings.