[verum ago]

maybe they aren't balloons per se, but just inflatable foil mirrors. but right -- why bother with anything inflatable when you can just unfold a big framework/foil mirror

[anymouse]

Dynamic stability and control. You don't need much inflation to rigidize and stabilize an inflatable structure. A sheet of metalized mylar by itself will flop all over the place in space. Make it two layers sealed at the edges with some adhesive at strategic points and a few psi of any gas, and you now have a stable and controllable structure that can even be designed to focus the reflected sunlight to a specific point.