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As mentioned, "studies to prove DRM works" are not really possible. However, "studies to indicate that DRM doesn't cost sales" would be possible. They haven't been done. (Or they haven't been released because that's not what they prove.)
A mainstream publisher could, hypothetically, take a random set of its midlist titles, say twenty or so, and release half of them with DRM and the other half without, and track sales, physical and ebook, over the course of a year, and draw some conclusions. There'd be a lot of variables, and they wouldn't be able to make any absolute statements, but the data might show some kind of trends they could work with.
Over and over, what a casual look at the marketplace says is: Obscurity, not piracy, is the bane of authors. Books people have heard about, sell, and they sell faster if they're more accessible & cheaper. Books people have not heard about don't sell, regardless of what features they do or don't have.
Both types are pirated; there's been no evidence that piracy equates to less sales except for the ridiculous claim that every free download copy is a lost sale.
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