From my understanding Bitcoin solves the Byzantine's Generals Problem under the assumption that there's no attacker that control more than 50%+1 of the mining power and this has been studied and proved in academic papers.

But what about Proof-of-Stake? I see several crypto (not just Ethereum) using or planning to use PoS. But does it work? Has proof-of-stake been studied in scientific papers and proved to actually solve the same problem that Bitcoin solves? Under which assumptions?

I understand that PoS is not wasteful like PoW and that PoS apparently allows for faster blocks, but if PoW solves one problem that PoS doesn't solve, PoS ain't an option to replace PoW because they're simply not doing the same thing?

Bitcoin can actually be attacked by a 25% attack via selfish mining. The use of uncles in Ethereum make it less appealing to centralize and also less profitable to monopolize the blockchain with 25% of the hashing power.

There is at least this post by vbuterin in which the properties of the consensus-by-bet mechanism are analysed. This post is specifically the mechanism planned for Casper's proof of stake for securing the Ethereum blockchain and is applicable to other consensus-by-bet-based proofs-of-stake systems (but not all proofs-of-stake systems).

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