I can think of two ways to do this. The first way is a zero knowledge proof. This is a cryptography concept where you can prove that you know some secret (in this case the private key) without revealing the secret. I am unfammiliar with the specifics of Bitcoin, and suspect that the feasability of this method depends on the type of public/private keys they are using.
The other approach is publish a list of your public keys along with predictions of future transactions. Assuming you actually control the public keys you claim to, you should be able to make those transactions successfully.
Both of these approaches will run into some difficulty with a robbery or loss of keys kept in cold storage. Because your keys are (supposedly) in cold storage, it is not suspicious that you cannot prove control of them. However, if you were to suffer a harddrive failure (and not have backups), then you could simply claim that those keys were still in cold storage.
The other approach is publish a list of your public keys along with predictions of future transactions. Assuming you actually control the public keys you claim to, you should be able to make those transactions successfully.
Both of these approaches will run into some difficulty with a robbery or loss of keys kept in cold storage. Because your keys are (supposedly) in cold storage, it is not suspicious that you cannot prove control of them. However, if you were to suffer a harddrive failure (and not have backups), then you could simply claim that those keys were still in cold storage.