Is it permissible for a brother to pay Zakat on behalf of his imprisoned brother, and what is the ruling if the imprisoned person does not return the paid amount after his release, and is it permissible to pay on his behalf with the intention of not claiming the money back afterwards?

Paying Zakat (definition) on behalf of a prisoner without his permission is not valid for him, whether the payer intended it as a donation or to seek reimbursement from him. This is because Zakat is an act of worship that requires intention from the one obligated to pay it. If the person on whose behalf it is paid does not give permission, the payment is not valid on his behalf. Similarly, if someone pays Zakat on behalf of a person who has not authorized him, it is not valid for that person, based on the hadith (definition): "Actions are but by intentions."