What is the expiation for severely beating a younger brother due to his wrongful behavior, and what is the ideal way to deal with him and contain him?

Striking in this manner is impermissible for someone who has the authority to discipline, so how about for someone who does not possess this authority? You must seek forgiveness from Allah and repent to Him. Part of this entails seeking absolution for the aggressor from the victim, as the Messenger of Allah, peace and blessings be upon him, said: "Whoever has wronged his brother concerning an injustice, let him seek his forgiveness for it, for there will be no dinar or dirham then, before his good deeds are taken for his brother. And if he has no good deeds, then some of his brother's evil deeds will be taken and cast upon him." We believe that you should ask for forgiveness from your father and your brother. A brother like yours, who is approaching puberty, understands the meaning of sin and forgiveness for it.