During the centuries of Roman history from Republic to Empire the paterfamilias' authority?

You do not actually say what you want with your question. You just make a statement.

Pater familias means father of the family in Latin. It referred to the oldest man in the family being the head of the household. He had complete control over the whole family. He was the proprietor of the assets of the household, had sole legal right to carry out household property transactions and was responsible for household debts. He had authority over all household members. He also had responsibilities to his family and the community. He had the duty to ensure the moral propriety of his household and maintain its well-being, raise healthy children and honour the ancestral god of the family. He had the power of life and death over family members, but it was seldom exercised.

The authority (podestas) he had over his wife depended on the form of marriage they contracted. In the Early Republic in a marriage cum manus (with hand) the wife was 'handed over from the podestas of her father to that of her husband. If the man divorced her, her dowry went back to her and her family. By the Late Republic this type of marriage was rare and the woman remained part of her family. Women could be emancipated from the potestas of a paterfamilias and were legally independent (sui juris) but had a male guardian appointed to them.

The filii familias (children on the family) remained under the podestas of their father and could not acquire the rights of a pater familias while he lived. Any property they acquired was for the family estate. The pater familias had the power to sell his children into slavery, but if a child had been sold as a slave three times, he was free from his posestas. The sons succeeded to the status of pater familias over their own households (pater familias sui iuris) when the father died even if they were only in their teens. Children emancipated by a pater familias were effectively disinherited.

Over time, the authority of the pater familias decreased. His rights became theoretical and were not enforced. The power over life and death was abolished, the right of punishment was moderated, and the sale of children was restricted to cases of extreme necessity.