Final answer:
The castles built in Northumbria by William I after 1069 served the purpose of establishing and maintaining Norman control over the region.
Step-by-step explanation:
The purpose of the castles built in Northumbria by William I after 1069 was primarily to establish and maintain Norman control over the region. These castles served as military fortifications to protect the newly conquered territory from potential rebellions and invasions. Additionally, the castles were used as administrative centers to exert Norman authority and collect taxes from the local population. Examples of such castles include Newcastle Castle and Durham Castle, which were strategically located near the river Tyne and Durham Cathedral respectively.