I cannot speak to the specific example of the Roman Empire under Trajan, but in the case of places/times like ancient Egypt and various powers in the ancient Near East there are references in texts from various time periods that specify locations of battles, places that were required to pay taxes, etc., which show that those places were under the control of the Egyptians (or whomever). In addition, archaeological evidence can show influences on material culture (for example changes in styles of pottery, burial practices) due to political changes (and/or trade, migration, etc.).