Dear brother Nine_Two,
In relation to the Syrian Orthodox and the Armenian Apostolic Churches, administratively speaking, no control whatsoever.
In relation to the Ethopian and Eritrean Orthodox Churches, the Coptic Pope has the same role as the Pope of Rome has to the many Catholic Churches. The Coptic Pope is looked to in questions of doctrine, and the patriarchs of those Churches explicitly seek communion from him upon election (e.g., during circumstance of the irregular patriarchal election in the Eritrean Church, the question on everyone’s mind was "who will HH Pope Shenoute extend communion to?). Otherwise, they are autocephalous - but in reality, they function autonomously.
This autocephaly-as-functional-autonomy is the general paradigm of the OO Churches.
For the Armenians, the Catholicos of All Armenians in Etchmiadzin is the Supreme head of the Catholicos of Cilicia, and the Armenian Patriarchs of Jerusalem and Constantinople.
The Syrian Orthodox recognize the Patriarch of Antioch as the Supreme head, and gives the title to no other heads of local Churches. The highest local head is the Catholicos of the Malankara Syriac Church; all other local Churches are headed by Metropolitans.
I hope that helps.
Blessings,
Marduk