In every iteration of Logan's Run, society is ruled by a computer which controls the population of the world, and tracks the runners who run from the required death time.
In the novels, the computer is called the Thinker and is based within the completed Crazy Horse Monument, in the Dakota Mountains of what was America. The Thinker controls maze cars, hell cars and Deep Sleep Operatives.
In The Movie, a feminine-voiced computer controls the city. The computer regularly communicates with Sandmen, who are required to check in with the computer after successfully completing a mission.
In the film, the computer tasks Logan with a special mission: to infiltrate and destroy the Runners from within, and to find and destroy Sanctuary. To that end, the computer alters Logan's lifeclock so that he can become a Runner himself. The process is referred to as "Procedure 033-03".
When Logan returns to the City and reports that there is no such thing as Sanctuary, the computer attempts to interrogate Logan (using holographic surrogates) but is ultimately unable to deal with this logical contradiction, and self-destructs.
In the TV series, the city is under the control the cabal of elderly citizens. The council is made of Elite citizens of the city who have been given the privilege to live beyond the age of 30. In the Pilot of the TV series the cabal of elderly citizens offers Francis 7 life after the age of 30 as an elder.