There are similarities and differences between this TV series and the original film of Logan's Run. The film was set in 2274 and the TV series in 2319.
The best theory I have to link the two is this: At the end of the film, the city was partly damaged after the AI system was damaged by Logan. Many people fled the city and saw the old man standing there. But not all the people left. A small group of men decided that they did not want to die in Carousel, that the city could be changed and made better, so they set about getting it repaired and things were back to normal, but still with Carousel. After leaving the city the crystals in their hands stopped working for most people, so they abolished the crystals.
This small group of men who were then the leaders became the Elders we see in episode 1 of the TV series. They took over controlling the city where the AI had failed.
So now we see them 45 years older, and things are under their control, but still some people believe in Sanctuary and running away. A new Logan 5 and a new Jessica 6, and a new Francis 7, are the stars of the show. Logan and Jessica choose to leave the city, which is now much easier than it had been in 2274.
The best theory I have to link the two is this: At the end of the film, the city was partly damaged after the AI system was damaged by Logan. Many people fled the city and saw the old man standing there. But not all the people left. A small group of men decided that they did not want to die in Carousel, that the city could be changed and made better, so they set about getting it repaired and things were back to normal, but still with Carousel. After leaving the city the crystals in their hands stopped working for most people, so they abolished the crystals.
This small group of men who were then the leaders became the Elders we see in episode 1 of the TV series. They took over controlling the city where the AI had failed.
So now we see them 45 years older, and things are under their control, but still some people believe in Sanctuary and running away. A new Logan 5 and a new Jessica 6, and a new Francis 7, are the stars of the show. Logan and Jessica choose to leave the city, which is now much easier than it had been in 2274.