"This is complete utter nonsense. A quick back-of-envelop calculation: The sun has an angular diameter of 0.0093 radians, so "focusing" the sun's rays from a distance is optically impossible"
Hmmm - never heard of parabolic reflectors???
It is perfectly possible to make a parabolic reflector that has a focal distance of several hundred kilometres. Besides, it has already been done - see the earlier reference in these comments to the Russian experiment.
Smoke & Mirrors
Concerns of oriental overlords with death rays aside, a dish that size would be awfully good at receiving comms transmissions. Be they from low power devices such as cell-phones or tight beam microwave links as demonstrated way back in 1975 by the Aquacade/Rhyolite spy-sats
He needs to get a hat....
This does seem to be attention seeking behaviour, and with not much chance of success, to say the least.
If it would help the guy deal with what is clearly a real problem, I'd be prepared to design a hat with a high brightness display screen, (possibly wrapped round into a cylinder, like a top hat?) that could be used to display meassages along the lines of:
Look at me!
The Tw*t in the Hat
Depending on the depth and optical density of the clouds, cloudy nights would just mean that the target area was lit by a diffuse glow - as would, to a lesser degree, a wide area around it.
Practicalities aside, I'd be intrigued to know what effect regular sunlight at the planned intensity would have on local flora. Would photosynthesis still occur, albeit at a much reduced rate (bearing in mind that its quantum effects that are involved here, so a photon of the right energy should still do the trick even if there are way less than usual numbers of the things incoming)? Would it play hob with plants natural cycles?
I still think it's a bad idea, albeit not utterly unfeasible, anyway!
Even if you ignore the issues of orbital mechanics, keeping a giant mirror aimed at one spot on earth, and the logistics of getting the mirror up there, 310 miles up is not high enough. That's only 4% of the Earth's diameter. So when it's night time in Chengdu, the "moon" will be in the Earth's shadow, except for a few minutes around dusk and dawn.
310 miles is too low
Geo synchronous is too high - and a silly idea anyway
What you want is something mid range, - GPS have about a 6 hour orbital period, I imagine an 8 hour period would work well so long as you use multiple units to hand off duties as the geometry progressed.
The whole system would need to be dynamic, not a static mirror sitting over the target but multiple satellites radically changing orientation continually through out their journey. Think intricate clockwork.
This would have the advantage of being useful for more than a single city
but it is still a really bad idea from an environmental perspective - it will burn out every organism that regulates its life from a lunar cycle - and the lightspill pollution will be far more widespread than any ground level illuminated site.