For a given spacecraft, in a given LEO orbit, assuming you have all the required data, how do you choose the needed burnout speed and timing (or position in orbit), in order to reach a certain point on earth with a ballistic trajectory?
In reality one re-entry burn would never suffice to reach a specific point. The most you can aim for is within a certain area (e.g. splash down in the pacific) but the uncertainties in your actual impact position will be very large. Either you have to regularly perform corrective manoeuvres or you make sure your spacecraft just burns up and never reaches the ground.
Now assuming you have a VERY accurate model of the atmosphere (including wind) and precise knowledge of the attitude of your satellite, the process of determining the required burn duration and direction is purely numerical optimization. The equations of motion of a satellite re-entering are very non-linear and there is no analytical solution.
In reality it will be a combination of numerical optimization to calculate the re-entry burn and then a sophisticated control system to keep the satellite as close to the reference trajectory as possible.
Edit: In case of spaceplanes this is not valid as they can just glide towards the spot they want to go to and have much more control over their trajectory.
Not really a scientific answer, but it should help with the question:
An uncontrolled re-entry is mostly unpredictable. If a satellite on a low earth orbit experiences drag from atmospheric friction, predicting re-entry is almost impossible. The drag depends on tons of factors, such as sun activity and weather, and changes constantly. Since the satellite orbits around the whole earth once every ~90 minutes, if you are wrong by 45 minutes you're already on the other side of the planet (plus some shift from the earth's rotation). The closer you get the better you can calculate, often based on live measurements. That's why it's impossible to know where a decaying satellite will hit even the day before it crashes.
Controlled is a bit different. Assuming you don't have a strongly elliptic orbit, if you burn retrograde (i.e. against the orbital flight direction - basically "braking"), you create a perigee (i.e. lowest point in orbit) on the exact other side of the object you are orbiting. With a strong enough retrograde burn, you can lower that point deep into the atmosphere or into the planet. Time it right and you can create a point of re-entry by your choosing - it's not very precise, but you can at least reliably hit a specific area of an ocean of your choosing or, say, siberia.
Space Shuttles do that and add control over atmospheric flight to the mix, which gives them enough control to land on a runway.
The way spacecraft do this is via one reentry burn using the best estimates of the atmosphere, and doing some slight corrections to the path while reentering. The best example of this is the Space Shuttle, which could precisely land due to the aerodynamic surfaces, however, every spacecraft has some ability to steer itself inside the atmosphere. It is very rare to have a purely "Ballistic reentry", however, they were sometimes done, particularly in the early days of the space program.
Basically orienting the capsule will allow it to slightly change its direction. Soyuz can land with a 28 km accuracy. See How does the landing accuracy of Dragon (under parachutes) compare to Soyuz? for some comparisons to Dragon, and some raw values.