How do GPS/Glonass/Galileo satellite know their position?

There are two different issues here.

The position of the satellite is a complex calculation based on a network of 400+ ground stations (IGS) and orbital characteristics (from keppler's laws). The orbite is computed with a precision of a few decimeters based on 25 stations, and this information is sent back to the satellite.

The base stations are not used to send information to the GPS satellites. However, it allows for the correction of the errors of propagation through the ionosphere and the troposphere, as well as the residual errors on the orbites. Those errors are systematic and can be considered as identical if the base station is close to the receiver. Because the position of the base station is known, you can infer the errors and remove them from the mobile receiver signal.

I recently came across an article in GPS World stating that some of the more recent vehicles launched into space had small special mirrors attached to the satellite's exterior. Repeated laser beams sent from the earth's surface would reflect off the mirror, providing a more accurate estimate of the vehicle's location.