But on the the physics! Capacitors discharge exponentially when put to a load. That is to say, the lose their charge very rapidly! And if that load is, say, a short circuit, they lose their charge explosively (literally). How does one use capacitors for energy that must be used over a long period of time? I'd have to guess that some sort of voltage controlled impedance load that would vary the load between a very high state to an operating state to allow the capacitor to discharge in pulses of current or some such thing. Probably something completely different would actually work but that's my gut reaction. Still some science to be done!
Most or all of these applications will feature a DC-DC regulated converter that isolates the ultracap from its load. When discharging, the converter ‘makes up the difference’ as the cap decreases in voltage.
However, the efficiency of power conversion, which may be as high as 90% at high cap voltages, does indeed decrease as the cap voltage decreases, so there is a limit where the converter has to say, “no more juice available—shutting down now.”
Regenerative recharging of the cap requires yet another DC-DC converter, hooked up opposite the the discharge one. Their mutual operation must, of course, be carefully controlled. [They probably have designs which can share components between the two functions to decrease cost and size.]