has any work been done on the iPad? Screen replacement? Any water damage or drops?
Every time I have seen the issue you are describing, the digitizer or the IC chip was bad. An easy way to tell is when the screen goes dark and unresponsive, hold down the power button. If the screen lights up with the "power off" slider, you know you have a bad digitizer or IC chip.
If that's not it, then try the usual things. Hold down home and power until the device resets. If the issue persists, connect it to your computer, hold down power for 5 seconds, press the home button (while still holding down power) for 15 seconds, then release the power button (while still holding down home) until the "connect to itunes" screen appears. Restore the Ipad and see if that fixes it. If you STILL have the problem, follow the same steps but set it up as a new iPad.