Please share with me the model number of your Hitachi television.
Can you tell me if the power light blinks and if so how many times when it shuts down? I have my suspicion what is causing your issue but this would assist with my final assessment.
I think you are facing two potential issues but because your TV still has video and sound sporadically it's harder to pinpoint. I believe you have an overheating issue which is causing the shutdowns. Your TV has cooling fans inside, very much like the ones in computers. These go bad, get clogged with dust and stop working, causing your boards to overheat. Unfortunately your TV was also manufactured with a long batch of TVs that had bad power supply boards. Not just Hitachi but many other plasma makers since they all share similar manufacturing plants. Most likely you have capacitors slowly dying on your power supply board, they would look bloated and/or leaking. These CAPS were undersized for the high voltages.
If you are having a cooling issue this will also most likely accelerate your failing power supply board. I would recommend you contact Hitachi for the nearest authorized repair center. I believe this issue will compound to the point where it will no longer turn on. Certainly you can attempt this yourself at your own risk but more often I see many folks wreck their TVs further and since your TV is not totally dead, I think its at least worth having them diagnose it. Then you can try and fix it yourself or let them repair the unit. I would start with Hitachi and go from there.
Parts can be purchased typically at www.partstore.com or www.shopjimmy.com if you decide to pursue it yourself. If you try home remedies be extremely careful and unplug your TV for at least 2 hours before opening. There is VERY high voltages inside that unit. It could harm or kill you if touched incorrectly. The power board can be inspected for bad CAPS as I mentioned above but if you honestly do not know the difference between a CAP and a resistor then you should seek professional assistance. Some professionals have been able to desolder and solder new CAPS on the power board but you would require some excellent soldering skills. Most replace the board.
Also I want to add, I suspect your DVD player sees this more frequently as the draw from your DVD player sending video through your input connections is stressing the boards further. Therefore, the components on your TV overheat and shutdown faster. That is mostly a hunch on my part but since these inputs are tied to the same board, that additional workload could definitely cause it to shut down quicker.