So does it lock up with the current app still displayed on the screen, just frozen? Or does it appear to instantly power off, and require power to be applied before it will go through the boot up process? The iPads are not jailbroken, right? I have never heard of this, or seen it on my devices. Your iPad 2 must be a couple versions back; 9.3.5? What is the version on the iPad Air gen 1? My guess is that there is one really sketchy, misbehaving app on both devices, and that is the linkage to them both having this problem.
Try this: use your devices for their usual purposes, but on one of them be super attentive to clearing out apps after using them (double click home button, then swipe up the unused apps to clear them out of memory). Also, on that test device try turning off Background App Refresh (found under Settings > General). See if that makes a difference.