There is absolutely no solution to this problem. I've open my controller about 7 times, replacing the analog itself and a bunch of other tomfoolery (didn't work, unless the replacement was as messed up as the last one); the problem is most likely the sisterboard inside the controller itself. You can try opening the controller yourself and taking some canned air to it if it's just dirty from dust or something, maybe replacing the analog if you're skilled in soldering and have another one available.
Otherwise, your best bet is return the one you have now, and get another controller. There is no solution for this problem beyond that.
I am having this issue with the left analog stick on the Dual Shock 4 controller I received at console launch in the US. It's very bad and I won't be able to play any more games until I get a new controller. I've tried resetting the controller, I've tried using it only while plugged into the console with the usb cable.
The left stick just seems to alternate beteen jittery response or intermittent non-responsiveness. Noticed this while playing through Infamous, but it's occuring on all games. You really canot notice this within the menu system or on the dashboard, this occurs mostly within games.
I started having this issue today as well. One of my controllers stops responding when I hold it back then around the 3 oclock or forward position. My character completly stops responding unless I move back again.
Yeah I've just noticed this problem a few days ago. It's very noticable in games such as BF4 and COD: Ghosts when I'm tyring to sprint and my character stops all of sudden after only a few metres. Makes playing my PS4 a nightmare at this moment in time and I'm now unsure if Sony will fix or replace the controller.