DualShock 4 suddenly turns off after connecting to iPhone

Solution 1:

It works now! I’ve found out that the problem was the controller battery.

For some reason, I cannot charge the DualShock 4 with my phone charger... I’ve tried with an iPhone (and also with an iPad) charger for a couple of hours and the battery was not charge. Then, I connected it to my Mac USB, and an orange light was lit on the controller. I left it charging overnight, and the next morning I was able to successfully connect it via Bluetooth to my iPhone.

So that’s why the controller would quickly disconnect and turn off, because the battery was low.