I develop a device for my purposes which based on Android Things, but I've got a problem with integration with a smartphone application because I want to control it by smartphone. Maybe someone has solved this problem, what is a better way to integrate it? Maybe using Firebase Realtime DB or push notifications, because the writing of back-end for this device will be overengineering?
You can do it in a few different ways:
You can see some examples on this repo, under chapter 7: https://github.com/PacktPublishing/Android-Things-Quick-Start-Guide
That chapter of the book covers the 3 options in details and goes over explaining the code of the examples.