At launch, Google's Stadia Controller will not be usable with a PC or compatible mobile device without the use of a USB connection. While the controller will work wirelessly when playing through a Chromecast Ultra, PCs and mobile devices will require a USB HID connection for the Stadia Controller to work properly. This is odd, as the Stadia Controller connects directly to Google's servers over WiFi and sends inputs to the cloud rather than a local device.
https://www.notebookcheck.net/Google-s-Stadia-wireless-controller-will-require-physical-cable-for-PCs-and-mobile-devices-at-launch.439773.0.html