Is the page going to be offering the same exact content? If yes, then the template should be responsive itself, meaning, your HTML and JS should work in a way indicative to the device viewing the page. Here are some articles on the subject.
(Note: the articles are a bit outdated, so just take the overall gist).
If you are going to offer different functionality for a mobile device, that is when you’d possibly need different controllers. To do that, you could filter for the device’s attributes with something like this, and go to the “mobile” controller(s).
or this
https://github.com/piwik/device-detector
Scott