Arduino Control over the internet
You have two choices:


The Arduino regularly communicates with an external web server, sending data and asking if there is anything to do. This is the way Mitsubishi heatpumps can be controlled from the web. The advantage is that your internet connection does not need to publish a web server. Drawback is that the communication is slow (being dependent on the polling interval), plus the need for a web server to talk to. For instance, to control your Mitsubishi heatpump, you need to create an account on one of their servers for the purpose.
The Arduino exposes a web interface that is then exposed to the internet. This needs configuration of your internet router and preferably a static IP address on the internet. Security is a problem, and I'm not sure that even the Due can do HTTPS plus some form of authentication. I used this method, but fronted my Arduino with an old laptop running Linux and Apache web server. The laptop does the authentication and HTTPS processing. Advantage is near real-time control; disadvantage is the amount of hardware and configuration required.