First of all, no such thing as border patrols in Italy or any other Schengen country, unless you are close to a border (e.g. on the German side of the border to the Netherlands, customs "patrols" are checking pedestrians for illegal cannabis!). Then, the idea of the Schengen visa is that you can travel within the zone. I've never heard that you are free to travel during the day but that you're not allowed to sleep in a different country.

Bottom line, be honest while telling them your plans and, once you get the visa, stop worrying.