The House of Savoy ruled across the Alps a decently unfied, if relatively small, domain, for many centuries. So did the Swiss Confederacy, the Three Leagues and of course the Habsburgs, among others. Control of the passes was critical and very often polities stretched on both sides of them (as Switzerland still does in Ticino) though it was rarely a very centralized thing. A long-lasting, centralized state across the Alps is very possible IMHO, considering the regularity with which the Holy Roman Emperors and the French, among others, crossed them.