A stupid question: whether ther exists a universally recognized definition of when two differential graded algebras should be Morita equivalent? I mean a sort of equivalence which would incorporate the usual Morita equivalence of algebras (when there is no differential) and the usual homotopy equivalence of differential graded algebras induced by an $A_\infty$/-quasi-isomorphism. In particular, I would like to believe, that the cyclic cohomology of such algebras should be canonically isomorphic via a generalised trace map.
I believe, this should exist, but a short search on Google rendered only papers on derived Morita equivalence, i.e. as much as I can judge, on equivalence between derived categories of algebras (non-differential). If I am mistaken and this is actually what I need, please, do explain it to me! Or if you know a good reference on the question I am asking, please, do share it with me!