the system E is time-invariant if for each E, each E, and each E, if E is admissible for E then the translation E is admissible for E, and E.
for systems with outputs, it is also required that D be independent of D.
when dealing with time-invariant systems, it is customary to identify controls with their translations.
thus, for instance, given D and D, one thinks of their concatenation D as an element of D, meaning the concatenation D, where D is the translation of D to D.
a system that is not necessarily time-invariant is sometimes called, to emphasize that fact, a time-varying system.