Distinguishing a LTI from not with unknown inputs

Linear time invariant (LTI) systems are a staple of physics. They appear in many situations. But how do you know a system is a LTI?

In particular, if you are provided with a black box which responds to some input I(t) yielding some output O(t), and you do not know I(t) or O(t) for t=t0, can you determine whether the black box is an LTI? (You may use your observations to craft later inputs: I(t) may depend on O(x) for x on [t0, t) )

If it cannot be done, are there theorems which provide bounds on the nonlinear effects?

• Are you allowed to observe for an infinite amount of time? If not, then any non-LTI system which behaves exactly like an LTI for some arbitrarily long amount of time (say, 7 billion years) before suddenly switching its behavior would be indistinguishable from an LTI system. – probably_someone Mar 2 at 17:29
• Can you apply inputs of your choice? – user45664 Mar 2 at 21:15
• @user45664 yes, after some specified t0. After that, you can construct any input you please – Cort Ammon Mar 2 at 21:17
• If you are looking for mathematical rigor, the answer to your question is certainly no. Practically speaking however, the way you could check in a reasonable manner is to find its impulse response and then use that to invert the action of the black box. This will fail if the system is not an LTI, but would work if it is one. – KF Gauss Mar 2 at 21:52