What is the formal definition of "memory" in an LTI, continuous dynamical system?

Question

I've heard that dynamical systems have "memory". In discrete time this is represented by the current state depending on the prior states, so that makes sense. But in the continuous time an Linear, Time-Invariant (LTI) system is also said to have "memory". I know that this somehow relates to the fact that convolution is used to solve it, but I'd like to know if there is a deeper definition.

Answer 1

Using you wording, an LTI system clearly has no memory in any reasonable sense.

In mathematical terms, this LTI notion (which, without exaggeration, corresponds to basically $0$% of the theory) means that you have a linear dynamics of the form $x\mapsto e^{At}x$ for some matrix $A$.

In order to have examples with memory in continuous time, we should consider other types of dynamics. For example, something such as $$x'(t)=x(t-1).$$ Note that the equation is linear (first property of an LTI), but it has memory because you need to know what happens in $[-1,0]$ to find out what happens at $0$.

These ideas go back to Vito Volterra (unfortunately not always recognized), and marvelously put in the context of dynamical systems by Jack Hale with the consequent exponential development.

And going back to your question: we usually say that a dynamics has memory if one cannot determine it at time $t$ if we don't know it at some former time.