How to Reconstruct the System's Dynamics by Differentiating Interval-Valued and Set-Valued Functions

To predict the future state of a physical system, we must know the differential equations ẋ = f(x) that describe how this state changes with time. In many practical situations, we can observe individual trajectories x(t). By differentiating these trajectories with respect to time, we can determine the values of f(x) for different states x; if we observe many such trajectories, we can reconstruct the function f(x). However, in many other cases, we do not observe individual systems, we observe a set X of such systems. We can observe how this set X changes, but not how individual states change. In such situations, we need to reconstruct the function f(x) based on the observations of such “set trajectories” X(t). In this paper, we show how to extend the standard differentiation techniques of reconstructing f(x) from vector-valued trajectories x(t) to general set-valued trajectories X(t).