#### Title

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

#### Publication Date

2011

#### Abstract

To predict the future state of a physical system, we must know the differential equations *x *= *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)*