In this paper, fuzzy initial value problems for modelling aspects of uncertainty in dynamical systems are introduced and interpreted from a probabilistic point of view. Due to the uncertainty incorporated in the model, the behavior of dynamical systems modelled in this way will generally not be unique. Rather, we obtain a large set of trajectories which are more or less compatible with the description of the system. We propose so-called fuzzy reachable sets for characterizing the (fuzzy) set of solutions to a fuzzy initial value problem. Loosely spoken, a fuzzy reachable set is defined as the (fuzzy) set of possible system states at a certain point of time, with given constraints concerning the initial system state and the system evolution. The main-part of the paper is devoted to the development of the numerical methods for the approximation of such sets. Algorithms for precise as well as outer approximations are presented. It is shown that fuzzy reachable sets can be approximated to any degree of accuracy under certain assumptions. Our method is illustrated by means of an example from the field of economics.
Kaa: A Python Implementation of Reachable Set Computation Using Bernstein Polynomials
