Reachability of Black-Box Nonlinear Systems after Koopman Operator Linearization

Abstract

Abstract Reachability analysis of nonlinear dynamical systems is a challenging and computationally expensive task. Computing the reachable states for linear systems, in contrast, can often be done efficiently in high dimensions. In this paper, we explore verification methods that leverage a connection between these two classes of systems based on the concept of the Koopman operator. The Koopman operator links the behaviors of a nonlinear system to a linear system embedded in a higher dimensional space, with an additional set of so-called observable variables. Although the new dynamical system has linear differential equations, the set of initial states is defined with nonlinear constraints. For this reason, existing approaches for linear systems reachability cannot be used directly. We propose the first reachability algorithm that deals with this unexplored type of reachability problem. Our evaluation examines several optimizations, and shows the proposed workflow is a promising avenue for verifying behaviors of nonlinear systems.