A Validated Integration Algorithm for Nonlinear ODEs using Taylor Models and Ellipsoidal Calculus

This paper presents a novel algorithm for bounding the reachable set of parametric nonlinear differential equations. This algorithm is based on a first-discretize-then-bound approach to enclose the reachable set via propagation of a Taylor model with ellipsoidal remainder, and it accounts for truncation errors that are inherent to the discretization. In contrast to existing algorithms that proceed in two phases—an a priori enclosure phase, followed by a tightening phase—theproposed algorithm first predicts a continuous-time enclosure and then seeks a maximal step-size for which validity of the predicted enclosure can be established. It is shown that this reversed approach leads to a natural step-size control mechanism, which no longer relies on the availability of an a priori enclosure. Also described in the paper is an open-source implementation of the algorithm in ACADO Toolkit. A simple numerical case study is presented to illustrate the performance and stability of the algorithm.