# Lagrangian Reachabililty

## TL;DR

LRT is a new Lagrangian-based algorithm for conservative reachability analysis of nonlinear systems, utilizing the Cauchy-Green stretching factor to produce tight over-approximations of reachable states.

## Contribution

The paper introduces LRT, a novel Lagrangian approach that improves reachability analysis by using the Cauchy-Green stretching factor for tighter enclosures.

## Key findings

- LRT outperforms CAPD and Flow* in benchmarks.
- LRT provides tighter reachability over-approximations.
- Prototype implementation demonstrates practical efficiency.

## Abstract

We introduce LRT, a new Lagrangian-based ReachTube computation algorithm that conservatively approximates the set of reachable states of a nonlinear dynamical system. LRT makes use of the Cauchy-Green stretching factor (SF), which is derived from an over-approximation of the gradient of the solution flows. The SF measures the discrepancy between two states propagated by the system solution from two initial states lying in a well-defined region, thereby allowing LRT to compute a reachtube with a ball-overestimate in a metric where the computed enclosure is as tight as possible. To evaluate its performance, we implemented a prototype of LRT in C++/Matlab, and ran it on a set of well-established benchmarks. Our results show that LRT compares very favorably with respect to the CAPD and Flow* tools.