Demonstrating Numerical Convergence to the Analytic Solution of some Backwards Reachable Sets with Sharp Features∗

We examine the convergence properties of a level set algorithm designed to track evolving interfaces; in particular, its convergence properies on a series of two and three dimensional backwards reachable sets whose flow fields involve kink formation (sharp features) and, in some cases, rarefaction fans introduced by input parameters in the dynamics. The chosen examples have analytic solutions to facilitate the convergence analysis. We describe the error analysis method, the formulation of reachability in terms of a Hamilton-Jacobi equation, and our implementation of the level set method in some detail. In addition to the convergence analysis presented here, these techniques and examples could be used to validate either other nonlinear reachability algorithms or other level set implementations. keywords: nonlinear reachability, Hamilton-Jacobi equations, convergence analysis ∗Research supported by ONR under MURI contract N00014-02-1-0720 and by DARPA under the Software Enabled Control Program (AFRL contract F33615-99-C-3014)