UBC Theses and Dissertations

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UBC Theses and Dissertations

Solving reachable sets on a manifold Cross, Elizabeth Ann

Abstract

First, this thesis explores the implementation of the fast marching method as part of the toolbox of level set methods. This method uses Dijkstra's algorithm to approximate the solution to the non-linear Eikonal equation. Functions for calculating signed distances and extension velocities are also implemented. These functions use the fast marching method in their implementation. Second, it explores a method for computing reachable sets on a manifold; in other words, the dynamics governing these reachable sets can be described by a Differential Algebraic Equation. It uses level set methods to solve the underlying Hamilton Jacobi equation of the reachable set and it ensures an accurate solution on the manifold by using the closest point method. The closest point method guarantees that the reachable set is perpendicular to the manifold at the points of intersection. Several two and three dimensional toy problems and a real-life power generator problem are explored in order to test the method.

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