Morse Properties for the Minimum Time Function on 2-D Manifolds

Abstract

Given a two-dimensional smooth manifold M and two smooth vector fields X and Y on M, we want to steer a point p e M to a point q e M in minimum time using only intergral curves of the vector fields X and Y. Fixing p, we define the minimum time function {\Bbb T}_p(q) to reach q. We prove that, generically, {\Bbb T}_p(q) is a Morse function in topological sense giving a positive answer to a question of V.I. Arnold.