Cut-Loci and Cusp Singularities in Parametrized Families of Extremals

Abstract

We analyze the simple cusp singularity arising in the flow of a parametrized family of extremals. The simple cusp point generates a region in the state space which is covered 3:1 with both locally minimizing and maximizing branches. The changes from the locally minimizing to the maximizing branch away from the simple cusp point occur at fold-loci and there trajectories lose strong local optimality. However, the two minimizing branches intersect and generate a cut-locus which limits the optimality of the close-by trajectories and eliminates these trajectories from optimality near the cusp point prior to the conjugate point. In the language of PDE, the simple cusp in the parametrized flow of extremals generates a shock in the corresponding solution to the Hamilton-Jacobi-Bellman equation.