Parametrized families of extremals and singularities in solutions to the Hamilton-Jacobi-Bellman equation

We analyze the effect which a fold and simple cusp singularity in the flow of a parametrized family of extremal trajectories of an optimal control problem has on the corresponding parametrized cost or value function. A fold singularity in the flow of extremals generates an edge of regression of the value implying the well-known results that trajectories stay strongly locally optimal until the fold-locus is reached, but lose optimality beyond. Thus fold points correspond to conjugate points. A simple cusp point in the parametrized flow of extremals generates a swallowtail in the parametrized value. More specifically, there exists 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 occur at the fold-loci and there trajectories lose strong local optimality. However, the branches intersect and generate a cut-locus which limits the optimality of close-by trajectories and eliminates these trajectories from optimality near the cusp point prior to the conjugate point. In the language of partial differential equations, a simple cusp point generates a shock in the solutions to the Hamilton-Jacobi-Bellman equation while fold points will not be part of the synthesis of optimal controls near the simple cusp point.