Local minimizers for variational obstacle avoidance on Riemannian manifolds

This paper studies a variational obstacle avoidance problem on complete Riemannian manifolds. That is, we minimize an action functional, among a set of admissible curves, which depends on an artificial potential function used to avoid obstacles. In particular, we generalize the theory of bi-Jacobi fields and biconjugate points and present necessary and sufficient conditions for optimality. Local minimizers of the action functional are divided into two categories-called Q-local minimizers and Omega-local minimizers-and subsequently classified, with local uniqueness results obtained in both cases.