Curvature of optimal control: Deformation of classical planar systems

Consider the problem of deciding whether a trajectory pair (u∗(t), x∗(t)), t ∈ [0, T ] of a generally nonlinear system ẋ = F (x, u), x ∈ Mn is a time-optimal solution connecting the endpoints x(0) and x(T ), or whether the system is locally controllable about this trajectory. The classical approach analyzes the endpoint map u 7→ x(T, u) (for fixed T and x(0)) and determine whether or not it is locally an open map. The Pontryagin Maximum Principle and high-order open-mapping theorems provide necessary conditions for a trajectorycontrol-pair to be optimal. Sufficient conditions for optimality (and necessary conditions for nonlinear controllability) are harder to obtain. Like the Legendre-Clebsch condition, they generally take the form of tests of definiteness for second order derivatives. Recently Agrachev introduced an attractive alternative by developing a notion of curvature of optimal control that generalizes classical Gauss (and Ricci) curvatures. That theory has been developed for systems whose controls take values in a circle or sphere u ∈ Rn−1.