Curvature of optimal control: Deformation of scalar-input planar systems

The Pontryagin Maximum Principle and high-order open-mapping theorems generalize elementary first-derivative tests to nonlinear optimal control. They provide necessary conditions for a trajectory-control-pair to be optimal, or sufficient conditions for local controllability. 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 for definiteness of 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. This theory naturally applies to systems whose controls take values on a circle or sphere. In this article we present initial studies of how this notion of curvature provides insight into the limiting case when the circles become degenerate ellipses in the form of closed intervals. Of particular interest are well studied accessible, but uncontrollable, nonlinear systems, and systems that exhibit conjugate points, in which the control takes values in a closed interval u = (u1, u2) ∈ [-1,1] x {0} ⊆ R². We focus on systems that are well-known models for the analysis of small-time local controllability and time-optimal control.