On the curvature of the trajectory manifold of nonlinear systems

This paper represents a preliminary contribution in the direction of characterizing geometric properties of the trajectory manifold of nonlinear systems. We introduce the notion of curvature of the trajectory manifold and define it by means of a nonlinear quadratic optimal control problem. The quadratic cost can be viewed as a weighted $L_2$ norm induced by a suitable inner product that provides a notion of orthogonality. The curvature at a given trajectory is defined in terms of the curves orthogonal to the tangent space at the given trajectory. We characterize the set of orthogonal curves. We show that it is a topological complement of the tangent space. We provide numerical techniques to compute orthogonal curves and to compute a lower bound of the curvature. We test these techniques on the inverted pendulum example.