Conjugate points of dynamic pairs and control systems

Abstract

We study the geometry of dynamic pairs $(X,\mathcal{V})$ on a manifold $M$, where $X$ is a vector field and $\mathcal{V}$ is a distribution on $M$, both satisfying a regularity condition. Special cases are pairs defined by systems of second order ODEs, geodesic sprays in Riemannian, Finslerian and Lagranian geometries, semi-Hamiltonian systems and control-affine systems. Analogs of conjugate points from the calculus of variations are defined for the pair $(X,\mathcal{V})$. The main results give estimates for the position of conjugate points in terms of a curvature operator, analogously to the Cartan--Hadamard and Bonet--Myers theorems. Contrary to classical cases, no metric is given a priori, the distribution $\mathcal{V}$ may be nonintegrable and the curvature operator is defined in terms of $(X,\mathcal{V})$.