Conjugate points of dynamic pairs and control systems

We study the geometry of dynamic pairs ( X , V ) on a manifold M , where X is a vector field and 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 , 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 V may be nonintegrable and the curvature operator is defined in terms of ( X , V ).