The maximum principle of Pontryagin is applied to control systems where the drift vector field is the geodesic spray corresponding to an affine connection. The result is a second-order differential equation whose right-hand side is the ``adjoint Jacobi equation.'' By choosing the cost function to be the square norm of the input with respect to a Riemannian metric, one generates equations which generalise spline equations in two directions: (1) the setting is that of manifolds with a general affine connection and (2) it is allowed to impose the cost only on those accelerations which live in a subbundle of the tangent bundle.

No online version avaliable.