Optimal control of affine connection control systems: a variational approach

We investigate the optimal control of affine connection control systems. The formalism of the affine connection can be used to describe geometrically the dynamics of mechanical systems, including those with nonholonomic constraints. In the standard variational approach to such problems, one converts an n-dimensional second-order system into a 2n-dimensional first-order system, and uses these equations as constraints on the optimization. An alternative approach, which we develop in this paper, is to include the system dynamics as second-order constraints of the optimization, and optimize relative to variations in the configuration space. Using the affine connection, its associated tensors, and the notion of covariant differentiation, we show how variations in the configuration space induce variations in the tangent space. In this setting, we derive second-order equations having a geometric formulation parallel to that of the system dynamics. They also specialize to results found in the literature