Optimal control of nilpotent systems: a sub-Riemannian approach

We present a general framework for the optimal control of driftless nonlinear systems defined by means of distributions of smooth vector fields that generate nilpotent Lie algebras. A smooth varying inner product on the planes of the distribution, yields the energy functional that allows to approach the optimal control problem as a sub-Riemannian geodesic problem. This class of systems is relevant because provides good models for nonholonomic systems in mechanics and automation as well as in classical particle physics. We discuss two examples of nonholonomic systems within this formalism, the Cartan geometry that corresponds to the problem of rolling without slipping or twisting, and the classical Foucault pendulum that is accepted as indisputable demonstration of the Earth’s rotation movement.