On the class of optimal control problems of 3-step nilpotent systems
This paper addresses the study of non-holonomic systems that are written by means of invariant distributions satisfying the property that the Lie algebra generated by the distribution is 3-step nilpotent. A system in that class can be written as an optimal control problem with a plant that is affine in the control parameters and a cost which is given by the kinetic energy of the system. Standard techniques in optimal control theory provide necessary conditions for the extremal trajectories. The paper presents a general theory of this class of systems along with detailed calculations for some low dimension cases.
CYBERNETICS AND PHYSICS, Vol. 2, No. 3, 151-158.