Optimal control of driftless nilpotent systems: some new results

This paper derives two new results on optimization of nilpotent systems without drift. These results are based on the observation that nilpotent systems can be transformed into polynomial systems using product of exponential representation. Hence, the nilpotent system is first extended to become fully actuated using Lie brackets of the system vector fields with additional inputs which are fictitious. Using the product of exponential representation, this extended system is transformed to a canonical form in Phillip Hall coordinates and is well known to have a polynomial structure. The new results exploit the structure of the governing equations in Phillip Hall coordinates. These results are: 1) in the absence of inequality constraints, a quadratic cost functional in the inputs can be guaranteed to be minimized by solving a sequence of quasi-linearized problems; and 2) in the presence of state and control constraints, the optimal solution of Mayer´s cost always lies on a constraint arc