Global Solution for the Optimal Feedback Control of the Underactuated Heisenberg System

We present a global solution for an optimal feedback controller of the underactuated Heisenberg system or nonholonomic integrator. Employing a recently developed technique based on generating functions appearing in the Hamilton-Jacobi theory, we circumvent a singularity caused by under actuation to develop a nonlinear optimal feedback control in an implicitly analytical form. The systematic procedure to deal with under actuation indicates that generating functions should be effective tools for solving general underactuated optimal control problems.

I. Introduction

We study an optimal feedback control (OFC) problem for an underactuated Heisenberg system, using generating functions appearing in the Hamilton-Jacobi theory [1], [2]. Related work on the use of canonical transformations and their associated generating functions for solving optimal control problems may be found in Fraeijs de Veubeke [3], Powers and Tapley [4], Hagedorn [5], etc. However, these papers rely on using special canonical transformations only valid for their specific dynamical systems of interest, and thus their techniques are not directly applicable to general optimal control problems. We view a general two point boundary value problem (TPBVP) for a Hamiltonian system as the problem of finding a suitable canonical transformation [6], [7], and derive a generic procedure for solving underactuated optimal control problems.