N ot Hi gh lig ht DISCRETE MECHANICS AND OPTIMAL CONTROL : AN ANALYSIS ∗

The optimal control of a mechanical system is of crucial importance in many application areas. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper is to directly discretize the variational description of the system’s motion. The resulting optimization algorithm lets the discrete solution directly inherit characteristic structural properties from the continuous one like symmetries and integrals of the motion. We show that the DMOC (Discrete Mechanics and Optimal Control) approach is equivalent to a finite difference discretization of Hamilton’s equations by a symplectic partitioned Runge-Kutta scheme and employ this fact in order to give a proof of convergence. The numerical performance of DMOC and its relationship to other existing optimal control methods are investigated. Mathematics Subject Classification. 49M25, 49N99, 65K10. Received October 8, 2008. Revised September 17, 2009. Published online March 31, 2010. Introduction In order to solve optimal control problems for mechanical systems, this paper links two important areas of research: optimal control and variational mechanics. The motivation for combining these fields of investigation is twofold. Besides the aim of preserving certain properties of the mechanical system for the approximated optimal solution, optimal control theory and variational mechanics have their common origin in the calculus of variations. In mechanics, the calculus of variations is also fundamental through the principle of stationary action; that is, Hamilton’s principle. When applied to the action of a mechanical system, this principle yields the equations of motion for that system – the Euler-Lagrange equations. In optimal control theory the calculus of variations also plays a fundamental role. For example, it is used to derive optimality conditions via the Pontryagin maximum principle. In addition to its importance in continuous mechanics and control theory, the discrete calculus of variations and the corresponding discrete variational principles play an important role in constructing efficient numerical schemes for the simulation of mechanical systems and for optimizing dynamical systems.