Optimal Control for Holonomic and Nonholonomic Mechanical Systems with Symmetry and Lagrangian Reduction

In this paper we establish necessary conditions for optimal control using the ideas of Lagrangian reduction in the sense of reduction under a symmetry group. The techniques developed here are designed for Lagrangian mechanical control systems with symmetry. The benefit of such an approach is that it makes use of the special structure of the system, especially its symmetry structure, and thus it leads rather directly to the desired conclusions for such systems. Lagrangian reduction can do in one step what one can alternatively do by applying the Pontryagin maximum principle followed by an application of Poisson reduction. The idea of using Lagrangian reduction in the sense of symmetry reduction was also obtained by Bloch and Crouch [Proc. 33rd CDC, IEEE, 1994, pp. 2584--2590] in a somewhat different context, and the general idea is closely related to those in Montgomery [Comm. Math. Phys., 128 (1990), pp. 565--592] and Vershik and Gershkovich [Dynamical Systems VII, V. Arnold and S. P. Novikov, eds., Springer-Verlag, 1994]. Here we develop this idea further and apply it to some known examples, such as optimal control on Lie groups and principal bundles (such as the ball and plate problem) and reorientation examples with zero angular momentum (such as the satellite with moveable masses). However, one of our main goals is to extend the method to the case of nonholonomic systems with a nontrivial momentum equation in the context of the work of Bloch, Krishnaprasad, Marsden, and Murray [Arch. Rational Mech. Anal., (1996), to appear]. The snakeboard is used to illustrate the method.