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

Wang-Sang Koon, Mathematics, UC Berkeley

Recently several papers have appeared exploring the symmetry reductio n of optimal control problems on configuration spaces such as Lie groups and principal bundles. The mechanical systems which they have modeled vary widely: ranging from the falling cat, the rigid body with two oscillators, to the plate-ball system as well as the (airport) landing tower problem. Since the Pontryagin Maximum Principle is such an important and powerful tool in optimal control theory, it is frequently employed as a first step in finding necessary conditions for the optimal controls. Finally, different variants of Poisson reduction on the cotangent bundle $T^*Q$ of the configuration space $Q$ are use d to obtain the reduced equations of motion for the optimal trajectories.

This talk will present a Lagrangian alternative to these problems. More importantly, our method can handle the optimal control of nonholonomic mechanical system such as the snakeboard, which has a nontrivial evolution equation for its nonholonomic momentum. Our key idea is to link the method of Lagrange multipliers with Lagrangian reduction. This procedure which will be referred to as ``reduced Lagrangian optimization'', is able to handle all the above cases including the snakeboard. We hope that it will complement other existing methods and may also have the advantage that it is easier to use in many situations and can solve many new problems.