Poisson Hamiltonian Pontryagin Dynamics and Optimal Control of Mechanical Systems on Lie Groupoids

We develop a Poisson Hamiltonian formulation of Pontryagin dynamics for optimal control of mechanical systems on Lie groupoids. The reduced dynamics is formulated intrinsically on the dual Lie algebroid endowed with its canonical linear Poisson structure and evolves on its symplectic leaves. The main result of this work shows that symplectic leaves, rather than coadjoint orbits, provide the natural reduced phase spaces for Pontryagin dynamics on Lie groupoids. Under suitable regularity assumptions, we prove the equivalence between the variational formulation of the optimal control problem and the associated Poisson Hamiltonian Pontryagin system, and we show that groupoid invariant Lagrangians lead to reduced optimality conditions of Euler Poincare type. Several mechanical examples, including systems with configuration dependent inertia and local symmetries, are presented to illustrate the theory.