Dirac Structures and Reduction of Optimal Control Problems with Symmetries

Abstract

We study the symmetry reduction of optimal control problems using Dirac structures. The first-order conditions of the Pontryagin Maximum Principle are written as an implicit Dirac system on the Pontryagin bundle. For a problem invariant under a Lie group, we reduce both the Lagrange-Pontryagin variational principle and the underlying Dirac structure, and prove that they yield the same reduced equations, which we identify with the Hamiltonian equations on the dual of the Atiyah algebroid. Reconstruction, independence of the principal connection, and the relation between criticality and optimality complete the picture. Dropping regularity, we give the Gotay-Nester-Hinds constraint algorithm a Dirac-geometric form and prove that symmetry reduction commutes with it. For control-linear systems whose admissible directions are horizontal for a principal connection, abnormal extremals lie on the leaf where the base costate vanishes, and the first secondary constraint is the degeneracy of the curvature pairing. We also give a reduced criterion for strict abnormality and a metric-free index form. Examples: a spacecraft with rotors (flat connection), the Heisenberg group (nowhere-zero curvature) and the Martinet problem (curvature vanishing on a hypersurface).