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

Ghorbanali Haghighatdoost

Abstract

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.

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

Abstract

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.
Paper Structure (21 sections, 4 theorems, 50 equations, 3 figures)

This paper contains 21 sections, 4 theorems, 50 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Poisson Geometry and Reduced Optimal Control
  3. Poisson Structure on the Dual Lie Algebroid
  4. Optimal Control Systems on Lie Algebroids
  5. Control systems
  6. Cost functional
  7. Pontryagin Hamiltonian
  8. Hamiltonian dynamics on $A^*$
  9. Geometric properties
  10. Relation to previous work.
  11. Two Optimal Control Problems
  12. Problem I: Variational optimal control on a Lie algebroid
  13. Problem II: Hamiltonian optimal control on the dual Lie algebroid
  14. Equivalence of the two formulations
  15. Main Equivalence Theorem
  16. ...and 6 more sections

Key Result

Theorem 1

Let $A \to M$ be a Lie algebroid with anchor $\rho : A \to TM$, and let be a smooth fiberwise surjective bundle map defining a control system on $A$. Let $L : E \to \mathbb{R}$ be a smooth Lagrangian, regular in the control variables. Then the following statements hold: Consequently, the variational and Hamiltonian formulations of the optimal control problem on the Lie algebroid $A$ are equivale

Figures (3)

  • Figure 1: Phase portrait of the Pontryagin dynamics on $T^*M$. Trajectories remain confined to symplectic leaves of the canonical Poisson structure, reflecting the Lie groupoid reduction.
  • Figure 2: Numerical evolution of a leaf invariant along an optimal trajectory. The constant value confirms confinement to a symplectic leaf of $T^*M$.
  • Figure 3: Costate--state relation along optimal trajectories, illustrating configuration--dependent coupling characteristic of Lie groupoid dynamics.

Theorems & Definitions (4)

  • Theorem 1: Equivalence of variational and Hamiltonian optimal control
  • Corollary 2: Euler--Poincaré form of the optimality conditions on Lie groupoids
  • Corollary 3
  • Corollary 4