[Paper Review] Optimal Control Theory on almost-Lie Algebroids

Michał Jóźwikowski|arXiv (Cornell University)|Nov 7, 2011

Homotopy and Cohomology in Algebraic Topology38 references3 citations

TL;DR

This paper develops optimal control theory on almost-Lie algebroids, generalizing classical optimal control to geometric structures that relax the integrability condition of Lie algebroids. By formulating the Pontryagin maximum principle in this framework, the authors derive necessary conditions for optimality using the Hamiltonian formalism on almost-Lie algebroids, extending variational methods to non-integrable geometric settings.

ABSTRACT

We extend the Pontryagin Maximum Principle (PMP) to the geometric setting of almost-Lie (AL) algebroids -- objects which generalize Lie algebroids. The result may be understood as a very general reduction scheme for optimal control problems (OCPs). It covers the standard PMP, as well as gives necessary optimality conditions for symmetric OCPs on Lie groups, principal bundles, and Lie groupoids. We do not assume the symmetry of boundary conditions. The ideas are based on a very general concept of homotopy of admissible paths on AL algebroids. Our framework works for OCPs with fixed-end-points and general boundary conditions.

Motivation & Objective

• To extend optimal control theory to the geometric setting of almost-Lie algebroids, which generalize Lie algebroids by relaxing the Jacobi identity.
• To formulate the Pontryagin maximum principle in the context of almost-Lie algebroids, providing necessary conditions for optimality.
• To generalize variational principles and Hamiltonian dynamics to non-integrable geometric structures arising in mechanics and control.
• To establish a framework for optimal control on singular or non-regular geometric structures where standard Lie algebroid theory does not apply.

Proposed method

• Formalizing the dynamics of control systems using the anchor map and bracket operation of an almost-Lie algebroid.
• Defining a Hamiltonian function on the dual bundle of the almost-Lie algebroid to describe the system's energy and evolution.
• Deriving the Hamiltonian vector field via the almost-Lie algebroid structure, even when the Jacobi identity fails.
• Applying the Pontryagin maximum principle in this geometric setting to obtain necessary optimality conditions.
• Using the coadjoint action and dual pairing to relate the control variables to the momentum space of the algebroid.
• Establishing a correspondence between extremal curves and solutions of a Hamiltonian system on the dual bundle.

Experimental results

Research questions

• RQ1 How can the Pontryagin maximum principle be generalized to systems defined on almost-Lie algebroids?
• RQ2 What are the necessary conditions for optimality in control systems governed by non-integrable geometric structures?
• RQ3 How does the failure of the Jacobi identity in almost-Lie algebroids affect the formulation of optimal control laws?
• RQ4 Can the Hamiltonian formalism be consistently extended to almost-Lie algebroids for variational and control problems?
• RQ5 What is the role of the dual bundle and coadjoint orbits in the geometric structure of optimal control on almost-Lie algebroids?

Key findings

• The Pontryagin maximum principle is successfully extended to almost-Lie algebroids, providing necessary conditions for optimality in systems with non-integrable geometric constraints.
• The Hamiltonian vector field on the dual bundle of an almost-Lie algebroid is well-defined even when the Jacobi identity fails, enabling the use of Hamiltonian methods.
• Optimal trajectories correspond to integral curves of a Hamiltonian vector field derived from the algebroid structure and control constraints.
• The coadjoint action on the dual bundle plays a crucial role in characterizing conserved quantities and symmetries in the control system.
• The framework allows for the analysis of control systems on singular or non-regular geometric structures where classical Lie algebroid theory is inapplicable.
• The theory provides a geometric unification of optimal control and variational principles in non-integrable settings.