Symmetry Reduction in Optimal Control of Multiagent Systems on Lie Groups

TL;DR

This paper develops a symmetry reduction method for optimal control problems of multiagent systems on Lie groups, simplifying the equations by exploiting physical symmetries, and applies it to energy-minimization for unicycles.

Contribution

It introduces a novel symmetry reduction framework for optimal control of multiagent systems on Lie groups, combining variational and Hamiltonian approaches.

Findings

Reduced optimality conditions derived using variational calculus.

Hamiltonian formalism applied via Pontryagin's maximum principle.

Successful application to energy-minimum control of unicycles.

Abstract

We study the reduction of degrees of freedom for the equations that determine necessary optimality conditions for extrema in an optimal control problem for a multiagent system by exploiting the physical symmetries of agents, where the kinematics of each agent is given by a left-invariant control system. Reduced optimality conditions are obtained using techniques from variational calculus and Lagrangian mechanics. A Hamiltonian formalism is also studied, where the problem is explored through an application of Pontryagin's maximum principle for left-invariant systems, and the optimality conditions are obtained as integral curves of a reduced Hamiltonian vector field. We apply the results to an energy-minimum control problem for multiple unicycles.