Optimal Control with Broken Symmetry of Multi-Agent Systems on Lie Groups

TL;DR

This paper develops a reduction framework for optimal control problems of multi-agent systems on Lie groups with partial symmetry breaking, using variational principles and Pontryagin's maximum principle, demonstrated on collision avoidance for unicycles.

Contribution

It introduces a novel reduction method for optimal control with partial symmetry breaking on Lie groups, leveraging variational principles and Pontryagin's maximum principle.

Findings

Effective reduction of optimality conditions for multi-agent systems

Application to collision avoidance demonstrates practical utility

Framework accommodates partial symmetry breaking in control problems

Abstract

In this paper we study reduction by symmetry for optimality conditions in optimal control problems of left-invariant affine multi-agent control systems, with partial symmetry breaking cost functions. Our approach emphasizes the role of variational principles. Specifically, we recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Hamiltonian and obtain the reduced optimality conditions from a reduced variational principle via Pontryagin Maximum Principle. We apply the results to a collision avoidance problem for multiple unicycles in the presence of an obstacle.