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We study the reduction of degrees of freedom for the equations that determinenecessary optimality conditions for extrema in an optimal control problem for amultiagent system by exploiting the physical symmetries of agents, where thekinematics of each agent is given by a left-invariant control system. Reducedoptimality conditions are obtained using techniques from variational calculusand Lagrangian mechanics. A Hamiltonian formalism is also studied, where theproblem is explored through an application of Pontryagin's maximum principlefor left-invariant systems, and the optimality conditions are obtained asintegral curves of a reduced Hamiltonian vector field. We apply the results toan energy-minimum control problem for multiple unicycles.

Symmetry Reduction in Optimal Control of Multiagent Systems on Lie Groups