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.

中文翻译：

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李群上多智能体系统最优控制中的对称约简

我们通过利用代理的物理对称性来研究确定多代理系统最优控制问题中极端必要最优条件的方程式的自由度降低，其中每个代理的运动学由左不变控制系统给出。使用变分微积分和拉格朗日力学的技术可以获得降低的最优条件。还研究了哈密顿形式，通过应用庞特里亚金最大原理对左不变系统探索问题，并将最优条件作为简化哈密顿向量场的积分曲线获得。我们将结果应用于多个单轮脚踏车的最小能量控制问题。