An imbalanced rotor is considered. A system of moving balancing masses is given. We determine the optimal movement of the balancing masses to minimize the imbalance on the rotor. The optimal movement is given by an open-loop control solving an optimal control problem posed in infinite time. By methods of the Calculus of Variations, the existence of the optimum is proved and the corresponding optimality conditions have been derived. Asymptotic behavior of the control system is studied rigorously. By Łojasiewicz inequality, convergence of the optima as time towards a steady configuration is ensured. An explicit estimate of the convergence rate is given. This guarantees that the optimal control stabilizes the system. In case the imbalance is below a computed threshold, the convergence occurs exponentially fast. This is proved by the Stable Manifold Theorem applied to the Pontryagin optimality system. Moreover, a closed-loop control strategy based on Reinforcement Learning is proposed. Numerical simulations have been performed, validating the theoretical results.

中文翻译：

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通过优化控制抑制转子不平衡

考虑了不平衡的转子。给出了一个移动平衡质量系统。我们确定平衡质量的最佳运动，以最大限度地减少转子上的不平衡。最优运动是通过开环控制解决无限时间提出的最优控制问题给出的。用变分法的方法证明了最优值的存在，并推导出了相应的最优值条件。严格研究控制系统的渐近行为。通过 Łojasiewicz 不等式，最优值随时间收敛确保配置稳定。给出了收敛速度的明确估计。这保证了最优控制使系统稳定。在不平衡低于计算阈值的情况下，收敛以指数速度发生。应用于 Pontryagin 最优系统的稳定流形定理证明了这一点。此外，提出了一种基于强化学习的闭环控制策略。已经进行了数值模拟，验证了理论结果。