  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 \L ojasiewicz inequality,
convergence of the optima as time $t\to +\infty$ 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.
