Dynamical feedback control with operator learning for the Vlasov-Poisson system

Abstract: To meet the demands of instantaneous control of instabilities over long time horizons in plasma fusion, we design a dynamic feedback control strategy for the Vlasov-Poisson system by constructing an operator that maps state perturbations to an external control field. In the first part of the paper, we propose learning such an operator using a neural network. Inspired by optimal control theory for linearized dynamics, we introduce a low-rank neural operator architecture and train it via adjoint state method. The resulting controller is effective at suppressing instabilities well beyond the training time horizon. To generalize control across varying initial data, we further introduce a novel cancellation-based control strategy that removes the destabilizing component of the electric field. This approach naturally defines an operator without requiring any training, ensures perturbation decay over infinite time, and demonstrates strong robustness under noisy feedback. Numerical experiments confirm the effectiveness of the method in both one- and multidimensional settings.