Reservoir Computing Causal Operators for Data-Driven Moment Control of Nonlinear Ensembles
This letter develops a data-driven control framework for nonlinear ensemble systems using reservoir computing (RC). We consider ensemble control problems, in which the objective is to regulate a large, potentially uncountable, population of systems with unknown dynamics. To address this challenge, we introduce a moment kernelization approach that yields a dual representation and enables a valid finite-dimensional approximation of ensemble dynamics. Building on this reduction, we cast ensemble control synthesis as the approximation of a causal operator that maps moment trajectories to control inputs. We show that continuous-time reservoir systems induce well-defined causal input-output operators with the fading-memory property, providing a principled foundation for learning these feedback operators from moment trajectory data. Based on this theory, we design an RC-based controller trained on input-output moment trajectories and deployed in a closed-loop configuration for tracking and stabilization of nonlinear ensemble systems.