An operator theoretic approach to linear ensemble control

Abstract

In this paper, we revisit the original least squares formulation of the ensemble control problem. Based on new considerations, we are able to resolve the longstanding problem of formulating general, and easily verifiable conditions for an ensemble to be controllable in the least-squares sense. The key is to take a purely operator theoretic approach from the very beginning, i.e. to study the ensemble control problem by virtue of the associated Fredholm integral equation. By establishing a direct connection to a recently introduced moment-based approach, the theoretical question of least-squares ensemble controllability is eventually settled. In the second part, we take the very same integral operator theoretic approach to consider the equally longstanding problem of synthesizing inputs that realize a desired steering between two ensemble states. By means of a suitable discretization of the integral operator, we obtain a computational procedure to synthesize control signals that steer ensembles in both a robust and minimum energy fashion. This yields a unified framework for the ensemble control of linear systems.