Ensemble Control on Lie Groups

TL;DR

This paper investigates the controllability of ensemble systems on semisimple Lie groups, introducing a novel covering method based on Lie algebra decomposition that simplifies the analysis by reducing it to controllability of subsystems on Lie subgroups.

Contribution

It develops a new covering method using Cartan decomposition to analyze ensemble controllability on Lie groups, establishing an equivalence with classical controllability.

Findings

Ensemble controllability is equivalent to classical controllability on semisimple Lie groups.

The covering method decomposes the problem into controllability of subsystems on Lie subgroups.

This approach simplifies the analysis of infinite-dimensional ensemble systems.

Abstract

Problems involving control of large ensmebles of structurally identical dynamical systems, called \emph{ensemble control}, arise in numerous scientific areas from quantum control and robotics to brain medicine. In many of such applications, control can only be implemented at the population level, i.e., through broadcasting an input signal to all the systems in the population, and this new control paradigm challenges the classical systems theory. In recent years, considerable efforts have been made to investigate controllability properties of ensemble systems, and most works emphasized on linear and some forms of bilinear and nonlinear ensemble systems. In this paper, we study controllability of a broad class of bilinear ensemble systems defined on semisimple Lie groups, for which we define the notion of ensemble controllability through a Riemannian structure of the state space Lie group. Leveraging the Cartan decomposition of semisimple Lie algebras in representation theory, we develop a \emph{covering method} that decomposes the state space Lie group into a collection of Lie subgroups generating the Lie group, which enables the determination of ensemble controllability by controllability of the subsystems evolving on these Lie subgroups. Using the covering method, we show the equivalence between ensemble and classical controllability, i.e., controllability of each individual system in the ensemble implies ensemble controllability, for bilinear ensemble systems evolving on semisimple Lie groups. This equivalence makes the examination of controllability for infinite-dimensional ensemble systems as tractable as for a finite-dimensional single system.