Ensemble Control of Finite Dimensional Time-Varying Linear Systems

TL;DR

This paper develops controllability conditions and optimal control laws for ensemble control of finite-dimensional, time-varying linear systems, with applications in quantum control and spectroscopy.

Contribution

It introduces necessary and sufficient controllability conditions and analytical optimal control solutions for ensemble control of time-varying linear systems.

Findings

Controllability linked to singular values of system operators

Optimal solutions relate to prolate spheroidal wave functions

Applications include quantum control and systems with parameter uncertainties

Abstract

In this article, we investigate the problem of simultaneously steering an uncountable family of finite dimensional time-varying linear systems. We call this class of control problems Ensemble Control, a notion coming from the study of spin dynamics in Nuclear Magnetic Resonance (NMR) spectroscopy and imaging (MRI). This subject involves controlling a continuum of parameterized dynamical systems with the same open-loop control input. From a viewpoint of mathematical control theory, this class of problems is challenging because it requires steering a continuum of dynamical systems between points of interest in an infinite dimensional state space by use of the same control function. The existence of such a control raises fundamental questions of ensemble controllability. We derive the necessary and sufficient controllability conditions and an accompanying analytical optimal control law for ensemble control of time-varying linear systems. We show that ensemble controllability is in connection with singular values of the operator characterizing the system dynamics. In addition, we study the problem of optimal ensemble control of harmonic oscillators to demonstrate our main results. We show that the optimal solutions are pertinent to the study of time-frequency limited signals and prolate spheroidal wave functions. A systematic study of ensemble control systems has immediate applications to systems with parameter uncertainties as well as to broad areas of quantum control systems as arising in coherent spectroscopy and quantum information processing. The new mathematical structures appearing in such problems are an excellent motivation for new developments in control theory.