Ensemble Control of Finite-Dimensional Time-Varying Linear Systems

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 systemswith 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.

The Mathematics of Networks of Linear Systems

“…Li (2011) has proposed an operator-theoretic characterization of L 2 -ensemble reachability for general time-varying linear multivariable systems. However, that characterization is stated in terms of the growth rates of singular values of the input-state operator and, thus, is difficult to verify, even for the timeinvariant linear systems (12.1).…”

From averaged to simultaneous controllability

“…In the next item, we mention some existing literature when considering the L ∞ -norm. (6) For parameter dependent systems, the notion of ensemble controllability is also commonly used (see for instance [5,6,13,17]). A system is said to be ensemble controllable in time T > 0 if, for every ε and every y…”

Riemann–Stieltjes Optimal Control Problems for Uncertain Dynamic Systems

“…Thus, although our development of the Riemann-Stieltjes optimal control problem was motivated by a need to manage uncertainties in engineering systems, it is clear from the preceding discussion that problem RS can be used to connect concepts from viability theory. It turns out that many more independently developed ideas across disparate disciplines [8,[51][52][53][54][55][56][57][58] can also be connected to a Riemann-Stieltjes optimal control problem. We briefly summarize these additional connections to illustrate some unification concepts and open problems while noting that the following discussions are no way meant to be exhaustive.…”