A new perspective on control of uncertain complex systems

In this article, we investigate a new class of control problems called Ensemble Control, a notion coming from the study of complex 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 signal. 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 introduce the basics of ensemble control and derive the necessary and sufficient controllability characterizations for ensemble control of finite-dimensional time-varying linear systems with an accompanying analytical optimal control law. We show that ensemble controllability is in connection with singular values of the operator characterizing the system dynamics. A systematic study of ensemble control systems has immediate applications to systems with parameter uncertainties as well as to broad areas of quantum control and systems biology. The new mathematical structures appearing in such problems are excellent motivation for new developments in control theory.