Control and manipulation of quantum mechanical systems using electromagnetic fields is a widely studied subject in areas of physics and chemistry, including spectroscopy, atomic molecular, and optical physics, and quantum chemistry. This article attempts to provide a glimpse into the rich class of bilinear control systems that are ubiquitous in these problems. In this article, we use control of spin systems in magnetic resonance as a model system to highlight characteristic feature of problems in quantum control. Background information is provided to enable the reader to appreciate new results and developments, where principled use of ideas from control theory have provided new insights into finding optimal ways to control and manipulate quantum mechanical systems. The study of deterministic and stochastic models that arise in problems in measurement and manipulation of quantum mechanical systems may foster new developments in control.

This paper discusses several recent results by the author and collaborators, which are united by the common goal of making nonlinear control theory more robust to imperfect information. These results are also united by common technical tools, centering around input-to-state stability (ISS), small-gain theorems, Lyapunov functions, and hybrid systems. The goal of this paper is to present an overview of these results which highlights their unifying features and which is more accessible to a general audience than the original technical articles.

The dynamics of hybrid systems with mode dynamics of different dimensions is described. The first part gives some deterministic examples of such multi-mode multi-dimensional $(M^3D)$ systems. The second part considers such models under sequential switching at random times. More specifically, the backward Kolmogorov equation is derived, and Lie-algebraic methods are used in the case where the modes are linear. For Poissonian switched equi-dimensional modes, the diffusion limit and its implication in vibrational stability are studied. The motion of a pebble on an elevator belt is given as an example.

The interaction of information and control has been a topic of interest to system theorists that can be traced back at least to the 1950’s when the fields of communications, control, and information theory were new but developing rapidly. Recent advances in our understanding of this interplay have emerged from work on the dynamical effect of state quantization together with results connecting communication channel data rates and system stability. Although this work has generated considerable interest, it has been centrally concerned with the relationship between control system performance and feedback information processing rates while ignoring the complexity (i.e. the cost of information processing). The concepts of communication and computation complexity of a controlled dynamical system based on digitized information lie in what is largely an uncharted area. In our recent work an attempt was made to explore this area by introducing a new measure of communication complexity for a two-player distributed control system. This complexity is named control communication complexity (CCC). It is based on the communication complexity concept defined in distributed computing and seeks to connect the complexity of information exchange over finite bandwidth channels with the control system dynamics. The purpose of the present paper is to extend the study of control communication complexity to an interesting class of continuous-time control systems that have appeared in the recent literature dealing with quantum communication and control systems. An interesting aspect of this extension is that it brings together heretofore independent research themes that have been prominent in the research career of Roger Brockett.