### Analysis, Control, and Learning of Dynamic Ensemble and Population Systems

#### Organisers

- Jr-Shin Li
- Shen Zheng

#### Speakers

- Jr-Shin Li, Washington University in St. Louis, USA
- Shen Zeng, Washington University in St. Louis, USA
- Ugo Boscain, CNRS, CMAP Ecole Polytechnique, Palaiseau, France
- Michael Schönlein, University of Würzburg, Germany
- Gunther Dirr, Institute for Mathematics, Germany
- Xudong Chen, Electrical, University of Colorado, Boulder, Colorado, USA
- Hiroya Nakao, Tokyo Institute of Technology, Japan
- Daoyi Dong, University of New South Wales, Canberra, Australia
- Karsten Kuritz and Frank Allgöwer, University of Stuttgart, Germany

#### Summary

A wide range of emerging and highly relevant problems found in nature, engineering, as well as societal structures are often plagued with overwhelmingly complex dynamical components and structures that are beyond the reach of current systems analysis and control design principles. Notable examples from these areas include decoding dynamic topology and functional connectivity of high-dimensional networked biological systems, exciting large quantum ensembles in applications of nuclear magnetic resonance spectroscopy, imaging, and computation, inducing synchronization patterns and behavior in brain and social networks, and establishing autonomous intelligent machines or factories in cyber-physical systems with interconnected spatiotemporally varying dynamics, as well as the problem of epidemic outbreaks witnessed in recent years. A common thread of these very challenging problems is the pervasive theme of having large populations of isolated or highly interacting dynamic units. On the top of the large-scale nature is the fundamental difficulty that control and observation can only be implemented at the population level, i.e., through broadcasting a single input signal to all the systems in the population and through receiving aggregated measurements of the systems in the population, respectively. This restriction gives rise to a new control paradigm of population-based control, called ensemble control. Interest in cutting-edge analysis, estimation, control, and learning algorithms and technologies suitable for these emerging sophisticated control systems have seen a stellar growth in recent years.

In this workshop, we will offer a survey of and initiate a dialogue about emerging techniques and research problems in this recently spurred, highly exciting field that inspires control theory’s new open challenges concerning with high-dimensional and very large-scale phenomena. Emphasis will be placed on both recent theoretical developments and emerging applications at the interface of systems science, data science, machine learning, quantum physics, neuroscience, and biology. We will introduce state-of-the-art methods for both theoretical and data-driven investigations of fundamental properties of complex population systems from both model-based and data-driven perspectives, including controllability, observability, and synchronizability, and discuss novel computational and learning methods for synthesizing optimal ensemble controls and decoding dynamics of ensemble systems. We will also describe the use of ensemble control and estimation theory in diverse applications, including characterization of neurons in diseased networks (e.g., Parkinson’s disease, epilepsy), entrainment of a population of nonlinear oscillators, transport of particles over networks, and learning in inhomogeneous quantum ensembles. Finally, various future directions and open questions regarding fundamental and practical problems in ensemble control will be articulated and discussed.

#### Programme

**Ensemble control for cellular oscillators: One ring to rule them all, Karsten Kuritz and Frank Allgöwer**

Many diseases including cancer, Parkinson’s disease and heart diseases are caused by loss or malfunction of regulatory mechanism of an oscillatory system. Successful treatment of these diseases might involve recovering the healthy behavior of the oscillators in the system, i.e., achieving synchrony or a desired distribution of the oscillators on their periodic orbit. We present the problem of controlling the distribution of a population of cellular oscillators described in terms of phase models. Different practical limitations on the observability and controllability of cellular states naturally lead to an ensemble control formulation in which a population-level feedback law for achieving a desired distribution is sought. Presented results and methods are readily applicable to the control of a wide range of other types of oscillating populations, such as circadian clocks, and spiking neurons.

**Learning Control for Inhomogeneous Quantum Ensembles, Daoyi Dong**

In this talk, I will discuss a couple of learning control methods for seeking for "smart" fields for controlling inhomogeneous quantum ensembles. A sampling-based learning control algorithm using gradient information is proposed for inhomogeneous quantum ensembles with unitary dynamics. Then a differential evolution algorithm is presented for control of inhomogeneous open quantum ensembles.

**Ensemble Controllability of Quantum Systems via Adiabatic Methods, Ugo Boscain**

In this talk we discuss how to control a parameter-dependent family of quantum systems. Our technique is based on adiabatic approximation theory and on the presence of curves of conical eigenvalue intersections of the controlled Hamiltonian. As particular cases, we recover chirped pulses for two-level quantum systems and counter-intuitive solutions for three-level stimulated Raman adiabatic passage (STIRAP). The proposed technique works for systems evolving both in finite-dimensional and infinite-dimensional Hilbert spaces. We show that the assumptions guaranteeing ensemble controllability are structurally stable with respect to perturbations of the parametrized family of systems.

**Moment-Based Ensemble Control and Learning using Aggregated Feedback, Jr-Shin Li**

An ensemble system refers to a large population (in the limit, a continuum) of dynamical systems with parametric variations. The manipulation of such systems is typically achieved by parameter-independent, open-loop control inputs, without exerting feedback controls due to practical limitations in collecting state feedback information. However, in many emerging applications, population-level, coarse measurements of ensemble systems are available, which can be exploited in an aggregated manner. In this talk, we will introduce a moment-based framework that utilizes the idea of statistical moments induced by aggregated measurements to synthesize a novel approach to controlling an ensemble system through its associated moment system. In particular, we present the connection between an ensemble control problem and a classical moment problem, by which ensemble control analysis, e.g., controllability, can be conducted through the study of the moment system. Moreover, we will illustrate how the proposed moment-based method enables a systematic design strategy to close the loop and design feedback laws for ensemble control systems.

**Ensemble Observability of Dynamical Systems and Sample-based Population Observers, Shen Zeng**

In solving applied problems related to population systems, such as in cell biology, a frequently met task is to first extract information about the parameter distribution within a population of systems from distributional measurement / population snapshot data. In this talk, we will focus on the theoretical core of this practical estimation problem, the ensemble observability problem, which consists of reconstructing a density of initial states from the evolution of the density of outputs under a finite-dimensional dynamical system. A fundamental result in the study of the ensemble observability problem is an inherent connection to mathematical tomography problems. In the first part of the talk, we aim to give a broad overview of the ensemble observability problem for both linear and nonlinear systems. The second part of the talk is aimed at the practical observer design for population systems. We review a sample-based implementation proposed recently by the speaker, in which the state of the population observer is described by a set of sample points and where the correction mechanism for updating the population state based on the recorded population output measurements leverages an interesting connection to optimal mass transport problems.

**Ensemble Reachability of linear parameter-dependent systems: Criteria and Constructive Methods, Michael Schönlein**

This talk considers a particular class of inﬁnite dimensional linear systems, deﬁned by a one-parameter family of linear control systems, commonly called an ensemble. Here, it is crucial that the control input is assumed to be independent from the parameter. The focus of this talk is on reachability properties for this class of systems. In particular, we present an abstract decomposition result and based on that we explore necessary and sufficient conditions for reachability in terms of the parameter-dependent matrices deﬁning the ensemble. Moreover, we discuss approaches that yield constructive methods for the computation of suitable open-loop inputs.

**Systems on Infinite Dimensional Lie Groups with Applications to Ensemble Control, Gunther Dirr**

Under the heading of “ensemble control” several quite diﬀerent control scenarios are subsumed. Yet, a common thread is the inﬁnite dimensionality of the underlying state spaces which results from the fact that in general “ensembles” consists of inﬁnitely many (weakly or non-interacting) subsystems. Particularly, in quantum control there are “ensembles” which can be modeled as invariant systems on inﬁnite dimensional Lie groups. We ﬁrst recall and introduce some basic notions and obstacles arising in the control of systems on inﬁnite dimensional Lie groups (compared to the ﬁnite dimensional case). In particular, we aim at a necessary and/or suﬃcient condition similar to the well-known Lie algebra rank condition (LARC) for ﬁnite dimensional systems. In the second part, we present some ﬁndings within the context of Banach Lie group and conclude with a successful applications to quantum control.

**Structure Theory for Ensemble Control and Estimation of Nonholonomic Systems, Xudong Chen**

Ensemble control deals with the problem of using a finite number of control inputs to simultaneously steer a large population (in the limit, a continuum) of individual control systems. As a dual, ensemble estimation deals with the problem of using a finite number of measurement outputs to estimate the initial state of every individual system in the (continuum) ensemble. We introduce in the talk a novel class of ensembles of nonlinear control systems, termed distinguished ensemble systems. Every such system has two key components, namely a set of finely structured control vector fields and a set of co-structured observation functions. In the first half of the talk, we demonstrate that the structure of a distinguished ensemble system can significantly simplify the analysis of ensemble controllability and observability. Moreover, such a structure can be used as a principle for ensemble system design. In the second half of the talk, we address the issue about existence of a distinguished ensemble system for a given manifold. We will focus on the case where the underlying space of every individual system is an arbitrary semi-simple Lie group or its homogeneous space.

**Reduced Dynamical Description of Collective Oscillations in Populations of Coupled Dynamical Elements, Hiroya Nakao**

There are various examples of real-world systems in which a population of dynamical elements with similar properties exhibits collective oscillations due to mutual interaction. A system of globally coupled nonlinear oscillators exhibiting collective oscillations is a typical example of such systems. In the large-population limit, a continuum description of the population using a probability density function can be introduced, which obeys a nonlinear Fokker-Planck equation (NLFPE). The collective oscillations of the population correspond to a stable limit-cycle orbit in the infinite-dimensional state space of the NLFPPE. In this talk, we will consider dimensionality reduction of such a limit-cycle solution to the NLFPE, which allows us to describe the dynamics of the population subjected to weak perturbations by a set of low-dimensional phase or phase-amplitude equations. Using the reduced lowdimensional equations, optimization of the external input to the population or mutual coupling between two populations for stable synchronization is analyzed as example.