In this thesis, the author introduces the concept of ensemble observability of dynamical systems and develop a theoretical framework in which this system property is characterized. An ensemble is a collection of nearly identical copies of one dynamical system, and it is said to be observable if the distribution of the nonidentical initial states can be reconstructed from observing the time evolution of a corresponding distribution of outputs. Similarly, a single dynamical system with output is said to be ensemble observable if a collection of copies of this system, with different initial states, is observable when considered as an ensemble. The consideration of ensemble observability is, in particular, motivated by recent efforts in the study of heterogeneous cell populations. Therein one aims to reconstruct a distribution of states within a population of cells, but is only given the time evolution of the distribution of certain measured quantities within the population. More generally, the motivation for introducing ensemble observability is rooted in the very concept of ensembles itself, in which a collection of individual systems may only be considered as a whole. A main result of this thesis illustrates a fundamental connection between the concept of ensemble observability and mathematical tomography problems. Another main result concerns a natural connection to polynomial systems, which is encountered in the course of a systems theoretic treatment of the ensemble observability problem through the consideration of moments of the distributions. The author also establishes a duality of both approaches for linear systems.

An ensemble system is a collection of nearly identical dynamical systems which admit a certain degree of heterogeneity, and which are subject to the restriction that they may only be manipulated or observed as a whole. This thesis presents analysis and control methods for cellular ensembles by considering reduced 1-dimensional dynamics of biological processes in high-dimensional single-cell data and models. To be more specific, we address the quest for real-time analysis of biological processes within single-cell data by introducing the measure-preserving map of pseudotime into real-time, in short MAPiT. MAPiT enables the reconstruction of temporal and spatial dynamics from single-cell snapshot experiments. In addition, we propose a PDE-constrained learning algorithm which allows for efficient inference of changes in cell cycle progression from time series single-cell snapshot data. The second part of this thesis, is devoted to controlling a heterogeneous cell population, in the sense, that we aim at achieving a desired distribution of cellular oscillators on their periodic orbit. A systems theoretic approach to the ensemble control problem provides novel necessary and sufficient conditions for the control of phase distributions in terms of the Fourier coefficients of the phase response curve. This thesis establishes a connection between the previously separate areas of single cell analysis and ensemble control. Our holistic view opens new perspectives for theoretic concepts in basic research and therapeutic strategies in precision medicine.

Hybrid systems, that is, heterogeneous systems that include discrete and continuous time subsystems, have been used to model applications in automotive such as engine, brake, and stability control, as well as air traffic control and manufacturing plant control. Because of their generality (they include as special cases continuous and discrete systems), deriving rigorous controller synthesis procedures is difficult. The most effective hybrid control algorithms are based on full state feedback. However, in most cases, only partial information about the internal state of the hybrid plant can be measured. Observability and detectability are concepts of fundamental importance that establish the conditions for reconstruction of the state of a system and have been thoroughly investigated in the continuous and discrete domain but not as systematically for hybrid systems. Hybrid systems' observability involves both the discrete structure and the continuous dynamics of the system. A hybrid system is said to be observable when it is possible to reconstruct the discrete as well as the continuous state of the system from the observed output information. This paper reviews and places in context how the continuous and the discrete dynamics, as well as their interactions, intervene in the observability property of a quite general class of hybrid systems: linear hybrid systems called H-systems. Our specific objective is to show how the hybrid characteristics of the system come into play and give rise to particular aspects and properties that do not simply generalize the ones that are well-known for traditional dynamical systems. This paper intends to provide a tutorial approach to hybrid systems observability in its various forms to students in control and its application as well as to practitioners in the field.

This book commemorates the centenary of the birth of Georges David Birhoff, the father of the theory of Dynamical Systems. It consists of a volume of dedicated papers, reflecting the intellectual revolution of his work. This book is divided into four parts: Fundamental Paradigms — Chaos, Turbulence, Attractors, Bifurcations; Dynamical Systems and Microphysics; Self-Organization and Biological Dynamical Systems; Epistemology and History.

This book is an introduction to the field of dynamical systems, in particular, to the special class of Hamiltonian systems. The authors aimed at keeping the requirements of mathematical techniques minimal but giving detailed proofs and many examples and illustrations from physics and celestial mechanics. After all, the celestial $N$-body problem is the origin of dynamical systems and gave rise in the past to many mathematical developments. Jurgen Moser (1928-1999) was a professor atthe Courant Institute, New York, and then at ETH Zurich. He served as president of the International Mathematical Union and received many honors and prizes, among them the Wolf Prize in mathematics. Jurgen Moser is the author of several books, among them Stable and Random Motions in DynamicalSystems. Eduard Zehnder is a professor at ETH Zurich. He is coauthor with Helmut Hofer of the book Symplectic Invariants and Hamiltonian Dynamics. Information for our distributors: Titles in this series are copublished with the Courant Institute of Mathematical Sciences at New York University.

Over the last four decades there has been extensive development in the theory of dynamical systems. This book aims at a wide audience where the first four chapters have been used for an undergraduate course in Dynamical Systems. Material from the last two chapters and from the appendices has been used quite a lot for master and PhD courses. All chapters are concluded by an exercise section. The book is also directed towards researchers, where one of the challenges is to help applied researchers acquire background for a better understanding of the data that computer simulation or experiment may provide them with the development of the theory.