Ensemble observability of dynamical systems

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This thesis introduces ensemble observability of dynamical systems and develops a theoretical framework to characterize this property. An ensemble consists of nearly identical copies of a dynamical system, deemed observable if the distribution of nonidentical initial states can be reconstructed from the time evolution of corresponding output distributions. A single system is ensemble observable if a collection of copies with different initial states is observable as a whole. The motivation for this concept arises from studies of heterogeneous cell populations, where the goal is to reconstruct a distribution of states within a cell population based on the time evolution of certain measured quantities. The introduction of ensemble observability is rooted in the idea of ensembles, where individual systems are considered collectively. A key result illustrates a fundamental link between ensemble observability and mathematical tomography problems. Additionally, the thesis establishes a connection to polynomial systems encountered during a systems-theoretic analysis of ensemble observability through the examination of distribution moments. It also demonstrates a duality between both approaches for linear systems.