We address the observability problem for ensembles that are described by probability distributions. The problem is to reconstruct a probability distribution of the initial state from the time-evolution of the probability distribution of the output under a classical finite-dimensional linear system. We present two solutions to this problem, one based on formulating the problem as an inverse problem and the other one based on reconstructing all the moments of the distribution. The first approach leads us to a connection between the reconstruction problem and mathematical tomography problems. In the second approach we use the framework of tensor systems to describe the dynamics of the moments which leads to a more systems theoretic treatment of the reconstruction problem. Furthermore we show that both frameworks are inherently related. The appeal of having two dual viewpoints, the first being more geometric and the second one being more systems theoretic, is illuminated in several examples of theoretical or practical importance.
In this paper we address an inverse problem on populations described by probability distributions. From a theoretical point of view, this problem can be seen as a natural extension to the classical observability problem. We consider a population that is described by a classical linear finite-dimensional system in which the initial state is a random vector subject to a non-parametric probability distribution. The problem is to reconstruct this initial state distribution from the time-evolution of the probability distribution of the output. We reveal as a novel viewpoint, that, at its core, this problem is a tomography problem which is a well-known subject in the field of inverse problems. Furthermore we show how this tomography problem is inherently linked with the observability properties of the finite-dimensional system thereby establishing a beautiful link between a control theoretic question and tomography problems.
53rd IEEE Conference on Decision and Control
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