COLLABORATIVE RESEARCH: DYNAMICS OF DENSITIES: MODELING, CONTROL AND ESTIMATION

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Present day advances in guidance and control, from aerospace applications to modern device technologies, require ever more sophisticated ways to deal with the deluge of data and to cope with the probabilistic nature of uncertainty. The proposed research aims at developing novel computational techniques and new mathematics that address problems in decision and control of uncertain systems and in regulating collections of dynamical systems in a unified manner. Typical examples include the steering of collections of dynamical systems such as a collection of unmanned area vehicles, a particle beam, micro-mechanical systems subject to thermal fluctuations, laser-driven molecular reactions, or the scheduling and control of the flow of resources across a network. Multiple engineering and scientific disciplines stand to gain from successful completion of the project.The basis of the proposed framework is the newly discovered Riemannian structure of Optimal Mass Transport (OMT) on probability densities. The ensuing geometry allows new techniques for the analysis of stochastic differential equations, a new perspective of collective dynamics of particles that leads to new modeling principles, and leads to a unified approach for designing suitable control laws that ensure stability and performance of collections of dynamical systems. Density functions become the primary object of study as these represent the state of the collection of dynamical systems. Data provide empirical snapshots, statistics, and histograms viewed as points on the Riemannian manifold of density functions. The description of system-states and uncertainty via density functions is compatible with novel mathematical concepts and tools in partial differential equations that can be now used for control purposes. This new viewpoint transcends control applications and may impact non-equilibrium statistical mechanics, medical imaging, filtering and signal analysis. The mathematics that we will develop aim to extend the current framework OMT to address multi-dimensional objects and tensor fields and, thereby, applications in diffusion tensor imaging, stochastically driven density flows of interacting dynamical particles, and network transport applications will be a focus of the research.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.