In a visionary outline of mathematics and its future at the beginning of the 20th century, Poincaré suggested that complicated dynamics governed by non-linear partial differential equations can be reduced to and analyzed by the linear infinite dimensional spectral methods advocated by Hilbert and Fredholm. Poincaré’s prophecy became reality after 1920 —due to not less visionary contributions of Carleman, Koopman and von Neumann. Although originally aimed at ergodic theory of measure-preserving systems, these linear operator reductions of dynamical systems have far reaching implications for dissipative systems of modern interest, and a much wider, unexpected area of applicability. But, it was only in the 1990s that potential for wider applications of the operator-theoretic approach were realized. Over the past two decades, the trend of applications of this approach has further evolved from the perspectives of theory, computation and numerical methods.
The hallmark of the work on the operator-theoretic approach in the last two decades is the linkage between geometrical properties of dynamical systems with the geometrical properties of the level sets of Koopman eigenfunctions in state space, and Koopman modes in configuration space. This approach also provides an opportunity for study of high-dimensional evolution equations in terms of dynamical systems concepts via a spectral decomposition, and links with associated numerical methods for such evolution equations. From this operator-theoretic framework, new data-drive algorithms are developed that give insight in the underlying processes even in the non-autonomous cases or in the presence of uncertainty. Applications to fluid dynamics, large power networks, climatology, physiology, pharmacology, disease dynamics, social precesses, synthetic and molecular biology, robotics, and numerous other fields of science and engineering have followed the theoretical and computational developments.
The workshop will bring together experts in different facets of Koopman operator perspective on dynamical and control systems. It will be an interplay between ergodic theory, operator theory, geometric dynamical systems, control theory and convex optimization, estimation, computational aspects of global optimization and applied linear algebra.
(Centre National de la Recherche Scientifique (CNRS), Laboratoire d'Analyse et d'Architecture des Systemes (LAAS))
Senka Maćešić (University of Rijeka)
Igor Mezic (University of California, Santa Barbara (UCSB), Mechanical Engineering)
Mihai Putinar (University of California, Santa Barbara (UCSB), Mathematics)