There is long history of use of mathematical decompositions to describe complex phenomena using simpler ingredients. One example is the decomposition of string vibrations into its primary, secondary, and higher modes. Recently, a spectral decomposition relying on Koopman operator theory has attracted interest in science and engineering communities. The spectral decomposition is based on an extension of the Koopman-von Neumann formalism to dissipative, possibly infinite-dimensional systems, including those describing flow of viscous fluids at the fundamental level, but also thermal flows in buildings, and power grid dynamics, at a more applied level. At its mathematical foundations, it is a spectral theory of composition operators. We will present the foundations of the theory, the numerical analysis approach, and review some of its applications in control and data analytics.
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