On Applications of the Spectral Theory of the Koopman Operator in Dynamical Systems and Control Theory

Recent contributions have extended the applicability of Koopman operator theory from dynamical systems to control. Stability theory got reformulated in terms of spectral properties of the Koopman operator [1], providing a nice link between the way we treat linear systems and nonlinear systems and opening the door for the use of classical linear e.g. pole placement theory in the fully nonlinear setting. New concepts such as isostables proved useful in the context of optimal control. Here, using Kato Decomposition we develop Koopman expansion for general LTI systems. We also interpret stable and unstable subspaces in terms of zero level sets of Koopman eigenfunctions. We then utilize conjugacy properties of Koopman eigenfunctions to extend these results to globally stable systems. In conclusion, we discuss how the classical Hamilton-Jacobi-Bellman setting for optimal control can be reformulated in operator-theoretic terms and point the applicability of spectral operator theory in max-plus algebra to it. Geometric theories such as differential positivity have been also related to spectral theories of the Koopman operator [2], in cases when the attractor is a fixed point or a limit cycle, pointing the way to the more general case of quasiperiodic and chaotic attractors.