The Koopman operator acts on observable functions defined over the state space of a dynamical system, thereby providing a linear global description of the system dynamics. A pointwise description of the system (e.g. in terms of lifted state dynamics) is recovered through a weak formulation, i.e. via the pointwise evaluation of observables at specific states. In this context, the use of reproducing kernel Hilbert spaces (RKHS) is of interest since the above evaluation can be represented as the duality pairing between the observables and (bounded) evaluation functionals. This representation emphasizes the relevance of a dual formulation for the Koopman operator, where a dual Koopman system governs the evolution of linear (evaluation) functionals. In this talk, we will report on two distinct works based on this general approach: global stability analysis and state estimation. First, we will present a systematic construction of Lyapunov functions based on the pointwise evaluation of Lyapunov functionals obtained for the dual Koopman system. For a specific RKHS (i.e. the Hardy space on the polydisc), this approach yields sufficient stability criteria that are directly expressed in terms of the Taylor expansion coefficients of the (analytic) vector field. Second, we will build a Luenberger observer that estimates the (infinite-dimensional) state of the Koopman dual system, and equivalently the (finite-dimensional) state of the nonlinear dynamics. This theoretical result supports numerical Koopman operator-based estimation techniques known from the literature. Moreover, we will introduce new concepts of observability and detectability for the dual Koopman system, which are shown to be equivalent to observability and detectability of the nonlinear system, and we will characterize them in terms of spectral properties.
A. Mauroy is an Associate Professor at the University of Namur, Belgium (Department of Mathematics and Namur Institute for Complex Systems). He received the M.S. degree in aerospace engineering and the Ph.D. degree in applied mathematics (under the supervision of R. Sepulchre) from the University of Liège, Belgium, in 2007 and 2011, respectively. Prior to joining UNamur, he was a postdoctoral researcher at UCSB from 2011 to 2013, at the University of Liège from 2013 to 2015, and at the University of Luxembourg in 2016. His research interests revolve around the application of (Koopman) operator-theoretic methods to dynamical systems, control, data-driven analysis, and network identification.
