This course objective is to help students to acquire the concepts and main results as well as the methods of the theory of infinite-dimensional dynamical systems (distributed parameter systems, nonlinear systems analysis via the Koopman operator). The various aspects of studying such systems (modelling, analysis, design, simulation) are discussed in lectures, tutorials and personal work.
Part 1. Study of linear differential equations where the state variable evolves in a Banach or Hilbert space of infinite dimension. Generalisation of the concept of matrix exponential. The homogeneous and controlled Cauchy problems. Applications to partial differential equations (PDE), such as the heat equation, the vibrating string or reaction-diffusion equations. Extensions to the analysis of semi-linear PDE on invariant domains.
Part 2. Definition of the semigroup of Koopman operators assoiciated with a nonlinear dynamical system. Properties in spaces of continuous functions and Lp spaces (strong continuity, contraction, dissipativity). Infinitesimal generator of the Koopman semigroup. Dual operator. Relationships with the properties of dynamical systems. Spectral properties.
Report, seminars, and oral presentations.
Jacob B. and Zwart H., Linear port-Hamiltonian systems on infinite-dimensional spaces, Birkhäuser, Basel, 2012
Lasota, A., & Mackey, M. C., Chaos, fractals, and noise: stochastic aspects of dynamics (Vol. 97). Springer Science & Business Media, 2013
Bátkai, András, M. Kramar Fijavž, and Abdelaziz Rhandi. Positive operator semigroups. Birkhauser Verlag Ag, 2017.