System theoretic properties of linear infinite-dimensional port-Hamiltonian systems

• Applicant: Dr. Anthony Hastir
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 532208976

Linear port-Hamiltonian systems (pHs) constitute an interesting class of dynamical systems since it comprises a lot of physical applications. The port-Hamiltonian formulation is constructed via an Hamiltonian which is directly linked to the energy in the system. Dynamical systems in which different coordinates (such as the time and a spatial coordinate) are considered, are described by partial-differential equations (PDEs). By using an operator theoretic setting, PDEs may be modeled on infinite-dimensional spaces, giving rise to infinite-dimensional systems. PHs as linear infinite-dimensional systems are useful thanks to their ability to model many different dynamical systems such as transmission lines, wave equations, beam equations, Schrödinger equations, etc . Linear infinite-dimensional pHs are considered in this project. For that class of systems, we start by investigating properties of the transfer functions, either for internal or boundary control and observation. By properties, we mean notably regularity such as for instance membership to some classes (see e.g. the Wiener or the Callier-Desoer classes). Two conjectures are addressed in this context. As a second part of the project, we consider the Linear-Quadratic (LQ) optimal control problem for linear infinite-dimensional pHs. We present the associated operator Riccati equation (ORE) in the case of internal and boundary control and observation. We investigate the solution of the ORE in these different situations. Moreover, we expect to characterize the Hamiltonian operator corresponding to this problem in terms of Riesz-spectral property notably. Furthermore, in the case when the control and observation are at the boundary, we address the LQ optimal control problem in the frequency domain by using spectral factorization techniques, for which the properties of the transfer functions highlighted above play an important role. We illustrate the theoretical results on several examples.

• DFG Programme: WBP Position