Positivity and monotonicity methods in infinite-dimensional systems theory

• Applicant: Sahiba Arora
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 523942526

A dynamical system is a mathematical model that describes the time evolution of one or several (or infinitely many) quantities. It is often reasonable to view such a system as a black box with only limited ability to exert influence from the outside (via so-called inputs) and perform measurements (via so-called outputs). Mathematical systems theory is the study of input/output functioning of dynamical systems, which e.g., enables one to regulate the behaviour of the system via a proper choice of input. Since its inception in the 1940s, systems theory -both in finite and infinite dimensions- has seen tremendous mathematical development with applications, for instance, in engineering, computing, and mathematical biology. Within the mathematical field of functional analysis, for instance in operator semigroups, positivity has had a significant impact, and in turn, has given rise to various applications. At the same time, positivity is an intrinsic property of various real-world systems. For instance, systems involving population dynamics, the concentration of a substance, or probability all have a positivity preserving property, which loosely means that positive input results in a positive output. This suggests that an analysis of positive systems is a worthwhile endeavour. However, even though systems theory is an established area of mathematics, the study of positive systems has mainly focused on finite-dimensions. This project aims at initiating an in-depth study of infinite-dimensional positive systems and applying the theory to a number of concrete examples. In addition to system theoretical methods, we will employ various tools from functional analysis -in particular operator semigroups and the theory of Banach lattices- to investigate various aspects of positive systems. The three main concepts that we focus on are admissibility, input-to-state stability, and spectral theory.

• DFG Programme: WBP Position