Positivity of infinite-dimensional linear systems

TL;DR

This paper studies the well-posedness and positivity of infinite-dimensional linear systems with unbounded operators, providing characterizations, conditions, and perturbation results, with applications to the linear Boltzmann equation.

Contribution

It offers a comprehensive characterization of positivity in infinite-dimensional systems, establishes equivalences between regularity notions, and applies findings to boundary value problems.

Findings

Weak and strong regularity are equivalent for positive systems.

Provided sufficient conditions for zero-class admissibility.

Established perturbation results for positive semigroups.

Abstract

In this paper, we investigate the well-posedness and positivity property of infinite-dimensional linear system with unbounded input and output operators. In particular, we characterize the internal and external positivity for this class of systems. This latter effort is motivated in part by a complete description of well-posed positive control/observation systems. An interesting feature of positive well-posed linear systems is that weak regularity and strong regularity, in the sense of Salamon and Weiss, are equivalent. Moreover, we provide sufficient conditions for zero-class admissibility for positive semigroups. In the context of positive perturbations of positive semigroups, we establish two perturbation results, namely the Desch-Schappacher perturbation and the Staffans-Weiss perturbation. As for illustration, these findings are applied to investigate the existence and uniqueness of a positive mild solution of the linear Boltzmann equation with non-local boundary conditions on finite network.