Admissibility of control operators for positive semigroups and robustness of input-to-state stability

Abstract: In this paper, we establish the well-posedness and stability of distributed parameter systems, focusing on linear positive control systems in a Banach lattice setting. We characterize well-posedness and derive a sufficient condition for admissibility based on a lower norm estimate of the resolvent operator on the positive cone. Furthermore, we analyze input-to-state stability (ISS) under boundary perturbations within the domain of the semigroup generator. Notably, we provide necessary and sufficient conditions for the robustness of ISS under Desch-Schappacher perturbations. Our theoretical results are demonstrated through a boundary value-controlled transport equation with non-local boundary conditions.