Explicit formulae for limit periodic flows on networks

Abstract

In recent papers it was shown that, under certain conditions, the C 0 -semigroup describing a flow on a network (metric graph) that contains terminal strong (ergodic) components converges to the direct sum of periodic semigroups generated by the flows on these components. In this note we shall provide an explicit description of these limit semigroups in terms of the components of the adjacency matrix of the line graph of the network. The result is based on the Frobenius-Perron theory and the estimates of long term behaviour of iterates reducible matrices.

Waves and diffusion on metric graphs with general vertex conditions

“…2.1], one needs to restrict to the case without sinks in the graph; that is, we have to assume that from every vertex the material is flowing out on at least one edge. In this case, the boundary conditions appearing in the description of the domain of the generator can be rewritten in the form Our results cover all situations where the boundary conditions can be written as in (3.2) which, apart from simple transport equation with constant coefficients like in [28,6,5,8], include also non-constant coefficients cases as in [30,4] or transport equations with scattering, see [31,15]. Yet another example of this kind are transmission line networks with nonconservative junction conditions as studied in [12].…”

Telegraph systems on networks and port-Hamiltonians. III. Explicit representation and long-term behaviour

“…In [6], the authors used this assumption to convert that problem to a transport problem with unit velocities. In [2,7],…”

“…) 1 The research has been partially supported by the National Science Centre of Poland Grant 2017/25/B/ST1/00051 and the National Research Foundation of South Africa Grant 82770 2 The research was completed while the author was a Doctoral Candidate in the Interdisciplinary Doctoral School at Lódź University of Technology, Poland.…”

Semigroups for dynamical processes on metric graphs

“…For a primitive matrix double-struckBw, the number of cells should stabilize at a certain level even though all the terminal strong components of G consist of cycles. For more details of this consideration, we refer to the explicit formulae of projection onto the eigenspace of double-struckBw associated with eigenvalue 1 computed in [27, Thm. 3.1].…”

“…The semigroup approach to linear transport equation on finite networks was initiated, independently of Barletti’s work, in 2005 by the author and Sikolya [22] and further pursued by Sikolya [23], Mátrai et al [24], Kunszenti–Kovács [25] and Banasiak et al [26,27]. Following the same line, Radl [28] considered the linear Boltzmann equation with scattering, Engel et al [29–31] and Boulite et al [32] vertex control problems, Klöss [33] wave equation, Bayazit et al [34,35] delay and non-autonomous transport problems, while Dorn et al [36,37], Kunszenti–Kovács [38], Namayanja [39] and Budde et al [40] studied transport problems in infinite networks.…”