Asymptotic periodicity of flows in time-depending networks

Abstract

We consider a linear transport equation on the edges of a network with time-varying coefficients. Using methods for non-autonomous abstract Cauchy problems, we obtain well-posedness of the problem and describe the asymptotic profile of the solutions under certain natural conditions on the network. We further apply our theory to a model used for air traffic flow management

Diffusion in Networks with Time-Dependent Transmission Conditions

“…This paper seems to be among the first studies of asymptotic behavior for non-autonomous evolution equation in a network. A recent manuscript [BDK13] treats a first order transport equation on the edges of a network with time-varying transmission conditions at the vertices using the approach of difference equations and evolution families. However, the nature of our problem and the techniques we use are completely different.…”

Bi-Continuous semigroups for flows on infinite networks

“…While the latter situation is the most common one (see e.g. [16,24,9]) the first one was considered for finite networks in [10,Sect. 5].…”

“…Such a problem was considered by Dorn et al [16,15,17] on the state space L 1 [0, 1] , 1 applying the theory of strongly continuous operator semigroups. A semigroup approach to flows in finite metric graphs was first presented by Kramar and Sikolya [24] and further used in [28,18,17,10,5,9] while transport processes in infinite networks were also studied in [6,8]. All these results were obtained in the L 1 -setting.…”

Semigroups for dynamical processes on metric graphs

“…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. New insights into the relation between network structure and dynamics were given in [41,42].…”