Asymptotic behavior of flows in networks

Abstract

Using functional analytical and graph theoretical methods, we extend the results of [12] to more general transport processes in networks allowing space dependent velocities and absorption. We characterize asymptotic periodicity and convergence to an equilibrium by conditions on the underlying directed graph and the (average) velocities.2000 Mathematics Subject Classification: 35F25; 34B15.Our aim is-based on the paper [12]-to handle more general transport processes allowing space dependent velocities and absorption in networks. We model the problem by a system of partial di¤erential equations on a directed graph where the vertices serve as linking points between the edges. We show that the flow is (up to §

Explicit formulae for limit periodic flows on networks

“…For constant speeds, this condition is the same as in [1,14], but it is less general than in [17,Definition 4.8]. Under (4), we rescale time as τ = ct and, for each j ∈ M, make the change v j = c j u j and introduce the spatial variable…”

“…We note that this theorem is valid under a more general condition than (4), that only involves commensurability of average times of traversing the ergodic cycles, [9,17]. For our purpose we need an explicit expression of (T A (t)) t≥0 , that is the basis of the proof in [1].…”

“…The problem lends itself to a semigroup theoretical approach which, in particular, allows for proving elegant results on the long term behaviour of the flow on the network, see e.g. [1,9,14,17]. The main result of these papers is that, under certain assumptions on the structure of the network and on the speeds of the flow along each edge, the flow is asymptotically periodic.…”

Modeling and Analysis of Modal Switching in Networked Transport Systems

“…Finally, a laborious but straightforward use of standard wellposedness theory for the ODEs (13), (15) shows that u(t, ·) depends continuously there on the data.…”

“…While well-posedness and asymptotic behavior of similar systems on graphs without switching modes have been considered in [10,15] using semigroup theory and optimal control of networked transport systems have been considered in [8,9,13,5,12] taking ω ij (t) or ϕ i (t) as a control, we will here consider the switching function µ(·) as a control of the system. We note that working with such discrete-continuous nature of systems governed by ordinary differential equations (ODEs) is a rapidly developing area; however, similar systems involving partial differential equations (PDEs) have seldom been considered in the literature so far, although noting [3].…”