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Networks & Heterogeneous Media

June 2018 , Volume 13 , Issue 2

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Functional model for extensions of symmetric operators and applications to scattering theory
Kirill D. Cherednichenko, Alexander V. Kiselev and Luis O. Silva
2018, 13(2): 191-215 doi: 10.3934/nhm.2018009 +[Abstract](43) +[HTML](25) +[PDF](571.43KB)
Abstract:

On the basis of the explicit formulae for the action of the unitary group of exponentials corresponding to almost solvable extensions of a given closed symmetric operator with equal deficiency indices, we derive a new representation for the scattering matrix for pairs of such extensions. We use this representation to explicitly recover the coupling constants in the inverse scattering problem for a finite non-compact quantum graph with $δ$-type vertex conditions.

A two-dimensional data-driven model for traffic flow on highways
Michael Herty, Adrian Fazekas and Giuseppe Visconti
2018, 13(2): 217-240 doi: 10.3934/nhm.2018010 +[Abstract](29) +[HTML](22) +[PDF](7915.97KB)
Abstract:

Based on experimental traffic data obtained from German and US highways, we propose a novel two-dimensional first-order macroscopic traffic flow model. The goal is to reproduce a detailed description of traffic dynamics for the real road geometry. In our approach both the dynamics along the road and across the lanes is continuous. The closure relations, being necessary to complete the hydrodynamics equation, are obtained by regression on fundamental diagram data. Comparison with prediction of one-dimensional models shows the improvement in performance of the novel model.

Stability and implementation of a cycle-based max pressure controller for signalized traffic networks
Leah Anderson, Thomas Pumir, Dimitrios Triantafyllos and Alexandre M. Bayen
2018, 13(2): 241-260 doi: 10.3934/nhm.2018011 +[Abstract](25) +[HTML](21) +[PDF](936.84KB)
Abstract:

Intelligent use of network capacity via responsive signal control will become increasingly essential as congestion increases on urban roadways. Existing adaptive control systems require lengthy location-specific tuning procedures or expensive central communications infrastructure. Previous theoretical work proposed the application of a max pressure controller to maximize network throughput in a distributed manner with minimal calibration. Yet this algorithm as originally formulated has unpractical hardware and safety constraints. We fundamentally alter the formulation of the max pressure controller to a setting where the actuation can only update once per multiple time steps of the modeled dynamics. This is motivated by the case of a traffic signal that can only update green splits based on observed link-counts once per "cycle time" of 60-120 seconds. Furthermore, we extend the domain of allowable actuations from a single signal phase to any convex combination of available signal phases to model intra-cycle signal changes dictated by pre-selected cycle green splits. We show that this extended max pressure controller will stabilize a vertical queueing network given restrictions on admissible demand flows that are slightly stronger than those suggested in the original formulation of max pressure. We ultimately apply our cycle-based extension of max pressure to a simulation of an existing arterial network and provide comparison to the control policy that is currently deployed at the modeled location.

Error bounds for Kalman filters on traffic networks
Ye Sun and Daniel B. Work
2018, 13(2): 261-295 doi: 10.3934/nhm.2018012 +[Abstract](21) +[HTML](20) +[PDF](1426.5KB)
Abstract:

This work analyzes the estimation performance of the Kalman filter (KF) on transportation networks with junctions. To facilitate the analysis, a hybrid linear model describing traffic dynamics on a network is derived. The model, referred to as the switching mode model for junctions, combines the discretized Lighthill-Whitham-Richards partial differential equation with a junction model. The system is shown to be unobservable under nearly all of the regimes of the model, motivating attention to the estimation error bounds in these modes. The evolution of the estimation error is investigated via exploring the interactions between the update scheme of the KF and the intrinsic physical properties embedded in the traffic model (e.g., conservation of vehicles and the flow-density relationship). It is shown that the state estimates of all the cells in the traffic network are ultimately bounded inside a physically meaningful interval, which cannot be achieved by an open-loop observer.

Uniform stability and mean-field limit for the augmented Kuramoto model
Seung-Yeal Ha, Jeongho Kim, Jinyeong Park and Xiongtao Zhang
2018, 13(2): 297-322 doi: 10.3934/nhm.2018013 +[Abstract](32) +[HTML](20) +[PDF](494.85KB)
Abstract:

We present two uniform estimates on stability and mean-field limit for the "augmented Kuramoto model (AKM)" arising from the second-order lifting of the first-order Kuramoto model (KM) for synchronization. In particular, we address three issues such as synchronization estimate, uniform stability and mean-field limit which are valid uniformly in time for the AKM. The derived mean-field equation for the AKM corresponds to the dissipative Vlasov-McKean type equation. The kinetic Kuramoto equation for distributed natural frequencies is not compatible with the frequency variance functional approach for the complete synchronization. In contrast, the kinetic equation for the AKM has a similar structural similarity with the kinetic Cucker-Smale equation which admits the Lyapunov functional approach for the variance. We present sufficient frameworks leading to the uniform stability and mean-field limit for the AKM.

Formation, stability and basin of phase-locking for Kuramoto oscillators bidirectionally coupled in a ring
Xiaoxue Zhao, Zhuchun Li and Xiaoping Xue
2018, 13(2): 323-337 doi: 10.3934/nhm.2018014 +[Abstract](20) +[HTML](22) +[PDF](417.49KB)
Abstract:

We consider the dynamics of bidirectionally coupled identical Kuramoto oscillators in a ring, where each oscillator is influenced sinusoidally by two neighboring oscillator. Our purpose is to understand its dynamics in the following aspects: 1. identify all the phase-locked states (or equilibria) with stability or instability; 2. estimate the basins for stable phase-locked states; 3. identify the convergence rate towards phase-locked states. The crucial tool in this work is the celebrated theory of Łojasiewicz inequality.

A conservation law with multiply discontinuous flux modelling a flotation column
Raimund Bürger, Stefan Diehl and María Carmen Martí
2018, 13(2): 339-371 doi: 10.3934/nhm.2018015 +[Abstract](26) +[HTML](23) +[PDF](5571.23KB)
Abstract:

Flotation is a unit operation extensively used in the recovery of valuable minerals in mineral processing and related applications. Essential insight to the hydrodynamics of a flotation column can be obtained by studying just two phases: gas and fluid. To this end, the approach based on the drift-flux theory, proposed in similar form by several authors, is reformulated as a one-dimensional non-linear conservation law with a multiply discontinuous flux. The unknown is the gas volume fraction as a function of height and time, and the flux function depends discontinuously on spatial position due to several feed inlets. The resulting model is similar, but not equivalent, to previously studied clarifier-thickener models for solid-liquid separation and therefore adds a new real-world application to the field of conservation laws with discontinuous flux. Steady-state solutions are studied in detail, including their construction by applying an appropriate entropy condition across each flux discontinuity. This analysis leads to operating charts and tables collecting all possible steady states along with some necessary conditions for their feasibility in each case. Numerical experiments show that the transient model recovers the steady states, depending on the feed rates of the different inlets.

Corrigendum to "(Almost) everything you always wanted to know about deterministic control problems in stratified domains"
Guy Barles and Emmanuel Chasseigne
2018, 13(2): 373-378 doi: 10.3934/nhm.2018016 +[Abstract](25) +[HTML](22) +[PDF](318.8KB)
Abstract:

The aim of this short note is: \begin{document} $(i)$ \end{document} to report an error in [1]; \begin{document} $(ii)$ \end{document} to explain why the comparison result of [1] lacks an hypothesis in the definition of subsolutions if we allow them to be discontinuous; \begin{document} $(iii)$ \end{document} to describe a simple counter-example; \begin{document} $(iv)$ \end{document} to show a simple way to correct this mistake, considering the classical Ishii's definition of viscosity solutions; \begin{document} $(v)$ \end{document} finally, to prove that this modification actually fixes the the comparison and stability results of [1].

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