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Networks & Heterogeneous Media

2017 , Volume 12 , Issue 3

Special issue on analysis and control on networks: Trends and perspectives. Part II

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Analysis and control on networks: Trends and perspectives
Fabio Ancona, Laura Caravenna, Annalisa Cesaroni, Giuseppe M. Coclite, Claudio Marchi and Andrea Marson
2017, 12(3): i-ii doi: 10.3934/nhm.201703i +[Abstract](87) +[HTML](0) +[PDF](130.1KB)
Traveling waves for degenerate diffusive equations on networks
Andrea Corli, Lorenzo di Ruvo, Luisa Malaguti and Massimiliano D. Rosini
2017, 12(3): 339-370 doi: 10.3934/nhm.2017015 +[Abstract](135) +[HTML](0) +[PDF](504.0KB)

In this paper we consider a scalar parabolic equation on a star graph; the model is quite general but what we have in mind is the description of traffic flows at a crossroad. In particular, we do not necessarily require the continuity of the unknown function at the node of the graph and, moreover, the diffusivity can be degenerate. Our main result concerns a necessary and sufficient algebraic condition for the existence of traveling waves in the graph. We also study in great detail some examples corresponding to quadratic and logarithmic flux functions, for different diffusivities, to which our results apply.

Coupling conditions for the transition from supersonic to subsonic fluid states
Martin Gugat, Michael Herty and Siegfried Müller
2017, 12(3): 371-380 doi: 10.3934/nhm.2017016 +[Abstract](235) +[HTML](0) +[PDF](597.1KB)

We discuss coupling conditions for the p-system in case of a transition from supersonic states to subsonic states. A single junction with adjacent pipes is considered where on each pipe the gas flow is governed by a general p-system. By extending the notion of demand and supply known from traffic flow analysis we obtain a constructive existence result of solutions compatible with the introduced conditions.

Nonlinear flux-limited models for chemotaxis on networks
Raul Borsche, Axel Klar and T. N. Ha Pham
2017, 12(3): 381-401 doi: 10.3934/nhm.2017017 +[Abstract](204) +[HTML](1) +[PDF](1358.5KB)

In this paper we consider macroscopic nonlinear moment models for the approximation of kinetic chemotaxis equations on a network. Coupling conditions at the nodes of the network for these models are derived from the coupling conditions of kinetic equations. The results of the different models are compared and relations to a Keller-Segel model on networks are discussed. For a numerical approximation of the governing equations an asymptotic well-balanced schemes is extended to directed graphs. Kinetic and macroscopic equations are investigated numerically and their solutions are compared for tripod and more general networks.

The Wigner-Lohe model for quantum synchronization and its emergent dynamics
Paolo Antonelli, Seung-Yeal Ha, Dohyun Kim and Pierangelo Marcati
2017, 12(3): 403-416 doi: 10.3934/nhm.2017018 +[Abstract](226) +[HTML](0) +[PDF](391.0KB)

We present the Wigner-Lohe model for quantum synchronization which can be derived from the Schrödinger-Lohe model using the Wigner formalism. For identical one-body potentials, we provide a priori sufficient framework leading the complete synchronization, in which L2-distances between all wave functions tend to zero asymptotically.

Optimal control of infinite-dimensional piecewise deterministic Markov processes and application to the control of neuronal dynamics via Optogenetics
Vincent Renault, Michèle Thieullen and Emmanuel Trélat
2017, 12(3): 417-459 doi: 10.3934/nhm.2017019 +[Abstract](123) +[HTML](0) +[PDF](682.4KB)

In this paper we define an infinite-dimensional controlled piecewise deterministic Markov process (PDMP) and we study an optimal control problem with finite time horizon and unbounded cost. This process is a coupling between a continuous time Markov Chain and a set of semilinear parabolic partial differential equations, both processes depending on the control. We apply dynamic programming to the embedded Markov decision process to obtain existence of optimal relaxed controls and we give some sufficient conditions ensuring the existence of an optimal ordinary control. This study, which constitutes an extension of controlled PDMPs to infinite dimension, is motivated by the control that provides Optogenetics on neuron models such as the Hodgkin-Huxley model. We define an infinite-dimensional controlled Hodgkin-Huxley model as an infinite-dimensional controlled piecewise deterministic Markov process and apply the previous results to prove the existence of optimal ordinary controls for a tracking problem.

Decay rates for elastic-thermoelastic star-shaped networks
Zhong-Jie Han and Enrique Zuazua
2017, 12(3): 461-488 doi: 10.3934/nhm.2017020 +[Abstract](117) +[HTML](0) +[PDF](2080.1KB)

This work discusses the asymptotic behaviour of a transmission problem on star-shaped networks of interconnected elastic and thermoelastic rods. Elastic rods are undamped, of conservative nature, while the thermoelastic ones are damped by thermal effects. We analyse the overall decay rate depending of the number of purely elastic components entering on the system and the irrationality properties of its lengths. First, a sufficient and necessary condition for the strong stability of the thermoelastic-elastic network is given. Then, the uniform exponential decay rate is proved by frequency domain analysis techniques when only one purely elastic undamped rod is present. When the network involves more than one purely elastic undamped rod the lack of exponential decay is proved and nearly sharp polynomial decay rates are deduced under suitable irrationality conditions on the lengths of the rods, based on Diophantine approximation arguments. More general slow decay rates are also derived. Finally, we present some numerical simulations supporting the analytical results.

Opinion dynamics on a general compact Riemannian manifold
Aylin Aydoǧdu, Sean T. McQuade and Nastassia Pouradier Duteil
2017, 12(3): 489-523 doi: 10.3934/nhm.2017021 +[Abstract](160) +[HTML](0) +[PDF](3287.3KB)

This work formulates the problem of defining a model for opinion dynamics on a general compact Riemannian manifold. Two approaches to modeling opinions on a manifold are explored. The first defines the distance between two points using the projection in the ambient Euclidean space. The second approach defines the distance as the length of the geodesic between two agents. Our analysis focuses on features such as equilibria, the long term behavior, and the energy of the system, as well as the interactions between agents that lead to these features. Simulations for specific manifolds, \begin{document} $\mathbb{S}^1, \mathbb{S}^2,$ \end{document} and \begin{document} $\mathbb{T}^2$ \end{document}, accompany the analysis. Trajectories given by opinion dynamics may resemble \begin{document} $n-$ \end{document}body Choreography and are called "social choreography". Conditions leading to various types of social choreography are investigated in \begin{document} $\mathbb{R}^2$ \end{document}.

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