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Flows on networks: recent results and perspectives. (English) Zbl 1301.35193

This well–written survey paper is on flows on networks and is organized as follows.
Section 2 presents results on existence of solutions and includes both the case of a single conservation law and systems of conversation laws. Section 3 provides various models to which the theory can be applied: vehicular traffic, gas pipelines, supply chains, data and telecommunication networks, irrigation channals and blood flows. Section 4 considers more complex solutions where a coupling of different solutions of equations (ODE–PDE or PDE–PDE) is needed. Also in this case various examples are provided, such as piston problems, sewer systems, supply chains and blood flow. Section 5 considers controlled systems, where the control acts within the coupling condition at the vertex \(\Psi\). The main examples are provided by vehicular traffic and gas pipelines. Section 6 presents some numerical methods for conservation laws on networks. Section 7 presents the specific applications of the developed methods: the study of optimal shape of vascular stent and vehicular traffic monitoring using GPS data from mobile sensors. The final section, section 8, presents open problems and future perspectives of this research.
The ample bibliography contains 125 items.

MSC:

35R02 PDEs on graphs and networks (ramified or polygonal spaces)
35L65 Hyperbolic conservation laws
34H05 Control problems involving ordinary differential equations
93C20 Control/observation systems governed by partial differential equations
93C15 Control/observation systems governed by ordinary differential equations
93B05 Controllability
49K15 Optimality conditions for problems involving ordinary differential equations
49K20 Optimality conditions for problems involving partial differential equations
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