Flows on networks: recent results and perspectives
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.
control problems, coupled PDE--ODE and PDE--PDE systems
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