Existence and Optimal Boundary Control of Classical Solutions to Networks of Quasilinear Hyperbolic Systems of Balance Laws

  • We study the existence, stability and optimal control of classical solutions of networked strictly hyperbolic systems of balance laws. Such networks arise, for instance, in the modeling of the gas dynamics in a network of gas pipelines, traffic networks, or networks of water channels. It is assumed that characteristic speeds are nonzero and do not change sign. Node conditions are stated in a general way by requiring that the boundary traces of all states adjacent to a vertex satisfy an algebraic equation. With a suitable assumption, this equation can be solved for outgoing characteristic variables as a function of the incoming ones. We prove the unique existence of a classical solution for arbitrarily large initial and boundary values on a possibly small time horizon. We derive continuity and differentiability properties of the solution operator of the quasilinear problem w.r.t. the control term located in the node conditions. Then we analyze an optimal control problem for the networked system, where the control is of boundary type. We prove the existence of optimal controls and the differentiability of the objective functional w.r.t. the controls.

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Author:Jannik Breitkopf, Martin Gugat, Stefan Ulbrich
Document Type:Preprint
Language:English
Date of Publication (online):2025/11/21
Date of first Publication:2025/11/21
Release Date:2025/11/21
Tag:networked systems, classical solutions, quasilinear hyperbolic systems, boundary control, conservation laws, nodal control, optimal nodal control
Page Number:42
Institutes:Friedrich-Alexander-Universität Erlangen-Nürnberg
Technische Universität Darmstadt
Subprojects:A02
C03
C05
Licence (German):License LogoCreative Commons - CC BY - Namensnennung 4.0 International
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