Boundary Control of Conservation Laws Exhibiting Shocks

Bayen, Alexandre M.; Monache, Maria Laura Delle; Garavello, Mauro; Goatin, Paola; Piccoli, Benedetto

doi:10.1007/978-3-030-93015-8_2

Control Problems for Conservation Laws With Traffic Applications

2021

DOI: 10.1007/978-3-030-93015-8_2

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Boundary Control of Conservation Laws Exhibiting Shocks

Alexandre M. Bayen

Maria Laura Delle Monache

Mauro Garavello

et al.

Abstract: This chapter focuses on control of systems of conservation laws with boundary data. Problems with one or two boundaries are considered and, in particular, we focus on cases where shocks may be developed by the solution. However, for completeness we briefly discuss in Sect. 2.2 other existing results where singularities are prevented via suitable feedback controls such as in [32].

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Cited by 2 publications

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“…In various fields, first-order (typically nonlinear) hyperbolic equations on networks like (1.4) have gained interest and led to a wide range of literature in recent years (see, e.g., [2,Chapt. 3], [6,14,19,26] for traffic models and [12] for models of gas flow, and the references therein).…”

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Asymptotic expansion for the solution of a convection-diffusion problem in a thin graph-like junction

Mel'nyk

Klevtsovskiy

2022

ASY

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A steady-state convection-diffusion problem with a small diffusion of order O ( ε ) is considered in a thin three-dimensional graph-like junction consisting of thin cylinders connected through a domain (node) of diameter O ( ε ), where ε is a small parameter. Using multiscale analysis, the asymptotic expansion for the solution is constructed and justified. The asymptotic estimates in the norm of Sobolev space H 1 as well as in the uniform norm are proved for the difference between the solution and proposed approximations with a predetermined accuracy with respect to the degree of ε.

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Asymptotic expansion for the solution of a convection-diffusion problem in a thin graph-like junction

Mel'nyk

Klevtsovskiy

2022

ASY

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Asymptotic expansion for convection-dominated transport in a thin graph-like junction

Mel’nyk,

Rohde

2024

Anal. Appl.

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We consider for a small parameter [Formula: see text] a parabolic convection–diffusion problem with Péclet number of order [Formula: see text] in a three-dimensional graph-like junction consisting of thin curvilinear cylinders with radii of order [Formula: see text] connected through a domain (node) of diameter [Formula: see text] Inhomogeneous Robin type boundary conditions, that involve convective and diffusive fluxes, are prescribed both on the lateral surfaces of the thin cylinders and the boundary of the node. The asymptotic behavior of the solution is studied as [Formula: see text] i.e. when the diffusion coefficients are eliminated and the thin junction is shrunk into a three-part graph connected in a single vertex. A rigorous procedure for the construction of the complete asymptotic expansion of the solution is developed and the corresponding energy and uniform pointwise estimates are proved. Depending on the directions of the limit convective fluxes, the corresponding limit problems [Formula: see text] are derived in the form of first-order hyperbolic differential equations on the one-dimensional branches with novel gluing conditions at the vertex. These generalize the classical Kirchhoff transmission conditions and might require the solution of a three-dimensional cell-like problem associated with the vertex to account for the local geometric inhomogeneity of the node and the physical processes in the node. The asymptotic ansatz consists of three parts, namely, the regular part, node-layer part, and boundary-layer one. Their coefficients are classical solutions to mixed-dimensional limit problems. The existence and other properties of those solutions are analyzed.

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Boundary Control of Conservation Laws Exhibiting Shocks

Cited by 2 publications

References 73 publications

Asymptotic expansion for the solution of a convection-diffusion problem in a thin graph-like junction

Asymptotic expansion for the solution of a convection-diffusion problem in a thin graph-like junction

Asymptotic expansion for convection-dominated transport in a thin graph-like junction

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