WIAS Preprint No. 3016, (2023)

Optimal boundary control of the isothermal semilinear Euler equation for gas dynamics on a network


Authors

  • Bongarti, Marcelo
    ORCID: 0000-0002-9027-7702
  • Hintermüller, Michael
    ORCID: 0000-0001-9471-2479

2020 Mathematics Subject Classification

  • 49J20 49K20 35AXX

Keywords

  • optimal boundary control, gas dynamics, gas networks, isothermal Euler equation, compressible fluid dynamics, nonlinear hyperbolic PDE's, pointwise state constraints, non-singular Lagrange multiplier

DOI

10.20347/WIAS.PREPRINT.3016

Abstract

The analysis and boundary optimal control of the nonlinear transport of gas on a network of pipelines is considered. The evolution of the gas distribution on a given pipe is modeled by an isothermal semilinear compressible Euler system in one space dimension. On the network, solutions satisfying (at nodes) the Kirchhoff flux continuity conditions are shown to exist in a neighborhood of an equilibrium state. The associated nonlinear optimization problem then aims at steering such dynamics to a given target distribution by means of suitable (network) boundary controls while keeping the distribution within given (state) constraints. The existence of local optimal controls is established and a corresponding Karush--Kuhn--Tucker (KKT) stationarity system with an almost surely non--singular Lagrange multiplier is derived.

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