Boundary feedback stabilization of a semilinear model for the flow in star-shaped gas networks^*

¹ Department Mathematik, Friedrich-Alexander Universität Erlangen-Nürnberg (FAU), Cauerstr. 11, 91058 Erlangen, Germany.
² Department of Mathematics, Technical University of Darmstadt, Dolivostr. 15, 64293 Darmstadt, Germany.

^** Corresponding author: martin.gugat@fau.de

Received: 27 May 2020
Accepted: 1 June 2021

Abstract

The flow of gas through a pipeline network can be modelled by a coupled system of 1-d quasilinear hyperbolic equations. In this system, the influence of certain source terms that model friction effects is essential. Often for the solution of control problems it is convenient to replace the quasilinear model by a simpler semilinear model. In this paper, we analyze the behavior of such a semilinear model on a star-shaped network. The model is derived from the diagonal form of the quasilinear model by replacing the eigenvalues by the sound speed multiplied by 1 or −1 respectively. Thus in the corresponding eigenvalues the influence of the gas velocity is neglected, which is justified in the applications since it is much smaller than the sound speed in the gas. For a star-shaped network of horizontal pipes for suitable coupling conditions we present boundary feedback laws that stabilize the system state exponentially fast to a position of rest for sufficiently small initial data. We show the exponential decay of the L²-norm for arbitrarily long pipes. This is remarkable since in general even for linear systems, for certain source terms the system can become exponentially unstable if the space interval is too long. Our proofs are based upon an observability inequality and suitably chosen Lyapunov functions. At the end of the paper, numerical examples are presented that include a comparison of the semilinear model and the quasilinear system.