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.
Highlights
The flow of gas through pipelines is governed by a quasilinear system of balance laws
We have considered the flow of gas in a star-shaped network of pipelines that is governed by a hyperbolic semilinear model of partial differential equations that can be understood as a simplification of the isothermal Euler equations
We have shown that locally around a state where the gas is at rest the system state can be steered towards this position of rest exponentially fast with a suitably chosen boundary feedback control
Summary
The flow of gas through pipelines is governed by a quasilinear system of balance laws (see for example [2]). We show that on a given finite time horizon [0, T ], the gas flow in a star-shaped network can be steered to a position of rest exponentially fast in the sense of the L2-norm if the L∞-norm of the initial state is sufficiently small. The exponential stabilization of the gas flow governed by the isothermal Euler equations in fan-shaped networks in the L2-sense has been studied in [12]. A strict H1-Lyapunov function and feedback stabilization for the quasilinear isothermal Euler equations with friction have been studied in [8]. We show simulations for the original quasilinear model with the suggested boundary feedback that indicate that in this case the system decays exponentially
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