Boundary feedback stabilization of the isothermal Euler-equations with uncertain boundary data
Submission Status: | revision under review |
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- In a gas transport system, the
customer behavior is uncertain.
Motivated by this situation, we consider
a boundary stabilization problem
for the flow through a gas pipeline,
where the outflow at one
end of the pipe
%that is governed by the customer's behavior
is uncertain.
The control action is located
at the
other end of the pipe.
The feedback law is
a classical
Neumann velocity feedback with a feedback parameter $k>0$.
We show that
as long as the
$H^1$-norm of the function that describes the noise in the customer's
behavior decays exponentially with
a rate that is sufficiently large,
the velocity of the gas can be stabilized exponentially
fast in the sense that a suitably chosen Lyapunov function decays exponentially.
For the exponential stability it is sufficient
that the feedback parameter $k$ is sufficiently large
and the stationary state to which the system is stabilized is
sufficiently small.
The stability result is local, that is it holds for initial
states that are sufficiently close to the stationary state.
This result is an example for the exponential boundary feedback stabilization of
a quasilinear hyperbolic system with uncertain boundary data.
The analysis is based upon the choice of a suitably Lyapunov function.
The decay of this Lyapunov function implies that
also the $L^2$-norm of the
difference of the system state and the stationary state decays exponentially.
MetadatenAuthor: | Martin Gugat, Rüdiger Schultz |
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Document Type: | Preprint |
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Language: | English |
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Date of Publication (online): | 2017/12/09 |
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Date of first Publication: | 2017/09/14 |
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Release Date: | 2017/09/14 |
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Subprojects: | C03 |
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