Boundary feedback stabilization problems for hyperbolic equations

2006 ◽  
pp. 238-245 ◽  
Author(s):  
I. Lasiecka ◽  
R. Triggiani
Keyword(s):  
Hyperbolic Equations ◽  
Feedback Stabilization ◽  
Boundary Feedback

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

2021 ◽  
Author(s):  
Martin Gugat ◽  
Jan Giesselmann
Keyword(s):  
Sound Speed ◽  
Lyapunov Functions ◽  
Hyperbolic Equations ◽  
Coupled System ◽  
Feedback Stabilization ◽  
Diagonal Form ◽  
Small Initial Data ◽  
Boundary Feedback ◽  
Feedback Laws ◽  
Space Interval

The flow of gas through a pipeline network can be modelled by a coupled system of 1-d quasilinear  hyperbolic equations. Often for the solution of control problems it is convenient  to replace the quasilinear model by a simpler semilinear model. 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 neglecting the influence of the gas velocity which is justified in the applications since it is much smaller than the sound speed. For a star-shaped network of pipes 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 $L^2$-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. Numerical examples including  a comparison of the semilinear and the  quasilinear model are presented.

Boundary Feedback Stabilization and Control of Star-shaped Open Canals

2013 ◽  
Vol 46 (13) ◽  
pp. 100-104 ◽  
Author(s):  
Lihui Cen ◽  
Yugeng Xi ◽  
Dewei Li
Keyword(s):  
Feedback Stabilization ◽  
Boundary Feedback ◽  
And Control
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