We consider a model describing the flow of a fluid inside an elastic tube that is connected to two tanks. We study the linearized system through semigroup theory. Controlling the pressures in the tanks renders a hyperbolic PDE with boundary control. The linearization induces a one-dimensional linear manifold of equilibria; when those are factored out, the corresponding semigroup is exponentially stable. The location of the eigenvalues in dependence on the viscosity is discussed. Exact boundary controllability of the system is achieved by the Riesz basis approach including generalized eigenvectors. A minimal time for controllability is given. The corresponding result for internal distributed control is stated.
DOI : 10.1051/cocv/2014039
Keywords: Flow in elastic tube, semigroup, exponential stability, boundary control system, exact controllability, Riesz basis
Peralta, Gilbert 1, 2 ; Propst, Georg 2
@article{COCV_2015__21_2_583_0,
author = {Peralta, Gilbert and Propst, Georg},
title = {Stability and {Boundary} {Controllability} of a {Linearized} {Model} of {Flow} in an {Elastic} {Tube}},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {583--601},
year = {2015},
publisher = {EDP Sciences},
volume = {21},
number = {2},
doi = {10.1051/cocv/2014039},
zbl = {1330.35033},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2014039/}
}
TY - JOUR AU - Peralta, Gilbert AU - Propst, Georg TI - Stability and Boundary Controllability of a Linearized Model of Flow in an Elastic Tube JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 583 EP - 601 VL - 21 IS - 2 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2014039/ DO - 10.1051/cocv/2014039 LA - en ID - COCV_2015__21_2_583_0 ER -
%0 Journal Article %A Peralta, Gilbert %A Propst, Georg %T Stability and Boundary Controllability of a Linearized Model of Flow in an Elastic Tube %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 583-601 %V 21 %N 2 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2014039/ %R 10.1051/cocv/2014039 %G en %F COCV_2015__21_2_583_0
Peralta, Gilbert; Propst, Georg. Stability and Boundary Controllability of a Linearized Model of Flow in an Elastic Tube. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 583-601. doi: 10.1051/cocv/2014039
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