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
Mots clés : Flow in elastic tube, semigroup, exponential stability, boundary control system, exact controllability, Riesz basis
@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},
publisher = {EDP-Sciences},
volume = {21},
number = {2},
year = {2015},
doi = {10.1051/cocv/2014039},
zbl = {1330.35033},
language = {en},
url = {http://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 - http://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 http://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. http://www.numdam.org/articles/10.1051/cocv/2014039/
A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, 2nd ed. Birkhäuser, Boston (2007). | Zbl
and , Numerical investigation of the Liebau phenomenon. Z. Angew. Math. Phys. 54 (2003) 1050–1072. | DOI | Zbl
, , and , Stabilization through viscoelastic boundary damping: a semigroup approach. Semigroup Forum 80 (2010) 405–415. | DOI | Zbl
K.J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations. Springer-Verlag, New York (2000). | Zbl
, and , Analysis of a geometrical multiscale blood flow model based on the coupling of ODEs and Hyperbolic PDEs. Multiscale Model. Simul. 4 (2005) 215–236. | DOI | Zbl
, Riesz basis approach to the stabilization of a flexible beam with a tip mass. SIAM J. Control Optim. 39 (2001) 1736–1747. | DOI | Zbl
, Riesz basis property and exponential stability of controlled Euler-Bernoulli beam equations with variable coefficients. SIAM J. Control Optim. 40 (2002) 1905–1923. | DOI | Zbl
, and , On the -semigroup generation and exponential stabilization resulting from a shear force feedback on a rotating beam. Syst. Control Lett. 54 (2005) 557–574. | DOI | Zbl
and , The Riesz basis property of discrete operators and applications to a Euler-Bernoulli beam equations with boundary linear feedback control. IMA J. Math. Control Inf. 18 (2001) 241–251. | DOI | Zbl
, Some trigonometrical inequalities with applications to the theory of series. Math. Z. 41 (1936) 367–379. | DOI | JFM | Zbl
B. Jacob and H. Zwart, Linear Port Hamiltonian Systems on Infinite-Dimensional Spaces. Birkhäuser, Basel (2012). | Zbl
V. Komornik, Exact Controllability and Stabilization: the Multiplier Method. Wiley-Masson, Paris-Chicester (1994). | Zbl
V. Komornik and P. Loreti, Fourier Series in Control Theory. Springer-Verlag, New York (2005). | Zbl
and , Regularity of hyperbolic equations under boundary terms. Appl. Math. Optim. 10 (1983) 275–286 | DOI | Zbl
J.-L. Lions, Contrôle des systèmes distribués singuliers. Gauthiers-Villars, Paris (1968). | Zbl
M. Miklavc˘ic˘,Applied Functional Analysis and Partial Differential Equations. World Scientific, Singapore (1998). | Zbl
, Valveless pumping in a fluid-filled closed elastic tube-system: one-dimensional theory with experimental validation. J. Math. Biol. 46 (2003) 309–332. | DOI | Zbl
and , Der Fördereffekt in ventillosen, elastischen Leitungen. Z. Angew. Math. Phys. 29 (1978) 123–133. | DOI
, A coupled system of ODEs and quasilinear hyperbolic PDEs arising in a multiscale blood flow model. J. Math. Anal. Appl. 343 (2008) 778–798. | DOI | Zbl
O. Staffans, Well-Posed Linear Systems. Cambridge University Press (2005). | Zbl
and , Simultaneous exact controllability and some applications. SIAM J. Control Optim. 38 (2000) 1408–1427. | DOI | Zbl
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups. Birkhäuser-Verlag, Basel (2009). | Zbl
H.J. von Bredow, Untersuchung eines ventillosen Pumpprinzips. Fortschr. Ber. VDI-Zeitschr. Reihe 7 (1968).
and , Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation. SIAM J. Control Optim. 42 (2003) 966–984. | DOI | Zbl
and , Eigenvalues and eigenvectors of semigroup generators obtained from diagonal generators by feedback. Commun. Inf. Syst. 11 (2011) 71–104. | Zbl
R. Young, An Introduction to Nonharmonic Fourier Analysis. Academic Press, New York (1980).
Cité par Sources :
