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Multivariable boundary PI control and regulation of a fluid flow system
Observability and controllability analysis of blood flow network
1. | Department of Mathematics, Tianjin University, Tianjin, 300072, China, China |
References:
[1] |
L. Formaggia, F. Nobile, A. Quarteroni and A. Venezini, Multiscale modelling of the circulatory system: A preliminary analysis,, Computing and visualization in science., 2 (1999), 75.
doi: 10.1007/s007910050030. |
[2] |
A. Quarteroni and A. Venezini, Analysis of a geometrical multiscale model based on the coupling of ODE's and PDE's for blood flow simulation,, Multiscale Modeling & Simulation., 1 (2003), 173.
doi: 10.1137/S1540345902408482. |
[3] |
L. Formaggia, J. F. Gerbeau, F. Nobile and A. Quarteroni, On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels,, Comp. Meth. Appl. Mech. Engng., 191 (2001), 561.
doi: 10.1016/S0045-7825(01)00302-4. |
[4] |
S. Čanić and E. H. Kim, Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi-symmetric vessels,, Math. Methods Appl. Sci., 26 (2003), 1161.
doi: 10.1002/mma.407. |
[5] |
T. Li and K. Zhao, Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model,, Netw. Heterog. Media., 6 (2011), 625.
doi: 10.3934/nhm.2011.6.625. |
[6] |
S. J. Sherwin, V. Franke, J. Peiró and K. Parker, One-dimensional modelling of a vascular network in space-time variables,, J. Engineering Mathematics., 47 (2003), 217.
doi: 10.1023/B:ENGI.0000007979.32871.e2. |
[7] |
A. Harloff, F. Albrecht, J. Spreer, A. F. Stalder, J. Bock, A. Frydrychowicz, J. Schöllhorn, A. Hetzel, M. Schumacher, J. Hennig and M. Markl, 3D blood blow characteristics in the carotid artery bifurcation assessed by flow-sensitive 4D MRI at 3T,, Magnetic Resonance in Medicine., 61 (2009), 65. Google Scholar |
[8] |
F. N. van de Vosse and N. Stergiopulos, Pulse Wave Propagation in the Arterial Tree,, Annu. Rev. Fluid Mech., 43 (2011), 467.
doi: 10.1146/annurev-fluid-122109-160730. |
[9] |
J. O. Barber, I. P. Alberding, J. M. Restrepo and T. W. Secomb, Simulated two-dimensional red blood cell motion, deformation and partitioning in microvessel bifurcations,, Ann Biomed Eng., 36 (2008), 1690.
doi: 10.1007/s10439-008-9546-4. |
[10] |
L. Formaggia, D. Lamponi, M. Tuveri and A. Veneziani, Numerical modeling of 1D arterial networks coupled with a lumped parameters description of the heart,, Compute Methods Biomech Biomed Engin., 9 (2006), 273.
doi: 10.1080/10255840600857767. |
[11] |
J. J. Wang and K. Parker, Wave propagation in a model for arterial circulation,, J. Biomech., 37 (2004), 457.
doi: 10.1016/j.jbiomech.2003.09.007. |
[12] |
S. Z. Zhao, X. Y. Xu, A. D. Hughes, S. A. Thom, A. V. Stanton, B. Ariff and Q. Long, Blood flow and vessel mechanics in a physiologically realistic model of a human carotid arterial bifurcation,, J. Biomech., 33 (2000), 975.
doi: 10.1016/S0021-9290(00)00043-9. |
[13] |
M. A. Fernández, V. Milišić and A. Quarteroni, Analysis of a geometrical multiscale blood flow model based on the coupling of ODE's and hyperbolic PDE's,, SIAM J. Multiscale Mod. Sim., 4 (2005), 215.
doi: 10.1137/030602010. |
[14] |
V. Milišić and A. Quarteroni, Analiysis of lumped parameter models for blood flow simulations and their relation with 1D models,, ESAIM: Mathematical Modelling and Numerical Analysis., 38 (2004), 613.
doi: 10.1051/m2an:2004036. |
[15] |
A. Quarteroni, M. Tuveri and A. Veneziani, Computational vascular fluid dynamics: Problems, models and methods,, Comput. Visual Sci., 2 (2000), 163.
doi: 10.1007/s007910050039. |
[16] |
I. C. Gohberg and M. G. Kreĭn, Introduction to the Theory of Linear Non-Selfadjoint Operators,, Translations of Mathematical Monographs, (1969).
|
[17] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Berlin: Springer-Verlag, (1983).
doi: 10.1007/978-1-4612-5561-1. |
[18] |
R. Curtain and G. Weiss, Exponential stabilization of well-posed systems by collocated feedback,, SIAM J. Control & Optim., 45 (2006), 273.
doi: 10.1137/040610489. |
[19] |
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups,, Birkhäuser Adv anced Texts: Basler Lehrbücher., (2009).
doi: 10.1007/978-3-7643-8994-9. |
[20] |
R. Mennicken and M. Möller, Non-self-adjoint Boundary Eigenvalue Problems,, North-Holland Mathematics Studies, (2003).
|
[21] |
S. A. Avdonin and S. A. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter System,, Cambridge University Press, (1995).
|
[22] |
G. Q. Xu and Y. X. Zhang, Vector-valued hyperbolic system and applications to complex networks of strings,, Joint 48th IEEE CDC and 28th CCC, (2009), 6448.
doi: 10.1109/CDC.2009.5400250. |
[23] |
R. M. Young, An Introduction to Nonharmonic Fourier Series,, Pure and Applied Mathematics, (1980).
|
[24] |
G. Q. Xu, C. Liu and S. P. Yung, Necessary conditions for the exact observability of systems on Hilbert spaces,, Systems & Control Letters, 57 (2008), 222.
doi: 10.1016/j.sysconle.2007.08.006. |
[25] |
G. Q. Xu and Y. F. Shang, Characteristic of left invertible semigroups and admissibility of observation operators,, Systems & Control Letters, 58 (2009), 561.
doi: 10.1016/j.sysconle.2009.03.006. |
[26] |
G. Q. Xu and S. P. Yung, The expansion of semigroup and criterion of Riesz Basis,, J. Diff. Equat., 210 (2005), 1.
doi: 10.1016/j.jde.2004.09.015. |
[27] |
G. Q. Xu, Z. J. Han and S. P. Yung, Riesz basis property of serially connected Timoshenko beams,, International Journal of Control, 80 (2007), 470.
doi: 10.1080/00207170601100904. |
[28] |
G. Q. Xu and B. Z. Guo, Riesz basis property of Evolution equations in Hilbert space and application to a coupled string equation,, SIAM J. Control & Optim., 42 (2003), 966.
doi: 10.1137/S0363012901400081. |
[29] |
B. Z. Guo and Y. Xie, A sufficient condition on Riesz basis with parentheses of non-self-adjoint operator and application to a serially connected string system under joint feedbacks,, SIAM J.Control & Optim., 43 (2004), 1234.
doi: 10.1137/S0363012902420352. |
show all references
References:
[1] |
L. Formaggia, F. Nobile, A. Quarteroni and A. Venezini, Multiscale modelling of the circulatory system: A preliminary analysis,, Computing and visualization in science., 2 (1999), 75.
doi: 10.1007/s007910050030. |
[2] |
A. Quarteroni and A. Venezini, Analysis of a geometrical multiscale model based on the coupling of ODE's and PDE's for blood flow simulation,, Multiscale Modeling & Simulation., 1 (2003), 173.
doi: 10.1137/S1540345902408482. |
[3] |
L. Formaggia, J. F. Gerbeau, F. Nobile and A. Quarteroni, On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels,, Comp. Meth. Appl. Mech. Engng., 191 (2001), 561.
doi: 10.1016/S0045-7825(01)00302-4. |
[4] |
S. Čanić and E. H. Kim, Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi-symmetric vessels,, Math. Methods Appl. Sci., 26 (2003), 1161.
doi: 10.1002/mma.407. |
[5] |
T. Li and K. Zhao, Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model,, Netw. Heterog. Media., 6 (2011), 625.
doi: 10.3934/nhm.2011.6.625. |
[6] |
S. J. Sherwin, V. Franke, J. Peiró and K. Parker, One-dimensional modelling of a vascular network in space-time variables,, J. Engineering Mathematics., 47 (2003), 217.
doi: 10.1023/B:ENGI.0000007979.32871.e2. |
[7] |
A. Harloff, F. Albrecht, J. Spreer, A. F. Stalder, J. Bock, A. Frydrychowicz, J. Schöllhorn, A. Hetzel, M. Schumacher, J. Hennig and M. Markl, 3D blood blow characteristics in the carotid artery bifurcation assessed by flow-sensitive 4D MRI at 3T,, Magnetic Resonance in Medicine., 61 (2009), 65. Google Scholar |
[8] |
F. N. van de Vosse and N. Stergiopulos, Pulse Wave Propagation in the Arterial Tree,, Annu. Rev. Fluid Mech., 43 (2011), 467.
doi: 10.1146/annurev-fluid-122109-160730. |
[9] |
J. O. Barber, I. P. Alberding, J. M. Restrepo and T. W. Secomb, Simulated two-dimensional red blood cell motion, deformation and partitioning in microvessel bifurcations,, Ann Biomed Eng., 36 (2008), 1690.
doi: 10.1007/s10439-008-9546-4. |
[10] |
L. Formaggia, D. Lamponi, M. Tuveri and A. Veneziani, Numerical modeling of 1D arterial networks coupled with a lumped parameters description of the heart,, Compute Methods Biomech Biomed Engin., 9 (2006), 273.
doi: 10.1080/10255840600857767. |
[11] |
J. J. Wang and K. Parker, Wave propagation in a model for arterial circulation,, J. Biomech., 37 (2004), 457.
doi: 10.1016/j.jbiomech.2003.09.007. |
[12] |
S. Z. Zhao, X. Y. Xu, A. D. Hughes, S. A. Thom, A. V. Stanton, B. Ariff and Q. Long, Blood flow and vessel mechanics in a physiologically realistic model of a human carotid arterial bifurcation,, J. Biomech., 33 (2000), 975.
doi: 10.1016/S0021-9290(00)00043-9. |
[13] |
M. A. Fernández, V. Milišić and A. Quarteroni, Analysis of a geometrical multiscale blood flow model based on the coupling of ODE's and hyperbolic PDE's,, SIAM J. Multiscale Mod. Sim., 4 (2005), 215.
doi: 10.1137/030602010. |
[14] |
V. Milišić and A. Quarteroni, Analiysis of lumped parameter models for blood flow simulations and their relation with 1D models,, ESAIM: Mathematical Modelling and Numerical Analysis., 38 (2004), 613.
doi: 10.1051/m2an:2004036. |
[15] |
A. Quarteroni, M. Tuveri and A. Veneziani, Computational vascular fluid dynamics: Problems, models and methods,, Comput. Visual Sci., 2 (2000), 163.
doi: 10.1007/s007910050039. |
[16] |
I. C. Gohberg and M. G. Kreĭn, Introduction to the Theory of Linear Non-Selfadjoint Operators,, Translations of Mathematical Monographs, (1969).
|
[17] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Berlin: Springer-Verlag, (1983).
doi: 10.1007/978-1-4612-5561-1. |
[18] |
R. Curtain and G. Weiss, Exponential stabilization of well-posed systems by collocated feedback,, SIAM J. Control & Optim., 45 (2006), 273.
doi: 10.1137/040610489. |
[19] |
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups,, Birkhäuser Adv anced Texts: Basler Lehrbücher., (2009).
doi: 10.1007/978-3-7643-8994-9. |
[20] |
R. Mennicken and M. Möller, Non-self-adjoint Boundary Eigenvalue Problems,, North-Holland Mathematics Studies, (2003).
|
[21] |
S. A. Avdonin and S. A. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter System,, Cambridge University Press, (1995).
|
[22] |
G. Q. Xu and Y. X. Zhang, Vector-valued hyperbolic system and applications to complex networks of strings,, Joint 48th IEEE CDC and 28th CCC, (2009), 6448.
doi: 10.1109/CDC.2009.5400250. |
[23] |
R. M. Young, An Introduction to Nonharmonic Fourier Series,, Pure and Applied Mathematics, (1980).
|
[24] |
G. Q. Xu, C. Liu and S. P. Yung, Necessary conditions for the exact observability of systems on Hilbert spaces,, Systems & Control Letters, 57 (2008), 222.
doi: 10.1016/j.sysconle.2007.08.006. |
[25] |
G. Q. Xu and Y. F. Shang, Characteristic of left invertible semigroups and admissibility of observation operators,, Systems & Control Letters, 58 (2009), 561.
doi: 10.1016/j.sysconle.2009.03.006. |
[26] |
G. Q. Xu and S. P. Yung, The expansion of semigroup and criterion of Riesz Basis,, J. Diff. Equat., 210 (2005), 1.
doi: 10.1016/j.jde.2004.09.015. |
[27] |
G. Q. Xu, Z. J. Han and S. P. Yung, Riesz basis property of serially connected Timoshenko beams,, International Journal of Control, 80 (2007), 470.
doi: 10.1080/00207170601100904. |
[28] |
G. Q. Xu and B. Z. Guo, Riesz basis property of Evolution equations in Hilbert space and application to a coupled string equation,, SIAM J. Control & Optim., 42 (2003), 966.
doi: 10.1137/S0363012901400081. |
[29] |
B. Z. Guo and Y. Xie, A sufficient condition on Riesz basis with parentheses of non-self-adjoint operator and application to a serially connected string system under joint feedbacks,, SIAM J.Control & Optim., 43 (2004), 1234.
doi: 10.1137/S0363012902420352. |
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