American Institute of Mathematical Sciences

Advanced
December  2014, 4(4): 521-554. doi: 10.3934/mcrf.2014.4.521

Observability and controllability analysis of blood flow network

Chun Zong 1, and  Gen Qi Xu 1,

1. 

Department of Mathematics, Tianjin University, Tianjin, 300072, China, China

Received  September 2013 Revised  January 2014 Published  September 2014

In this paper, we consider the initial-boundary value problem of a binary bifurcation model of the human arterial system. Firstly, we obtain a new pressure coupling condition at the junction based on the mass and energy conservation law. Then, we prove that the linearization system is interior well-posed and $L^2$ well-posed by using the semigroup theory of bounded linear operators. Further, by a complete spectral analysis for the system operator, we prove the completeness and Riesz basis property of the (generalized) eigenvectors of the system operator. Finally, we present some results on the boundary exact controllability and the boundary exact observability for the system.
Keywords: pressure and mass coupling condition, Controllability, separability., observability, vascular bifurcation.
Mathematics Subject Classification: Primary: 93B05, 93B07; Secondary: 76Z05, 93B6.
Citation: Chun Zong, Gen Qi Xu. Observability and controllability analysis of blood flow network. Mathematical Control & Related Fields, 2014, 4 (4) : 521-554. doi: 10.3934/mcrf.2014.4.521
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

I. C. Gohberg and M. G. Kreĭn, Introduction to the Theory of Linear Non-Selfadjoint Operators,, Translations of Mathematical Monographs, (1969).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

R. Mennicken and M. Möller, Non-self-adjoint Boundary Eigenvalue Problems,, North-Holland Mathematics Studies, (2003).   Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[23]

R. M. Young, An Introduction to Nonharmonic Fourier Series,, Pure and Applied Mathematics, (1980).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

I. C. Gohberg and M. G. Kreĭn, Introduction to the Theory of Linear Non-Selfadjoint Operators,, Translations of Mathematical Monographs, (1969).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

R. Mennicken and M. Möller, Non-self-adjoint Boundary Eigenvalue Problems,, North-Holland Mathematics Studies, (2003).   Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[23]

R. M. Young, An Introduction to Nonharmonic Fourier Series,, Pure and Applied Mathematics, (1980).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Imen Benabbas, Djamel Eddine Teniou. Observability of wave equation with Ventcel dynamic condition. Evolution Equations & Control Theory, 2018, 7 (4) : 545-570. doi: 10.3934/eect.2018026
[2]

Lingyang Liu, Xu Liu. Controllability and observability of some coupled stochastic parabolic systems. Mathematical Control & Related Fields, 2018, 8 (3&4) : 829-854. doi: 10.3934/mcrf.2018037
[3]

Abdelmouhcene Sengouga. Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints. Evolution Equations & Control Theory, 2020, 9 (1) : 1-25. doi: 10.3934/eect.2020014
[4]

Tatsien Li, Bopeng Rao, Zhiqiang Wang. Exact boundary controllability and observability for first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 243-257. doi: 10.3934/dcds.2010.28.243
[5]

Takahisa Inui, Nobu Kishimoto, Kuranosuke Nishimura. Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6299-6353. doi: 10.3934/dcds.2019275
[6]

Ait Ben Hassi El Mustapha, Fadili Mohamed, Maniar Lahcen. On Algebraic condition for null controllability of some coupled degenerate systems. Mathematical Control & Related Fields, 2019, 9 (1) : 77-95. doi: 10.3934/mcrf.2019004
[7]

Peter Šepitka. Riccati equations for linear Hamiltonian systems without controllability condition. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1685-1730. doi: 10.3934/dcds.2019074
[8]

Sergei Avdonin, Jeff Park, Luz de Teresa. The Kalman condition for the boundary controllability of coupled 1-d wave equations. Evolution Equations & Control Theory, 2020, 9 (1) : 255-273. doi: 10.3934/eect.2020005
[9]

Radu C. Cascaval, Ciro D'Apice, Maria Pia D'Arienzo, Rosanna Manzo. Flow optimization in vascular networks. Mathematical Biosciences & Engineering, 2017, 14 (3) : 607-624. doi: 10.3934/mbe.2017035
[10]

Meihua Wei, Yanling Li, Xi Wei. Stability and bifurcation with singularity for a glycolysis model under no-flux boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5203-5224. doi: 10.3934/dcdsb.2019129
[11]

Feng Zhang, Wei Zhang, Pan Meng, Jianzhong Su. Bifurcation analysis of bursting solutions of two Hindmarsh-Rose neurons with joint electrical and synaptic coupling. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 637-651. doi: 10.3934/dcdsb.2011.16.637
[12]

Satoshi Masaki. A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1481-1531. doi: 10.3934/cpaa.2015.14.1481
[13]

Larissa V. Fardigola. Transformation operators in controllability problems for the wave equations with variable coefficients on a half-axis controlled by the Dirichlet boundary condition. Mathematical Control & Related Fields, 2015, 5 (1) : 31-53. doi: 10.3934/mcrf.2015.5.31
[14]

Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations & Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325
[15]

Jongmin Han, Chun-Hsiung Hsia. Dynamical bifurcation of the two dimensional Swift-Hohenberg equation with odd periodic condition. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2431-2449. doi: 10.3934/dcdsb.2012.17.2431
[16]

Jiayue Zheng, Shangbin Cui. Bifurcation analysis of a tumor-model free boundary problem with a nonlinear boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020103
[17]

Guohua Zhang. Variational principles of pressure. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1409-1435. doi: 10.3934/dcds.2009.24.1409
[18]

V. Lanza, D. Ambrosi, L. Preziosi. Exogenous control of vascular network formation in vitro: a mathematical model. Networks & Heterogeneous Media, 2006, 1 (4) : 621-637. doi: 10.3934/nhm.2006.1.621
[19]

Alberto M. Gambaruto, João Janela, Alexandra Moura, Adélia Sequeira. Sensitivity of hemodynamics in a patient specific cerebral aneurysm to vascular geometry and blood rheology. Mathematical Biosciences & Engineering, 2011, 8 (2) : 409-423. doi: 10.3934/mbe.2011.8.409
[20]

Russell Betteridge, Markus R. Owen, H.M. Byrne, Tomás Alarcón, Philip K. Maini. The impact of cell crowding and active cell movement on vascular tumour growth. Networks & Heterogeneous Media, 2006, 1 (4) : 515-535. doi: 10.3934/nhm.2006.1.515

2019 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences