Observability and controllability analysis of blood flow network

Previous Article
MCRF Home
This Issue
Next Article
Multivariable boundary PI control and regulation of a fluid flow system

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

Abstract
Related Papers

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 and 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), 75-83. 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), 173-195. 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), 561-582. 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), 1161-1186. 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), 625-646. 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), 217-250. 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), 65-74.
[8]	F. N. van de Vosse and N. Stergiopulos, Pulse Wave Propagation in the Arterial Tree, Annu. Rev. Fluid Mech., 43 (2011), 467-499. 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), 1690-1698. 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), 273-288. doi: 10.1080/10255840600857767.
[11]	J. J. Wang and K. Parker, Wave propagation in a model for arterial circulation, J. Biomech., 37 (2004), 457-470. 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), 975-984. 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), 215-236. 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), 613-632. 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), 163-197. 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, Vol. 18 American Mathematical Society, Providence, R.I. 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), 273-297. doi: 10.1137/040610489.
[19]	M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Adv anced Texts: Basler Lehrbücher., Birkhäuser Verlag, Basel, 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, 192. North-Holland Publishing Co., Amsterdam, 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, Cambridge, 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, Shanghai, (2009), 6448-6453. doi: 10.1109/CDC.2009.5400250.
[23]	R. M. Young, An Introduction to Nonharmonic Fourier Series, Pure and Applied Mathematics, 93. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 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), 222-227. 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), 561-566. 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), 1-24. 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), 470-485. 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), 966-984. 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), 1234-1252. 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), 75-83. 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), 173-195. 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), 561-582. 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), 1161-1186. 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), 625-646. 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), 217-250. 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), 65-74.
[8]	F. N. van de Vosse and N. Stergiopulos, Pulse Wave Propagation in the Arterial Tree, Annu. Rev. Fluid Mech., 43 (2011), 467-499. 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), 1690-1698. 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), 273-288. doi: 10.1080/10255840600857767.
[11]	J. J. Wang and K. Parker, Wave propagation in a model for arterial circulation, J. Biomech., 37 (2004), 457-470. 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), 975-984. 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), 215-236. 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), 613-632. 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), 163-197. 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, Vol. 18 American Mathematical Society, Providence, R.I. 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), 273-297. doi: 10.1137/040610489.
[19]	M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Adv anced Texts: Basler Lehrbücher., Birkhäuser Verlag, Basel, 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, 192. North-Holland Publishing Co., Amsterdam, 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, Cambridge, 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, Shanghai, (2009), 6448-6453. doi: 10.1109/CDC.2009.5400250.
[23]	R. M. Young, An Introduction to Nonharmonic Fourier Series, Pure and Applied Mathematics, 93. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 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), 222-227. 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), 561-566. 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), 1-24. 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), 470-485. 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), 966-984. 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), 1234-1252. doi: 10.1137/S0363012902420352.

[1]	Imen Benabbas, Djamel Eddine Teniou. Observability of wave equation with Ventcel dynamic condition. Evolution Equations and 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 and 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 and Control Theory, 2020, 9 (1) : 1-25. doi: 10.3934/eect.2020014
[4]	Zhaoqiang Ge. Controllability and observability of stochastic implicit systems and stochastic GE-evolution operator. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 339-351. doi: 10.3934/naco.2021009
[5]	Ali Wehbe, Marwa Koumaiha, Layla Toufaily. Boundary observability and exact controllability of strongly coupled wave equations. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1269-1305. doi: 10.3934/dcdss.2021091
[6]	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 and Continuous Dynamical Systems, 2010, 28 (1) : 243-257. doi: 10.3934/dcds.2010.28.243
[7]	Takahisa Inui, Nobu Kishimoto, Kuranosuke Nishimura. Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6299-6353. doi: 10.3934/dcds.2019275
[8]	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
[9]	Peter Šepitka. Riccati equations for linear Hamiltonian systems without controllability condition. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1685-1730. doi: 10.3934/dcds.2019074
[10]	Sergei Avdonin, Jeff Park, Luz de Teresa. The Kalman condition for the boundary controllability of coupled 1-d wave equations. Evolution Equations and Control Theory, 2020, 9 (1) : 255-273. doi: 10.3934/eect.2020005
[11]	Ait Ben Hassi El Mustapha, Fadili Mohamed, Maniar Lahcen. On Algebraic condition for null controllability of some coupled degenerate systems. Mathematical Control and Related Fields, 2019, 9 (1) : 77-95. doi: 10.3934/mcrf.2019004
[12]	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 and Continuous Dynamical Systems - B, 2011, 16 (2) : 637-651. doi: 10.3934/dcdsb.2011.16.637
[13]	Meihua Wei, Yanling Li, Xi Wei. Stability and bifurcation with singularity for a glycolysis model under no-flux boundary condition. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 5203-5224. doi: 10.3934/dcdsb.2019129
[14]	Satoshi Masaki. A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1481-1531. doi: 10.3934/cpaa.2015.14.1481
[15]	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 and Related Fields, 2015, 5 (1) : 31-53. doi: 10.3934/mcrf.2015.5.31
[16]	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 and Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325
[17]	Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control and Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034
[18]	Muslim Malik, Anjali Rose, Anil Kumar. Controllability of Sobolev type fuzzy differential equation with non-instantaneous impulsive condition. Discrete and Continuous Dynamical Systems - S, 2022, 15 (2) : 387-407. doi: 10.3934/dcdss.2021068
[19]	Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations and Control Theory, 2022, 11 (2) : 605-619. doi: 10.3934/eect.2021016
[20]	Jongmin Han, Chun-Hsiung Hsia. Dynamical bifurcation of the two dimensional Swift-Hohenberg equation with odd periodic condition. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2431-2449. doi: 10.3934/dcdsb.2012.17.2431

2020 Impact Factor: 1.284

Tools

Metrics

Other articles
by authors

on AIMS
Chun Zong Gen Qi Xu
on Google Scholar
Chun Zong Gen Qi Xu

[Back to Top]