# American Institute of Mathematical Sciences

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

## Observability and controllability analysis of blood flow network

 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.
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.
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