The exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls

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September 2016, 11(3): 527-543. doi: 10.3934/nhm.2016008

The exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls

Yaru Xie ^1, and Genqi Xu ^1,

1.	Department of Mathematics, Tianjin University, Haihe Education Park, Tianjin, Tianjin, MO 300000, China, China

Received October 2014 Revised February 2016 Published August 2016

Abstract
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In this paper, we study the exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls. For the networks, there are some results on the exponential stability, but no result on estimate of the decay rate. The present work mainly estimates the decay rate for these systems, including signal wave equation, serially connected wave equations, and generic tree of 1-d wave equations. By defining the weighted energy functional of the system, and choosing suitable weighted functions, we obtain the estimation value of decay rate of the systems.

Keywords: weighted energy functional, Wave networks, boundary feedback control, exponential decay rate..

Mathematics Subject Classification: Primary: 35L05, 37L45; Secondary: 37L15, 90B1.

Citation: Yaru Xie, Genqi Xu. The exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls. Networks and Heterogeneous Media, 2016, 11 (3) : 527-543. doi: 10.3934/nhm.2016008

References:

[1]	K. Ammari and M. Jellouli, Stabilization of star-shaped tree of elastic strings, Differential & Integral Equations, 17 (2004), 1395-1410.
[2]	K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings, Journal of Dynamical & Control Systems, 11 (2005), 177-193. doi: 10.1007/s10883-005-4169-7.
[3]	K. Ammari and D. Mercier, Boundary feedback stabilization of a chain of serially connected strings, Evol. Equ. Control Theory, 4 (2015), 1-19. doi: 10.3934/eect.2015.4.1.
[4]	K. Ammari, D. Mercier and V. Régnier, Spectral analysis of the Schrodinger operator on binary tree-shaped networks and applications, Journal of Differential Equations, 259 (2015), 6923-6959. doi: 10.1016/j.jde.2015.08.017.
[5]	K. Ammari, A. Henrot and M. Tucsnak, Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string, Asymptotic Analysis, 28 (2001), 215-240.
[6]	K. Bartecki, A general transfer function representation for a class of hyperbolic distributed parameter systems, International Journal of Applied Mathematics & Computer Science, 23 (2013), 291-307. doi: 10.2478/amcs-2013-0022.
[7]	A. V. Balakrishnan, On superstable semigroup of operators, Dynamic Systems & Applications, 5 (1996), 371-384.
[8]	J. M. Coron, B. D'Andrea-Novel and G. Bastin, A lyapunov approach to control irrigation canals modeled by the Saint Venant equations, Control Conference. IEEE, 1999.
[9]	J. M. Coron, J. De Halleux and G. Bastin, On boundary control design for quasi-linear hyperbolic systems with entropies as Lyapunov functions, Proceedings of the IEEE Conference on Decision and Control, 3 (2003), 3010-3014.
[10]	J. M. Coron, B. d'Andrea-Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, Proceedings of the IEEE Conference on Decision & Control, 52 (2007), 2-11. doi: 10.1109/TAC.2006.887903.
[11]	R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer-Verlag, New York, Berlin, Heidelberg, 1995. doi: 10.1007/978-1-4612-4224-6.
[12]	S. Cox and E. Zuazua, The rate at which energy decays in a damped string, Communication Partial Differential Equations, 19 (1994), 213-243. doi: 10.1080/03605309408821015.
[13]	R. Dager and E. Zuazua, Controllability of star-shaped networks of strings, C. R. Acad. Sci. Paris, Series I, 332 (2001), 621-626. doi: 10.1016/S0764-4442(01)01876-6.
[14]	R. Dager, Observation and control of vibrations in tree-shaped networks of strings, SIAM J. Control & Optim, 43 (2004), 590-623. doi: 10.1137/S0363012903421844.
[15]	R. Dager and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures, Springer Berlin, 2006. doi: 10.1007/3-540-37726-3.
[16]	A. Diagne, G. Bastin and J. M. Coron, Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws, Automatica, 48 (2012), 109-114. doi: 10.1016/j.automatica.2011.09.030.
[17]	Y. N. Guo and G. Q. Xu, Stability and Riesz basis property for general network of strings, Journal of dynamical & control systems, 15 (2009), 223-245. doi: 10.1007/s10883-009-9064-1.
[18]	Y. N. Guo and G. Q. Xu, Exponential stabilization of a tree shaped network of strings with variable coefficients, Glasgow Mathematical Journal, 53 (2011), 481-499. doi: 10.1017/S0017089511000085.
[19]	Y. N. Guo and G. Q. Xu, Exponential stabilization of variable coefficient wave equations in a generic tree with small time-delays in the nodal feedbacks, Journal of Mathematical Analysis & Applications, 395 (2012), 727-746. doi: 10.1016/j.jmaa.2012.05.079.
[20]	B. Z. Guo and Y. Xie, A sufficient condition on Riesz basis with parentheses of non-selfadjoint operator and application to a serially connected string system under joint feedbacks, SIAM journal on control & optimization, 43 (2004), 1234-1252. doi: 10.1137/S0363012902420352.
[21]	M. Gugat and M. Herty, Existence of classical solutions and feedback stabilization for the flow in gas networks, ESAIM Control Optimisation & Calculus of Variations, 17 (2011), 28-51. doi: 10.1051/cocv/2009035.
[22]	F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.
[23]	Z. J. Han and L. Wang, Riesz basis property and stability of planar networks of controlled strings, Acta Applicandae Mathematicae, 110 (2010), 511-533. doi: 10.1007/s10440-009-9459-8.
[24]	Z. J. Han and G. Q. Xu, Spectrum and dynamical behavior of a kind of planar network of non-uniform strings with non-collocated feedbacks, Networks & Heterogeneous Media, 5 (2010), 315-334. doi: 10.3934/nhm.2010.5.315.
[25]	Z. J. Han and G. Q. Xu, Stabilization and SDG condition of serially connected vibrating strings system with discontinuous displacement, Asian Journal of Control, 14 (2012), 95-108. doi: 10.1002/asjc.218.
[26]	M. Jellouli, Spectral analysis for a degenerate tree and applications, International Journal of Control, 88 (2015), 1647-1662. doi: 10.1080/00207179.2015.1012652.
[27]	M.Krstic, B. Z. Guo and A. Balogh, Output-feedback stabilization of an unstable wave equation, Automatica, 44 (2008), 63-74. doi: 10.1016/j.automatica.2007.05.012.
[28]	J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modelling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Systems & Control: Foundations & Applications, 1994. doi: 10.1007/978-1-4612-0273-8.
[29]	J. E. Lagnese, Recent progress and open problems in control of multi-link elastic structures, Contemp. Math, 209 (1997), 161-175. doi: 10.1090/conm/209/02765.
[30]	J. L. Lions, Exact controllability, stabilization and perturbations for distributed parameter system, SIAM Review, 30 (1988), 1-68. doi: 10.1137/1030001.
[31]	K. S. Liu, F. L. Huang and G. Chen, Exponential stability analysis of a long chain of coupled vibrating strings with dissipative linkage, SIAM Journal on Applied Mathematics, 49 (1989), 1694-1707. doi: 10.1137/0149102.
[32]	G. Leugering and E. Zuazua, On exact controllability of generic trees, Esaim Proceedings, 8 (2000), 95-105. doi: 10.1051/proc:2000007.
[33]	G. Leugering, Dynamic domain decomposition of optimal control problems for networks of strings and Timoshenko beams, SIAM Journal on Control & Optimization, 37 (1999), 1649-1675. doi: 10.1137/S0363012997331986.
[34]	D. Y. Liu, Y. F. Shang and G. Q. Xu, Design of controllers and compensators of a serially connected string system and its Riesz basis, Kongzhi Lilun Yu Yinyong/Control Theory & Applications, 5 (2008), 815-818.
[35]	M. Najafi, G. R. Sarhangi and H. Wang, Stabilizability of coupled wave equations in parallel under various boundary conditions, Automatic Control IEEE Transactions on, 42 (1997), 1308-1312. doi: 10.1109/9.623099.
[36]	S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay-term in the feedbacks, Networks & Heterogeneous Media, 2 (2007), 425-479. doi: 10.3934/nhm.2007.2.425.
[37]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Berlin, Germany: Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.
[38]	D. L. Rusell, Mathematical models for the elastic beam and their control-theoretic implications, Semigroups, Theory and Applications, 152 (1986), 177-216.
[39]	S. Rolewicz, On controllability of systems of strings, Studia Math, Studia Mathematica, 36 (1970), 105-110.
[40]	Y. F. Shang, D. Y. Liu and G. Q. Xu, Super-stability and the spectrum of one-dimensional wave equations on general feedback controlled networks, IMA Journal of Mathematical Control & Information, 31 (2014), 73-99. doi: 10.1093/imamci/dnt003.
[41]	M. A. Shubov, Basis property of eigenfunctions of nonselfadjoint operator pencils generated by the equations of nonhomogeneous damped string, Integral Equations & Operator Theory, 25 (1996), 289-328. doi: 10.1007/BF01262296.
[42]	E. J. P. G. Schmidt, On the modelling and exact controllability of networks of vibrating strings, Siam Journal on Control & Optimization, 30 (1992), 229-245. doi: 10.1137/0330015.
[43]	L. Wang, Z. J. Han and G. Q. Xu, Exponential stability of serially connected thermoelastic system of type II with nodal damping, Applicable Analysis, 93 (2014), 1495-1514. doi: 10.1080/00036811.2013.836596.
[44]	H. Wang and G. Q. Xu, Exponential stabilization of 1-d wave equation with input delay, Wseas Transactions on Mathematics, 10 (2013), 1001-1013.
[45]	G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optimisation & Calculus of Variations, 12 (2006), 770-785. doi: 10.1051/cocv:2006021.
[46]	G. Q. Xu, Stabilization of string system with linear boundary feedback, Nonlinear Analysis Hybrid Systems, 1 (2007), 383-397. doi: 10.1016/j.nahs.2006.07.003.
[47]	G. Q. Xu and B. Z. Guo, Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation, SIAM Journal on Control & Optimization, 42 (2003), 966-984. doi: 10.1137/S0363012901400081.
[48]	G. Q. Xu, D. Y. Liu and Y. Q. Liu, Abstract second order hyperbolic system and applications to controlled network of strings, SIAM Journal on Control & Optimization, 47 (2008), 1762-1784. doi: 10.1137/060649367.
[49]	G. Q. Xu and N. E. Mastorakis, Differential Equations on Metric Graph, Athens: World Scientific and Engineering Academy and Society, 2010.
[50]	G. Q. Xu and Y. X. Zhang, The exponential stability of complex differential networks, Journal of Systems Science & Mathematical Sciences, 29 (2009), 1399-1418.
[51]	E. Zuazua, Control and stabilization of waves on 1-d networks, Modelling and Optimisation of Flows on Networks, Springer Berlin Heidelberg, 2062 (2013), 463-493. doi: 10.1007/978-3-642-32160-3_9.
[52]	Y. X. Zhang and G. Q. Xu, Controller design for Bush-type 1-D wave neworks, ESAIM Control Optimisation & Calculus of Variations, 18 (2012), 208-228. doi: 10.1051/cocv/2010050.
[53]	Y. X. Zhang and G. Q. Xu, A new approach for the stability analysis of wave networks, Abstract and Applied Analysis, 2014 (2014), Art. ID 724512, 10 pp. doi: 10.1155/2014/724512.
[54]	Y. X. Zhang and G. Q. Xu, Exponential and super stability of a wave network, Acta Applicandae Mathematicae An International Survey Journal on Applying Mathematics & Mathematical Applications, 124 (2013), 19-41. doi: 10.1007/s10440-012-9768-1.

show all references

References:

[1]	K. Ammari and M. Jellouli, Stabilization of star-shaped tree of elastic strings, Differential & Integral Equations, 17 (2004), 1395-1410.
[2]	K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings, Journal of Dynamical & Control Systems, 11 (2005), 177-193. doi: 10.1007/s10883-005-4169-7.
[3]	K. Ammari and D. Mercier, Boundary feedback stabilization of a chain of serially connected strings, Evol. Equ. Control Theory, 4 (2015), 1-19. doi: 10.3934/eect.2015.4.1.
[4]	K. Ammari, D. Mercier and V. Régnier, Spectral analysis of the Schrodinger operator on binary tree-shaped networks and applications, Journal of Differential Equations, 259 (2015), 6923-6959. doi: 10.1016/j.jde.2015.08.017.
[5]	K. Ammari, A. Henrot and M. Tucsnak, Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string, Asymptotic Analysis, 28 (2001), 215-240.
[6]	K. Bartecki, A general transfer function representation for a class of hyperbolic distributed parameter systems, International Journal of Applied Mathematics & Computer Science, 23 (2013), 291-307. doi: 10.2478/amcs-2013-0022.
[7]	A. V. Balakrishnan, On superstable semigroup of operators, Dynamic Systems & Applications, 5 (1996), 371-384.
[8]	J. M. Coron, B. D'Andrea-Novel and G. Bastin, A lyapunov approach to control irrigation canals modeled by the Saint Venant equations, Control Conference. IEEE, 1999.
[9]	J. M. Coron, J. De Halleux and G. Bastin, On boundary control design for quasi-linear hyperbolic systems with entropies as Lyapunov functions, Proceedings of the IEEE Conference on Decision and Control, 3 (2003), 3010-3014.
[10]	J. M. Coron, B. d'Andrea-Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, Proceedings of the IEEE Conference on Decision & Control, 52 (2007), 2-11. doi: 10.1109/TAC.2006.887903.
[11]	R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer-Verlag, New York, Berlin, Heidelberg, 1995. doi: 10.1007/978-1-4612-4224-6.
[12]	S. Cox and E. Zuazua, The rate at which energy decays in a damped string, Communication Partial Differential Equations, 19 (1994), 213-243. doi: 10.1080/03605309408821015.
[13]	R. Dager and E. Zuazua, Controllability of star-shaped networks of strings, C. R. Acad. Sci. Paris, Series I, 332 (2001), 621-626. doi: 10.1016/S0764-4442(01)01876-6.
[14]	R. Dager, Observation and control of vibrations in tree-shaped networks of strings, SIAM J. Control & Optim, 43 (2004), 590-623. doi: 10.1137/S0363012903421844.
[15]	R. Dager and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures, Springer Berlin, 2006. doi: 10.1007/3-540-37726-3.
[16]	A. Diagne, G. Bastin and J. M. Coron, Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws, Automatica, 48 (2012), 109-114. doi: 10.1016/j.automatica.2011.09.030.
[17]	Y. N. Guo and G. Q. Xu, Stability and Riesz basis property for general network of strings, Journal of dynamical & control systems, 15 (2009), 223-245. doi: 10.1007/s10883-009-9064-1.
[18]	Y. N. Guo and G. Q. Xu, Exponential stabilization of a tree shaped network of strings with variable coefficients, Glasgow Mathematical Journal, 53 (2011), 481-499. doi: 10.1017/S0017089511000085.
[19]	Y. N. Guo and G. Q. Xu, Exponential stabilization of variable coefficient wave equations in a generic tree with small time-delays in the nodal feedbacks, Journal of Mathematical Analysis & Applications, 395 (2012), 727-746. doi: 10.1016/j.jmaa.2012.05.079.
[20]	B. Z. Guo and Y. Xie, A sufficient condition on Riesz basis with parentheses of non-selfadjoint operator and application to a serially connected string system under joint feedbacks, SIAM journal on control & optimization, 43 (2004), 1234-1252. doi: 10.1137/S0363012902420352.
[21]	M. Gugat and M. Herty, Existence of classical solutions and feedback stabilization for the flow in gas networks, ESAIM Control Optimisation & Calculus of Variations, 17 (2011), 28-51. doi: 10.1051/cocv/2009035.
[22]	F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.
[23]	Z. J. Han and L. Wang, Riesz basis property and stability of planar networks of controlled strings, Acta Applicandae Mathematicae, 110 (2010), 511-533. doi: 10.1007/s10440-009-9459-8.
[24]	Z. J. Han and G. Q. Xu, Spectrum and dynamical behavior of a kind of planar network of non-uniform strings with non-collocated feedbacks, Networks & Heterogeneous Media, 5 (2010), 315-334. doi: 10.3934/nhm.2010.5.315.
[25]	Z. J. Han and G. Q. Xu, Stabilization and SDG condition of serially connected vibrating strings system with discontinuous displacement, Asian Journal of Control, 14 (2012), 95-108. doi: 10.1002/asjc.218.
[26]	M. Jellouli, Spectral analysis for a degenerate tree and applications, International Journal of Control, 88 (2015), 1647-1662. doi: 10.1080/00207179.2015.1012652.
[27]	M.Krstic, B. Z. Guo and A. Balogh, Output-feedback stabilization of an unstable wave equation, Automatica, 44 (2008), 63-74. doi: 10.1016/j.automatica.2007.05.012.
[28]	J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modelling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Systems & Control: Foundations & Applications, 1994. doi: 10.1007/978-1-4612-0273-8.
[29]	J. E. Lagnese, Recent progress and open problems in control of multi-link elastic structures, Contemp. Math, 209 (1997), 161-175. doi: 10.1090/conm/209/02765.
[30]	J. L. Lions, Exact controllability, stabilization and perturbations for distributed parameter system, SIAM Review, 30 (1988), 1-68. doi: 10.1137/1030001.
[31]	K. S. Liu, F. L. Huang and G. Chen, Exponential stability analysis of a long chain of coupled vibrating strings with dissipative linkage, SIAM Journal on Applied Mathematics, 49 (1989), 1694-1707. doi: 10.1137/0149102.
[32]	G. Leugering and E. Zuazua, On exact controllability of generic trees, Esaim Proceedings, 8 (2000), 95-105. doi: 10.1051/proc:2000007.
[33]	G. Leugering, Dynamic domain decomposition of optimal control problems for networks of strings and Timoshenko beams, SIAM Journal on Control & Optimization, 37 (1999), 1649-1675. doi: 10.1137/S0363012997331986.
[34]	D. Y. Liu, Y. F. Shang and G. Q. Xu, Design of controllers and compensators of a serially connected string system and its Riesz basis, Kongzhi Lilun Yu Yinyong/Control Theory & Applications, 5 (2008), 815-818.
[35]	M. Najafi, G. R. Sarhangi and H. Wang, Stabilizability of coupled wave equations in parallel under various boundary conditions, Automatic Control IEEE Transactions on, 42 (1997), 1308-1312. doi: 10.1109/9.623099.
[36]	S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay-term in the feedbacks, Networks & Heterogeneous Media, 2 (2007), 425-479. doi: 10.3934/nhm.2007.2.425.
[37]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Berlin, Germany: Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.
[38]	D. L. Rusell, Mathematical models for the elastic beam and their control-theoretic implications, Semigroups, Theory and Applications, 152 (1986), 177-216.
[39]	S. Rolewicz, On controllability of systems of strings, Studia Math, Studia Mathematica, 36 (1970), 105-110.
[40]	Y. F. Shang, D. Y. Liu and G. Q. Xu, Super-stability and the spectrum of one-dimensional wave equations on general feedback controlled networks, IMA Journal of Mathematical Control & Information, 31 (2014), 73-99. doi: 10.1093/imamci/dnt003.
[41]	M. A. Shubov, Basis property of eigenfunctions of nonselfadjoint operator pencils generated by the equations of nonhomogeneous damped string, Integral Equations & Operator Theory, 25 (1996), 289-328. doi: 10.1007/BF01262296.
[42]	E. J. P. G. Schmidt, On the modelling and exact controllability of networks of vibrating strings, Siam Journal on Control & Optimization, 30 (1992), 229-245. doi: 10.1137/0330015.
[43]	L. Wang, Z. J. Han and G. Q. Xu, Exponential stability of serially connected thermoelastic system of type II with nodal damping, Applicable Analysis, 93 (2014), 1495-1514. doi: 10.1080/00036811.2013.836596.
[44]	H. Wang and G. Q. Xu, Exponential stabilization of 1-d wave equation with input delay, Wseas Transactions on Mathematics, 10 (2013), 1001-1013.
[45]	G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optimisation & Calculus of Variations, 12 (2006), 770-785. doi: 10.1051/cocv:2006021.
[46]	G. Q. Xu, Stabilization of string system with linear boundary feedback, Nonlinear Analysis Hybrid Systems, 1 (2007), 383-397. doi: 10.1016/j.nahs.2006.07.003.
[47]	G. Q. Xu and B. Z. Guo, Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation, SIAM Journal on Control & Optimization, 42 (2003), 966-984. doi: 10.1137/S0363012901400081.
[48]	G. Q. Xu, D. Y. Liu and Y. Q. Liu, Abstract second order hyperbolic system and applications to controlled network of strings, SIAM Journal on Control & Optimization, 47 (2008), 1762-1784. doi: 10.1137/060649367.
[49]	G. Q. Xu and N. E. Mastorakis, Differential Equations on Metric Graph, Athens: World Scientific and Engineering Academy and Society, 2010.
[50]	G. Q. Xu and Y. X. Zhang, The exponential stability of complex differential networks, Journal of Systems Science & Mathematical Sciences, 29 (2009), 1399-1418.
[51]	E. Zuazua, Control and stabilization of waves on 1-d networks, Modelling and Optimisation of Flows on Networks, Springer Berlin Heidelberg, 2062 (2013), 463-493. doi: 10.1007/978-3-642-32160-3_9.
[52]	Y. X. Zhang and G. Q. Xu, Controller design for Bush-type 1-D wave neworks, ESAIM Control Optimisation & Calculus of Variations, 18 (2012), 208-228. doi: 10.1051/cocv/2010050.
[53]	Y. X. Zhang and G. Q. Xu, A new approach for the stability analysis of wave networks, Abstract and Applied Analysis, 2014 (2014), Art. ID 724512, 10 pp. doi: 10.1155/2014/724512.
[54]	Y. X. Zhang and G. Q. Xu, Exponential and super stability of a wave network, Acta Applicandae Mathematicae An International Survey Journal on Applying Mathematics & Mathematical Applications, 124 (2013), 19-41. doi: 10.1007/s10440-012-9768-1.

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