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December 2018, 13(4): 585-607. doi: 10.3934/nhm.2018027

On boundary optimal control problem for an arterial system: First-order optimality conditions

1. 

Dipartimento di Scienze Aziendali-Management e Innovation Systems, University of Salerno, Via Giovanni Paolo Ⅱ, 132, Fisciano, SA, Italy

2. 

Department of System Analysis, National Mining University, Yavornitskii av., 19, 49005 Dnipro, Ukraine

3. 

Institute for Applied System Analysis of National Academy of Sciences and Ministry of Education and Science of Ukraine, Peremogy av., 37/35, IASA, 03056 Kyiv, Ukraine

4. 

Department of Information and Electrical Engineering and Applied Mathematics, University of Salerno, Via Giovanni Paolo Ⅱ, 132, Fisciano, SA, Italy

* Corresponding author: Rosanna Manzo

Received  December 2017 Revised  August 2018 Published  November 2018

We discuss a control constrained boundary optimal control problem for the Boussinesq-type system arising in the study of the dynamics of an arterial network. We suppose that the control object is described by an initial-boundary value problem for $ 1D $ system of pseudo-parabolic nonlinear equations with an unbounded coefficient in the principle part and the Robin-type of boundary conditions. The main question we study in this part of the paper is about the existence of optimal solutions and first-order optimality conditions.

Citation: Ciro D'Apice, Olha P. Kupenko, Rosanna Manzo. On boundary optimal control problem for an arterial system: First-order optimality conditions. Networks & Heterogeneous Media, 2018, 13 (4) : 585-607. doi: 10.3934/nhm.2018027
References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

[2]

J. Alastruey, Propagation in the Cardiovascular System: Development, Validation and Clinical Applications, Ph.D thesis, Imperial College London, 2006.

[3]

R. C. CascavalC. D'ApiceM. P. D'Arienzo and R. Manzo, Boundary control for an arterial system, J. of Fluid Flow, Heat and Mass Transfer, 3 (2016), 25-33. doi: 10.11159/jffhmt.2016.004.

[4]

R. C. CascavalC. D'ApiceM. P. D'Arienzo and R. Manzo, Flow optimization of the vascular networks, Mathematical Biosciences and Engineering, 14 (2017), 607-624. doi: 10.3934/mbe.2017035.

[5]

C. D'ApiceM. P. D'ArienzoP. I. Kogut and R. Manzo, On boundary optimal control problem for an arterial system: Existence of feasible solutions, Journal of Evolution Equations, (2018), 1-42. doi: 10.1007/s00028-018-0460-4.

[6]

C. D'ApiceP.I. Kogut and R. Manzo, On relaxation of state constrained optimal control problem for a PDE-ODE model of supply chains, Networks and Heterogeneous Media, 9 (2014), 501-518. doi: 10.3934/nhm.2014.9.501.

[7]

C. D'ApiceP. I. Kogut and R. Manzo, On optimization of a highly re-entrant production system, Networks and Heterogeneous Media, 11 (2016), 415-445. doi: 10.3934/nhm.2016003.

[8]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Mathods for Science and Technology, Vol. 5: Evolutional Problems I, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58090-1.

[9]

L. FormaggiaD. LamponiM. Tuveri and A. Veneziani, Numerical modeling of 1D arterial networks coupled with a lumped parameters, description of the heart, Comput. Methods Biomech. Biomed. Eng., 9 (2006), 273-288. doi: 10.1080/10255840600857767.

[10]

L. Formaggia, A. Quarteroni and A. Veneziani, Cardiovascular Mathematics: Modeling and Simulation of the Circulatory System, Springer Verlag, Berlin, 2010.

[11]

H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974.

[12]

F. C. Hoppensteadt and C. Peskin, Modeling and Simulation in Medicine and the Life Sciences, Springer-Verlag, New York, 2002. doi: 10.1007/978-0-387-21571-6.

[13]

M. O. Korpusov and A. G. Sveshnikov, Nonlinear Functional Analysis and Mathematical Modelling in Physics: Methods of Nonlinear Operators, KRASAND, Moskov, 2011 (in Russian).

[14]

A. Kufner, Weighted Sobolev Spaces, Wiley & Sons, New York, 1985.

[15]

F. Liang, D. Guan and J. Alastruey, Determinant factors for arterial hemodynamics in hypertension: Theoretical insights from a computational model-based study, ASME Journal of Biomechanical Engineering, 140 (2018), 031006. doi: 10.1115/1.4038430.

[16]

D. MitsotakisD. Dutykh and L. Qian, Asymptotic nonlinear and dispersive pulsatile flow in elastic vessels with cylindrical symmetry, Computers & Mathematics with Applications, 75 (2018), 4022-4027. doi: 10.1016/j.camwa.2018.03.011.

[17]

M. S. OlufsenJ. T. OttesenH. T. TranL. M. EllweinL. A. Lipsitz and V. Novak, Blood pressure and blood flow variation during postural change from sitting to standing: model development and validation, J. Appl. Physiol, 99 (2005), 1523-1537. doi: 10.1152/japplphysiol.00177.2005.

[18]

G. Pontrelli and E. Rossoni, Numerical modeling of the pressure wave propagation in the arterial flow, International Journal for Numerical Methods in Fluids, 43 (2003), 651-671. doi: 10.1002/fld.494.

[19]

A. QuarteroniA. Manzoni and C. Vergara, The cardiovascular system: Mathematical modelling, numerical algorithms and clinical applications, Acta Numerica, 16 (2017), 365-590. doi: 10.1017/S0962492917000046.

[20]

P. ReymondF. MerendaF. PerrenD. Rafenacht and N. Stergiopulos, Validation of a one-dimensional model of the systemic arterial tree, Am J Physiol Heart Circ Physiol., 297 (2009), H208-H222. doi: 10.1152/ajpheart.00037.2009.

[21]

S. J. SherwinL. FormaggiaJ. Peiro and V. Franke, Computational modeling of 1D blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system, Internat. J. for Numerical Methods in Fluids, 43 (2003), 673-700. doi: 10.1002/fld.543.

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

[2]

J. Alastruey, Propagation in the Cardiovascular System: Development, Validation and Clinical Applications, Ph.D thesis, Imperial College London, 2006.

[3]

R. C. CascavalC. D'ApiceM. P. D'Arienzo and R. Manzo, Boundary control for an arterial system, J. of Fluid Flow, Heat and Mass Transfer, 3 (2016), 25-33. doi: 10.11159/jffhmt.2016.004.

[4]

R. C. CascavalC. D'ApiceM. P. D'Arienzo and R. Manzo, Flow optimization of the vascular networks, Mathematical Biosciences and Engineering, 14 (2017), 607-624. doi: 10.3934/mbe.2017035.

[5]

C. D'ApiceM. P. D'ArienzoP. I. Kogut and R. Manzo, On boundary optimal control problem for an arterial system: Existence of feasible solutions, Journal of Evolution Equations, (2018), 1-42. doi: 10.1007/s00028-018-0460-4.

[6]

C. D'ApiceP.I. Kogut and R. Manzo, On relaxation of state constrained optimal control problem for a PDE-ODE model of supply chains, Networks and Heterogeneous Media, 9 (2014), 501-518. doi: 10.3934/nhm.2014.9.501.

[7]

C. D'ApiceP. I. Kogut and R. Manzo, On optimization of a highly re-entrant production system, Networks and Heterogeneous Media, 11 (2016), 415-445. doi: 10.3934/nhm.2016003.

[8]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Mathods for Science and Technology, Vol. 5: Evolutional Problems I, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58090-1.

[9]

L. FormaggiaD. LamponiM. Tuveri and A. Veneziani, Numerical modeling of 1D arterial networks coupled with a lumped parameters, description of the heart, Comput. Methods Biomech. Biomed. Eng., 9 (2006), 273-288. doi: 10.1080/10255840600857767.

[10]

L. Formaggia, A. Quarteroni and A. Veneziani, Cardiovascular Mathematics: Modeling and Simulation of the Circulatory System, Springer Verlag, Berlin, 2010.

[11]

H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974.

[12]

F. C. Hoppensteadt and C. Peskin, Modeling and Simulation in Medicine and the Life Sciences, Springer-Verlag, New York, 2002. doi: 10.1007/978-0-387-21571-6.

[13]

M. O. Korpusov and A. G. Sveshnikov, Nonlinear Functional Analysis and Mathematical Modelling in Physics: Methods of Nonlinear Operators, KRASAND, Moskov, 2011 (in Russian).

[14]

A. Kufner, Weighted Sobolev Spaces, Wiley & Sons, New York, 1985.

[15]

F. Liang, D. Guan and J. Alastruey, Determinant factors for arterial hemodynamics in hypertension: Theoretical insights from a computational model-based study, ASME Journal of Biomechanical Engineering, 140 (2018), 031006. doi: 10.1115/1.4038430.

[16]

D. MitsotakisD. Dutykh and L. Qian, Asymptotic nonlinear and dispersive pulsatile flow in elastic vessels with cylindrical symmetry, Computers & Mathematics with Applications, 75 (2018), 4022-4027. doi: 10.1016/j.camwa.2018.03.011.

[17]

M. S. OlufsenJ. T. OttesenH. T. TranL. M. EllweinL. A. Lipsitz and V. Novak, Blood pressure and blood flow variation during postural change from sitting to standing: model development and validation, J. Appl. Physiol, 99 (2005), 1523-1537. doi: 10.1152/japplphysiol.00177.2005.

[18]

G. Pontrelli and E. Rossoni, Numerical modeling of the pressure wave propagation in the arterial flow, International Journal for Numerical Methods in Fluids, 43 (2003), 651-671. doi: 10.1002/fld.494.

[19]

A. QuarteroniA. Manzoni and C. Vergara, The cardiovascular system: Mathematical modelling, numerical algorithms and clinical applications, Acta Numerica, 16 (2017), 365-590. doi: 10.1017/S0962492917000046.

[20]

P. ReymondF. MerendaF. PerrenD. Rafenacht and N. Stergiopulos, Validation of a one-dimensional model of the systemic arterial tree, Am J Physiol Heart Circ Physiol., 297 (2009), H208-H222. doi: 10.1152/ajpheart.00037.2009.

[21]

S. J. SherwinL. FormaggiaJ. Peiro and V. Franke, Computational modeling of 1D blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system, Internat. J. for Numerical Methods in Fluids, 43 (2003), 673-700. doi: 10.1002/fld.543.

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