March  2020, 9(1): 131-151. doi: 10.3934/eect.2020019

Optimization of the blood pressure with the control in coefficients

Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit 6, P.O. Box 1-764, 014700 Bucharest, Romania

Received  October 2018 Revised  August 2019 Published  October 2019

This article is devoted to the study of a distributed control problem, with the control in coefficients, inspired by a disease that can lead to serious health problems: high blood pressure. We are concerned with the determination of a viscosity function that realizes an optimal blood pressure configuration. Using the mathematical model for viscous fluid-elastic structure interaction problems, we present existence, uniqueness, regularity results and estimates for the three unknown functions of the problem: velocity and pressure of the fluid and displacement of the elastic medium. The weak regularity of the state provided by the variational approach of the problem as well as the choice of the control variable induce some difficulties in the proof of the existence of an optimal control. The choice of the cost functional leads to an adjoint system which is not a divergence free one. For analyzing it, we propose a method based on the construction of several functions with suitable properties. Finally, we establish the necessary conditions of optimality.

Citation: Ruxandra Stavre. Optimization of the blood pressure with the control in coefficients. Evolution Equations & Control Theory, 2020, 9 (1) : 131-151. doi: 10.3934/eect.2020019
References:
[1]

P. Blanchard and E. Brüning, Mathematical Methods in Physics: Distributions, Hilbert Space Operators and Variational Methods, Progress in Mathematical Physics, 26. Birkhäuser Boston, Inc., Boston, MA, 2003.  Google Scholar

[2]

H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Collection Mathématiques Appliquées pour la Maétrise, Masson, Paris, 1983.  Google Scholar

[3]

E. Casas, Optimal control in coefficients of elliptic equations with state constraints, Appl. Math. Optim., 26 (1992), 21-37.  doi: 10.1007/BF01218394.  Google Scholar

[4]

I. CiupercaM. El Alaoui Talibi and M. Jai, On the optimal control of coefficients in elliptic problems. Application to the optimization of the head slider, ESAIM: Control, Optim. Calc. Var., 11 (2005), 102-121.  doi: 10.1051/cocv:2004029.  Google Scholar

[5]

G. FragnelliG. MarinoschiR. M. Mininni and S. Romanelli, Identification of a diffusion coefficient in strongly degenerate parabolic equations with interior degeneracy, J. Evol. Equ., 15 (2015), 27-51.  doi: 10.1007/s00028-014-0247-1.  Google Scholar

[6]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearized Steady Problems, Springer Tracts in Natural Philosophy, 38. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[7]

P. I. Kogut and G. Leugering, Optimal $L^1$-Control in Coefficients for Dirichlet Elliptic Problems: $H$-Optimal Solutions, Z. Anal. Anwend., 31 (2012), 31-53.  doi: 10.4171/ZAA/1447.  Google Scholar

[8]

J.-L. Lions, Contrôle Optimal de Systèmes Gouvernés par des Équations aux Dérivées Partielles, Avant Propos de P. Lelong Dunod, Paris, Gauthier-Villars, Paris, 1968.  Google Scholar

[9]

I. Malakhova-ZiablovaG. Panasenko and R. Stavre, Asymptotic analysis of a thin rigid stratified elastic plate-viscous fluid interaction problem, Appl. Anal., 95 (2016), 1467-1506.  doi: 10.1080/00036811.2015.1132311.  Google Scholar

[10]

G. P. Panasenko and R. Stavre, Asymptotic analysis of a viscous fluid-thin plate interaction: Periodic flow, Math. Models Methods Appl. Sci., 24 (2014), 1781-1822.  doi: 10.1142/S0218202514500079.  Google Scholar

[11]

J. Sokolowski, Optimal control in coefficients for weak variational problems in Hilbert space, Appl. Math. Optim., 7 (1981), 283-293.  doi: 10.1007/BF01442121.  Google Scholar

[12]

R. Stavre, The control of the pressure for a micropolar fluid, Z. angew. Math. Phys. (ZAMP), 53 (2002), 912-922.  doi: 10.1007/PL00012619.  Google Scholar

[13]

R. Stavre, Optimization of the blood flow in venous insufficiency, Annals Univ. Bucharest, 5 (2014), 383-402.   Google Scholar

[14]

R. Stavre, A boundary control problem for the blood flow in venous insufficiency. The general case, Nonlinear Anal. Real World Appl., 29 (2016), 98-116.  doi: 10.1016/j.nonrwa.2015.11.003.  Google Scholar

[15]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, Rhode Island, 2001. doi: 10.1090/chel/343.  Google Scholar

show all references

References:
[1]

P. Blanchard and E. Brüning, Mathematical Methods in Physics: Distributions, Hilbert Space Operators and Variational Methods, Progress in Mathematical Physics, 26. Birkhäuser Boston, Inc., Boston, MA, 2003.  Google Scholar

[2]

H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Collection Mathématiques Appliquées pour la Maétrise, Masson, Paris, 1983.  Google Scholar

[3]

E. Casas, Optimal control in coefficients of elliptic equations with state constraints, Appl. Math. Optim., 26 (1992), 21-37.  doi: 10.1007/BF01218394.  Google Scholar

[4]

I. CiupercaM. El Alaoui Talibi and M. Jai, On the optimal control of coefficients in elliptic problems. Application to the optimization of the head slider, ESAIM: Control, Optim. Calc. Var., 11 (2005), 102-121.  doi: 10.1051/cocv:2004029.  Google Scholar

[5]

G. FragnelliG. MarinoschiR. M. Mininni and S. Romanelli, Identification of a diffusion coefficient in strongly degenerate parabolic equations with interior degeneracy, J. Evol. Equ., 15 (2015), 27-51.  doi: 10.1007/s00028-014-0247-1.  Google Scholar

[6]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearized Steady Problems, Springer Tracts in Natural Philosophy, 38. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[7]

P. I. Kogut and G. Leugering, Optimal $L^1$-Control in Coefficients for Dirichlet Elliptic Problems: $H$-Optimal Solutions, Z. Anal. Anwend., 31 (2012), 31-53.  doi: 10.4171/ZAA/1447.  Google Scholar

[8]

J.-L. Lions, Contrôle Optimal de Systèmes Gouvernés par des Équations aux Dérivées Partielles, Avant Propos de P. Lelong Dunod, Paris, Gauthier-Villars, Paris, 1968.  Google Scholar

[9]

I. Malakhova-ZiablovaG. Panasenko and R. Stavre, Asymptotic analysis of a thin rigid stratified elastic plate-viscous fluid interaction problem, Appl. Anal., 95 (2016), 1467-1506.  doi: 10.1080/00036811.2015.1132311.  Google Scholar

[10]

G. P. Panasenko and R. Stavre, Asymptotic analysis of a viscous fluid-thin plate interaction: Periodic flow, Math. Models Methods Appl. Sci., 24 (2014), 1781-1822.  doi: 10.1142/S0218202514500079.  Google Scholar

[11]

J. Sokolowski, Optimal control in coefficients for weak variational problems in Hilbert space, Appl. Math. Optim., 7 (1981), 283-293.  doi: 10.1007/BF01442121.  Google Scholar

[12]

R. Stavre, The control of the pressure for a micropolar fluid, Z. angew. Math. Phys. (ZAMP), 53 (2002), 912-922.  doi: 10.1007/PL00012619.  Google Scholar

[13]

R. Stavre, Optimization of the blood flow in venous insufficiency, Annals Univ. Bucharest, 5 (2014), 383-402.   Google Scholar

[14]

R. Stavre, A boundary control problem for the blood flow in venous insufficiency. The general case, Nonlinear Anal. Real World Appl., 29 (2016), 98-116.  doi: 10.1016/j.nonrwa.2015.11.003.  Google Scholar

[15]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, Rhode Island, 2001. doi: 10.1090/chel/343.  Google Scholar

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