Advertisement

Journal of Evolution Equations

, Volume 18, Issue 4, pp 1745–1786 | Cite as

On boundary optimal control problem for an arterial system: Existence of feasible solutions

  • Ciro D’Apice
  • Maria Pia D’Arienzo
  • Peter I. Kogut
  • Rosanna Manzo
Article
  • 22 Downloads

Abstract

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 a control object is described by an initial-boundary value problem for 1D system of pseudo-parabolic nonlinear equations with an unbounded coefficient in the principal part and Robin boundary conditions. The main question we discuss in this part of paper is about topological and algebraical properties of the set of feasible solutions. Following Faedo–Galerkin method, we establish the existence of weak solutions to the corresponding initial-boundary value problem and show that these solutions possess some special extra regularity properties which play a crucial role in the proof of solvability of the original optimal control problem.

Keywords

Boussinesq-type system Existence result Faedo–Galerkin method Feasible solutions 

Mathematics Subject Classification

35K51 35B45 49J20 58F15 58F17 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.zbMATHGoogle Scholar
  2. 2.
    J. Alastruey, Numerical modelling of pulse wave propagation in the cardiovascular system: development, validation and clinical applications, Imperial College London, PhD Thesis, 2006.Google Scholar
  3. 3.
    H. Attouch, G. Buttazzo, G. Michaille, Variational Analysis in Sobolev and BV Spaces: Application to PDE and Optimization, SIAM, Philadelphia, 2006.CrossRefGoogle Scholar
  4. 4.
    F. Bagagiolo, Ordinary Differential Equations, Dipartimento di Matematica, Università di Trento, Philadelphia, 2009.Google Scholar
  5. 5.
    R. C. Cascaval, A Boussinesq model for pressure and flow velocity waves in arterial segments, Math Comp Simulation 82 (6) (2012), 1047–1055.MathSciNetCrossRefGoogle Scholar
  6. 6.
    R. C. Cascaval, C. D’Apice, M.P. D’Arienzo, R. Manzo Boundary control for an arterial system, J. of Fluid Flow, Heat and Mass Transfer 3 (2016), 25–33.Google Scholar
  7. 7.
    C. D’Apice, P. I. Kogut, R. Manzo, On Approximation of Entropy Solutions for One System of Nonlinear Hyperbolic Conservation Laws with Impulse Source Terms, Journal of Control Science and Engineering 982369 (2010), 10 pp.Google Scholar
  8. 8.
    C. D’Apice, P. I. Kogut, R. Manzo, On relaxation of state constrained optimal control problem for a PDE-ODE model of supply chains, Networks and Heterogeneous Media 9(3) (2014), 501–518.MathSciNetCrossRefGoogle Scholar
  9. 9.
    C. D’Apice, P. I. Kogut, R. Manzo, On Optimization of a Highly Re-Entrant Production System, Networks and Heterogeneous Media 11(3) (2016), 415–445.MathSciNetCrossRefGoogle Scholar
  10. 10.
    C. D’Apice, R. Manzo, A fluid-dynamic model for supply chains, Networks and Heterogeneous Media 1(3) (2006), 379–398.MathSciNetCrossRefGoogle Scholar
  11. 11.
    R. Dautray, J.-L. Lions,Mathematical Analysis and Numerical Mathods for Science and Technology, Vol. 5, Evolutional Problems I, Springer-Verlag, Berlin, 1992.Google Scholar
  12. 12.
    P. Drabek, A. Kufner, F. Nicolosi, Non linear elliptic equations, singular and degenerate cases, University of West Bohemia, 1996.Google Scholar
  13. 13.
    L. C. Evans, Partial Differential Equations, Vol. 19, Series ”Graduate Studies in Mathematics”, AMS, New York, 2010.Google Scholar
  14. 14.
    L. Formaggia, D. Lamponi and A. Quarteroni, One-dimensional models for blood flow in arteries, J Eng Math. 47 (2003), 251–276.MathSciNetCrossRefGoogle Scholar
  15. 15.
    L. Formaggia, D. Lamponi, M. 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.CrossRefGoogle Scholar
  16. 16.
    L. Formaggia, A. Quarteroni and A. Veneziani, Cardiovascular Mathematics: Modeling and simulation of the circulatory system, Springer Verlag, Berlin, 2010.zbMATHGoogle Scholar
  17. 17.
    F. C. Hoppensteadt, C. Peskin, Modeling and Simulation in Medicine and the Life Sciences, Springer, New York, 2004.zbMATHGoogle Scholar
  18. 18.
    H. Gajewski, K. Gröger, K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, VI, 281 S., Akademie-Verlag, Berlin, 1974.zbMATHGoogle Scholar
  19. 19.
    P. I. Kogut, R. Manzo, Efficient Controls for One Class of Fluid Dynamic Models, Far East Journal of Applied Mathematics 46(2) (2010), 85–119.MathSciNetzbMATHGoogle Scholar
  20. 20.
    P. I. Kogut, R. Manzo, On Vector-Valued Approximation of State Constrained Optimal Control Problems for Nonlinear Hyperbolic Conservation Laws, Journal of Dynamical and Control Systems 19(3) (2013), 381–404.MathSciNetCrossRefGoogle Scholar
  21. 21.
    M. O. Korpusov, A. G. Sveshnikov, Nonlinear Functional Analysis and Mathematical Modelling in Physics: Methods of Nonlinear Operators, KRASAND, Moskov, 2011 (in Russian).Google Scholar
  22. 22.
    A. Kufner, Weighted Sobolev Spaces, Wiley & Sons, New York, 1985.zbMATHGoogle Scholar
  23. 23.
    A. S. Liberson, J. S. Lillie, D. A. Borkholder, Numerical Solution for the Boussinesq Type Models with Application to Arterial Flow, Journal of Fluid Flow, Heat and Mass Transfer 1 (2014), 9–15.Google Scholar
  24. 24.
    J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Masson, Paris, 1988.zbMATHGoogle Scholar
  25. 25.
    P. Reymond, F. Merenda, F. Perren, D. 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.CrossRefGoogle Scholar
  26. 26.
    L. Rowell, Human Cardiovascular Control, Oxford Univ Press, London, 1993.CrossRefGoogle Scholar
  27. 27.
    J. Simon, Compact sets in the space \(L^p(0,T;B)\), Annali. di Mat. Pure ed. Appl. CXLVI (IV) (1987), 65–96.zbMATHGoogle Scholar
  28. 28.
    S. J. Sherwin, L. Formaggia, J. 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.CrossRefGoogle Scholar
  29. 29.
    R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Vol. 68, Applied Mathematics Sciences, Springer-Verlag, New York, 1988.Google Scholar
  30. 30.
    T. C. Sideris, Ordinary Differential Equations and Dynamical Systems, Vol. 2, Atlantis Studies in Differential Equations, Atlantis Press, Paris, 2013.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Ciro D’Apice
    • 1
  • Maria Pia D’Arienzo
    • 2
  • Peter I. Kogut
    • 3
  • Rosanna Manzo
    • 2
  1. 1.Dipartimento di Scienze Aziendali-Management e Innovation SystemsUniversity of SalernoFiscianoItaly
  2. 2.Department of Information and Electrical Engineering and Applied MathematicsUniversity of SalernoFiscianoItaly
  3. 3.Department of Differential EquationsOles Honchar Dnipro National UniversityDniproUkraine

Personalised recommendations