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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
J. Alastruey, Numerical modelling of pulse wave propagation in the cardiovascular system: development, validation and clinical applications, Imperial College London, PhD Thesis, 2006.
H. Attouch, G. Buttazzo, G. Michaille, Variational Analysis in Sobolev and BV Spaces: Application to PDE and Optimization, SIAM, Philadelphia, 2006.
F. Bagagiolo, Ordinary Differential Equations, Dipartimento di Matematica, Università di Trento, Philadelphia, 2009.
R. C. Cascaval, A Boussinesq model for pressure and flow velocity waves in arterial segments, Math Comp Simulation 82 (6) (2012), 1047–1055.
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.
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.
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.
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.
C. D’Apice, R. Manzo, A fluid-dynamic model for supply chains, Networks and Heterogeneous Media 1(3) (2006), 379–398.
R. Dautray, J.-L. Lions,Mathematical Analysis and Numerical Mathods for Science and Technology, Vol. 5, Evolutional Problems I, Springer-Verlag, Berlin, 1992.
P. Drabek, A. Kufner, F. Nicolosi, Non linear elliptic equations, singular and degenerate cases, University of West Bohemia, 1996.
L. C. Evans, Partial Differential Equations, Vol. 19, Series ”Graduate Studies in Mathematics”, AMS, New York, 2010.
L. Formaggia, D. Lamponi and A. Quarteroni, One-dimensional models for blood flow in arteries, J Eng Math. 47 (2003), 251–276.
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.
L. Formaggia, A. Quarteroni and A. Veneziani, Cardiovascular Mathematics: Modeling and simulation of the circulatory system, Springer Verlag, Berlin, 2010.
F. C. Hoppensteadt, C. Peskin, Modeling and Simulation in Medicine and the Life Sciences, Springer, New York, 2004.
H. Gajewski, K. Gröger, K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, VI, 281 S., Akademie-Verlag, Berlin, 1974.
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.
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.
M. O. Korpusov, A. G. Sveshnikov, Nonlinear Functional Analysis and Mathematical Modelling in Physics: Methods of Nonlinear Operators, KRASAND, Moskov, 2011 (in Russian).
A. Kufner, Weighted Sobolev Spaces, Wiley & Sons, New York, 1985.
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.
J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Masson, Paris, 1988.
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.
L. Rowell, Human Cardiovascular Control, Oxford Univ Press, London, 1993.
J. Simon, Compact sets in the space \(L^p(0,T;B)\), Annali. di Mat. Pure ed. Appl. CXLVI (IV) (1987), 65–96.
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.
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Vol. 68, Applied Mathematics Sciences, Springer-Verlag, New York, 1988.
T. C. Sideris, Ordinary Differential Equations and Dynamical Systems, Vol. 2, Atlantis Studies in Differential Equations, Atlantis Press, Paris, 2013.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
D’Apice, C., D’Arienzo, M.P., Kogut, P.I. et al. On boundary optimal control problem for an arterial system: Existence of feasible solutions. J. Evol. Equ. 18, 1745–1786 (2018). https://doi.org/10.1007/s00028-018-0460-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-018-0460-4