Advertisement

Journal of Dynamical and Control Systems

, Volume 19, Issue 3, pp 381–404 | Cite as

On Vector-Valued Approximation of State Constrained Optimal Control Problems for Nonlinear Hyperbolic Conservation Laws

  • P. I. Kogut
  • R. Manzo
Article

Abstract

We study one class of nonlinear fluid dynamic models with controls in the initial condition and the source term. The model is described by a nonlinear inhomogeneous hyperbolic conservation law with state and control constraints. We consider the case when the greatest lower bound of the cost functional can be unattainable on the set Ξ of admissible pairs or the set Ξ is possibly empty. Using the methods of vector-valued optimization theory, we show that this optimal control problem admits the existence of the so-called weakened approximate solution which can be interpreted as generalized solution to some vector optimization problem of special form.

2000 Mathematics Subject Classification

46B40 49J45 90C29 49N90 76N15 

Key words and phrases

Nonlinear conservation laws control and state constraints vector optimization problem weakened approximate solutions entropy solutions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Ait Mansour, A. Metrane, and M. Théra, Lower semicontinuous regularization for vector-valued mappings. J. Global Optimization 35(2) (2006), 283–309.MATHCrossRefGoogle Scholar
  2. 2.
    J. P. Aubin, and H. Frankowska, Set-valued analysis. Birkhäuser, Cambridge, MA, (1990).MATHGoogle Scholar
  3. 3.
    J. M. Borwein, J. P. Penot, and M. Théra, Conjugate convex operators. J. Math. Anal. and Applic. 102 (1984), 399–414.MATHCrossRefGoogle Scholar
  4. 4.
    A. Bressan, Hyperbolic Systems of Conservation Laws - The One-dimensional Cauchy Problem. Oxford Univ. Press, 2000.Google Scholar
  5. 5.
    G. Bretti, C. D’Apice, R. Manzo, B. Piccoli, A continuum-discrete model for supply chains dynamics. Networks and Heterogeneous Media (NHM) 2 (2007), No. 4, 661–694.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    G. A. Chechkin, and A. Yu. Goritsky, S.N.Kruzhkov’s lectures on first-order quasilinear PDEs. Analytical and Numerical Aspects of PDEs (2010) (to appear).Google Scholar
  7. 7.
    G. Dal Maso, An Introduction to Γ-Convergence. Birkhäuser, Boston (1993).CrossRefGoogle Scholar
  8. 8.
    C. D’apice, P. I. Kogut, and R. Manzo, Efficient controls for one class of fluid dynamic models. Far East Journal of Applied Mathematics 46 (2010), No. 2, 85–119.MathSciNetMATHGoogle Scholar
  9. 9.
    ———, Efficient controls for traffic Flow on Networks. Dynamical and Control Systems 16 (2010), No. 3, 407–437.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    ———, On Approximation of Entropy Solutions for One System of Nonlinear Hyperbolic Conservation Laws with Impulse Source Terms. Journal of Control Science and Engineering Article ID 982369 (2010), 10 pp.Google Scholar
  11. 11.
    C. D’Apice, and R. Manzo, A fluid dynamic model for supply chains. Networks and Heterogeneous Media 1 (2006), No. 3, 379–398.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    C. D’Apice, R. Manzo, and B. Piccoli, Modelling supply networks with partial differential equations. Quarterly of Applied Mathematics 67 (2009), No. 3, 419–440.MathSciNetMATHGoogle Scholar
  13. 13.
    ———, Existence of solutions to Cauchy problems for a mixed continuum-discrete model for supply chains and networks. Journal of Mathematical Analysis and Applications 362 (2010), No. 2, 374–386.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    M. Garavello, and B. Piccoli, Traffic flow on networks. AIMS Series on Appl. Math. 1 (2006).Google Scholar
  15. 15.
    E. Giusti, Minimal surfaces and functions of bounded variation. Birkhäuser, Boston (1984).MATHCrossRefGoogle Scholar
  16. 16.
    J. Jahn, Vector Optimization. Theory, applications, and extensions. Springer-Verlag, Berlin (2004).MATHCrossRefGoogle Scholar
  17. 17.
    P. I. Kogut, G. Leugering, Optimal control problems for partial differential equations on reticulated domains: approximation and asymptotic analysis. Birkhäuser, Boston (2011).MATHCrossRefGoogle Scholar
  18. 18.
    P. I. Kogut, R. Manzo, and I. V. Nechay, On existence of efficient solutions to vector optimization problems in Banach spaces. Note di Matematica 30 (2010), No. 4, 45–64.MathSciNetGoogle Scholar
  19. 19.
    ———, Topological aspects of scalarization in vector optimization problems. Australian Journal of Mathematical Analysis and Applications 7 (2010), No. 2, 25–49.MathSciNetGoogle Scholar
  20. 20.
    S. Kruzhkov, First-order quasilinear equations in several independent variables. Math. USSR Sbornik 10 (1970), 217–243.MATHCrossRefGoogle Scholar
  21. 21.
    P. D. Lax, Hyperbolic System of Conservation Laws and the Mathematical Theory of Shock Waves. Society of Industrial and Applied Mathematics, Philadelfia, Pa. (1973).CrossRefGoogle Scholar
  22. 22.
    D. T. Luc, Theory of Vector Optimization. Springer-Verlag, New York (1989).CrossRefGoogle Scholar
  23. 23.
    J. P. Penot, and M. Théra, Semi-continuous mappings in general topology. Arch.Math. 38 (1982), 158–166.MATHCrossRefGoogle Scholar
  24. 24.
    A. I. Volpert, The spaces BV and quasilinear equations. Math. USSR Sbornik 2 (1967), 225–267.CrossRefGoogle Scholar
  25. 25.
    S. Ulbrich, Optimal Control of Nonlinear Hyperbolic Conservation Laws with Source Terms. Fakultät für Mathematik, Technische Universität München (2002).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Differential EquationsDnipropetrovsk National UniversityDnipropetrovskUkraine
  2. 2.Dipartimento di Ingegneria Elettronica e Ingegneria InformaticaUniversità degli Studi di SalernoFisciano (SA)Italy

Personalised recommendations