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Optimization of Cauchy Problem for Partial Differential Inclusions of Parabolic Type

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Abstract

Optimization of Cauchy problem for partial differential inclusions of parabolic type is considered, and sufficient condition for optimality is derived. For derivation of sufficient conditions both for convex and nonconvex partial differential inclusions, the apparatus of locally conjugate mappings is used. Besides, some special limiting conditions such as the equality of the coordinate-wise limits of the conjugate variables and their derivatives are formulated. The obtained results are generalized to the multidimensional cases. The linear problem illustrates the fact that some of the conjugate variables in concrete problems can be eliminated.

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References

  1. Agrachev AA, Sachkov YL. Control theory from the geometric viewpoint, control theory and optimization II. Berlin: Springer; 2004.

    Book  Google Scholar 

  2. Barbu V, Da Prato G. Hamilton–Jacobi equations in Hilbert spaces. Pitman Research Notes in Mathematics 86. London: Pitman; 1983.

    Google Scholar 

  3. Ekland I, Temam R. Analyse convexe et problemes variationelles. Paris: Dunod and Gauthier Villars; 1974.

    Google Scholar 

  4. Friedman A. Partial differential equations of parabolic type. New York: Prentice-Hall; 1964. p. 347.

    MATH  Google Scholar 

  5. Hale J.K., Raugel G. Partial differential equations on thin domains. Differential equations and mathematical physics (Birmingham, AL, 1990), 63–97

  6. Ioffe AD, Tikhomirov VM. Theory of extremal problems. Moscow: Nauk; 1978.

    Google Scholar 

  7. Kaplan W. Maxima and minima with applications practical optimization and duality. New York: Wiley; 1999. p. 284.

    MATH  Google Scholar 

  8. Loewen PD, Rockafellar RT. Bolza problems with general time constraints. SIAM J Control Optim. 1997;35(6):2050–69.

    Article  MATH  MathSciNet  Google Scholar 

  9. Mahmudov EN. Approximation and optimization of discrete and differential inclusions. New York: Elsevier; 2011.

    MATH  Google Scholar 

  10. Mordukhovich B. S., Variational analysis and generalized differentiation, I: Basic theory; II: applications. Grundlehren Series (Fundamental Principles of Mathematical Sciences), 2006; Vol. 330 and 331, Springer

  11. DE Pingco MDR, Ferreira MMA, Fontes FACC. An Euler–Lagrange inclusion for optimal control problems with state constraints. J Dynam Control Syst. 2002;8:23–45.

    Article  Google Scholar 

  12. Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, MISHCHENKO EF. The mathematical theory of optimal processes. New York: Wiley; 1965.

    Google Scholar 

  13. Pshenichnyi BN. Convex analysis and external problems. Moscow: Nauka; 1980 (Russian).

    Google Scholar 

  14. Rubinov AM, Yang X. Lagrange-type functions in constrained non-convex optimization. Massachusetts: Kluwer; 2003.

    Book  MATH  Google Scholar 

  15. Subbotin AI. Generalized solutions of partial differential equations of the first order. The invariance of graphs relative to differential inclusions. J Math Sci. 2006;78(5):594–611.

    Article  MathSciNet  Google Scholar 

  16. Tuan HD. Contingent and intermediate tangent cone in hyperbolic differential inclusions and necessary optimality conditions. J Math Anal Appl. 1994;185:86–106.

    Article  MATH  MathSciNet  Google Scholar 

  17. Aubin JP. Boundary-value problems for systems of first order partial differential inclusions. Nonlinear Differ Equat Appl. 2004;7(1):67–90.

    Article  MathSciNet  Google Scholar 

  18. Bencohra M, Ntouyas SK. Nonlocal Cauchy problems for neutral functional differential and integrodifferential inclusions in Banach spaces. J Math Anal Appl. 2001;258:573–90.

    Article  MathSciNet  Google Scholar 

  19. Gatsori E, Ntouyas SK, Sficas YG. On a nonlocal Cauchy problem for differential inclusions. Abstr Appl Anal. 2004;5:425–34.

    Article  MathSciNet  Google Scholar 

  20. Mahmudov EN. On duality in problems of optimal control described by convex differential inclusions of Goursat–Darboux type. J Math Anal Appl. 2005;307:628–40.

    Article  MATH  MathSciNet  Google Scholar 

  21. Mahmudov EN. Necessary and sufficient conditions for discrete and differential inclusions of elliptic type. J Math Anal Appl. 2006;323:768–89.

    Article  MATH  MathSciNet  Google Scholar 

  22. Mahmudov EN. Locally adjoint mappings and optimization of the first boundary value problem for hyperbolic type discrete and differential inclusions. Nonlinear Anal Theory Method Appl. 2007;67(10):2966–81.

    Article  MATH  MathSciNet  Google Scholar 

  23. Mahmudov EN. Sufficient conditions for optimality for differential inclusions of parabolic type and duality. J Glob Optim. 2008;41(1):31–42.

    Article  MATH  MathSciNet  Google Scholar 

  24. Mahmudov EN. Murat Ünal Engin, optimal control of discrete and differential inclusions with distributed parameters in gradient form. J Dyn Control Syst. 2012;18(1):83–101.

    Article  MATH  MathSciNet  Google Scholar 

  25. Melnikova IV. The Cauchy problem for a differential inclusion in Banach spaces and distribution spaces. Sib Math J. 2001;42(4):751–65.

    Article  MathSciNet  Google Scholar 

  26. Aubin JP, Frankowska H. Set-valued solutions to the Cauchy problem for hyperbolic system of partial differential inclusion. Nonlinear Differ Equat Appl. 1996;4:149–68.

    Article  MathSciNet  Google Scholar 

  27. O'Regan D, Agarwal RP. Set valued mappings with applications in nonlinear analysis. New York: Taylor& Francis; 2002.

    MATH  Google Scholar 

  28. Lysternik L.A., Sobolev V.I., Elemente der Funktional Analysis, Akademie Verlag (1968) (Translated from Russian)

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Authors and Affiliations

  1. Industrial Engineering Department Faculty of Management, Istanbul Technical University, 34367 , Maçka, Istanbul, Turkey

    Elimhan N. Mahmudov

  2. Azerbaijan National Academy of Sciences Institute of Cybernetics, 9. F. Agayev str, AZ1141, Baku, Azerbaijan

    Elimhan N. Mahmudov

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  1. Elimhan N. Mahmudov
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Correspondence to Elimhan N. Mahmudov.

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Mahmudov, E.N. Optimization of Cauchy Problem for Partial Differential Inclusions of Parabolic Type. J Dyn Control Syst 20, 167–180 (2014). https://doi.org/10.1007/s10883-013-9198-z

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  • DOI: https://doi.org/10.1007/s10883-013-9198-z

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