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Optimization Of discrete and differential inclusions of Goursat-Darboux type with state constraints

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Necessary and sufficient conditions of optimality under the most general assumptions are deduced for the considered and for discrete approximation problems. Formulation of sufficient conditions for differential inclusions is based on proved theorems of equivalence of locally conjugate mappings.



  1. 1.

    Agarwal RP, Grace SR, O'Regan D: Oscillation of higher order difference equations via comparison. Glasnik Matematički. Serija III 2004,39(59)(2):287–299.

  2. 2.

    Barbu V: The time optimal control of variational inequalities. Dynamic programming and the maximum principle. In Recent Mathematical Methods in Dynamic Programming (Rome, 1984), Lecture Notes in Math.. Volume 1119. Springer, Berlin; 1985:1–19. 10.1007/BFb0074777

  3. 3.

    Butkovskiĭ AG: Theory of Optimal Control of Systems with Distributed Parameters. Nauka, Moscow; 1965:474. English translation in Distributed control systems, American Elsevier, New York, 1969

  4. 4.

    Clarke FH, Ledyaev YuS, Radulescu ML: Approximate invariance and differential inclusions in Hilbert spaces. Journal of Dynamical and Control Systems 1997,3(4):493–518.

  5. 5.

    Demianov VF, Vasilev LV: Nondifferentiable Optimisation. Optimization Software, New York; 1985.

  6. 6.

    Ekeland I, Temam R: Convex Analysis and Variational Problems. MIR, Moscow; 1979:399.

  7. 7.

    Fornosini E, Marchesini G: Doubly indexed dynamical systems. Mathematical Systems Theory 1978.,12(1):

  8. 8.

    Ioffe AD, Tikhomirov VM: Theory of Extremal Problems. Nauka, Moscow; 1974:479. English translation in North-Holland, Amsterdam, 1978

  9. 9.

    Kaczorek T: Two-Dimensional Linear Systems, Lecture Notes in Control and Information Sciences. Volume 68. Springer, Berlin; 1985:x+398.

  10. 10.

    Kuang H-W: Minimum time function for differential inclusion with state constraints. Mathematica Applicata 2000,13(2):31–36.

  11. 11.

    Lions J-L: Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Dunod, Gauthier-Villars, Paris; 1968:xiii+426.

  12. 12.

    Mahmudov EN: On duality in optimal control problems described by convex discrete and differcutial inclusious. Avtomatika i Telemekhanika 1987, (2):13–25. English translation in Automation and Remote Control 48 (1987)

  13. 13.

    Mahmudov EN: Optimization of discrete inclusions with distributed parameters. Optimization 1990,21(2):197–207. 10.1080/02331939008843535

  14. 14.

    Mahmudov EN: Mathematical Analysis and Applications. Papatya, Istanbul; 2002:392.

  15. 15.

    Makarov VL, Rubinov AM: Mathematical Theory of Economic Dynamics and Equilibria. Nauka, Moscow; 1973:335. English translation in Springer, Berlin, 1977

  16. 16.

    Mordukhovich BS: Optimal Control of Nonconvex Discrete and Differential Inclusions. Sociedad Matematica Mexicana, Mexico; 1998:vi + 324.

  17. 17.

    Mordukhovich BS: Optimal control of difference, differential, and differential-difference inclusions. Journal of Mathematical Sciences (New York) 2000,100(6):2613–2632. 10.1007/BF02672708

  18. 18.

    Pšeničnyĭ BN: Convex Analysis and Extremal Problems, Series in Nonlinear Analysis and Its Applications. Nauka, Moscow; 1980:320.

  19. 19.

    Rockafellar RT: Convex Analysis. Princeton University Press, New Jersey; 1972.

  20. 20.

    Tikhonov AN, Samarskii AA: The Equations of Mathematical Physics. 3rd edition. Nauka, Moscow; 1966. English translation of 2nd ed., vols, 1, 2, Holden-Day, California, 1964, 1967

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Correspondence to Elimhan N Mahmudov.

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Mahmudov, E.N. Optimization Of discrete and differential inclusions of Goursat-Darboux type with state constraints. Adv Differ Equ 2006, 041962 (2006).

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  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Functional Equation