Skip to content


  • Research Article
  • Open Access

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

Advances in Difference Equations20062006:041962

  • Received: 14 October 2005
  • Accepted: 20 September 2006
  • Published:


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.


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


Authors’ Affiliations

Department of Industrial Engineering, Engineering Faculty, Istanbul University, Avcilar, Istanbul, 34850, Turkey


  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.MathSciNetView ArticleMATHGoogle Scholar
  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/BFb0074777View ArticleGoogle Scholar
  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, 1969Google Scholar
  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.MathSciNetMATHGoogle Scholar
  5. Demianov VF, Vasilev LV: Nondifferentiable Optimisation. Optimization Software, New York; 1985.View ArticleGoogle Scholar
  6. Ekeland I, Temam R: Convex Analysis and Variational Problems. MIR, Moscow; 1979:399.MATHGoogle Scholar
  7. Fornosini E, Marchesini G: Doubly indexed dynamical systems. Mathematical Systems Theory 1978.,12(1):Google Scholar
  8. Ioffe AD, Tikhomirov VM: Theory of Extremal Problems. Nauka, Moscow; 1974:479. English translation in North-Holland, Amsterdam, 1978Google Scholar
  9. Kaczorek T: Two-Dimensional Linear Systems, Lecture Notes in Control and Information Sciences. Volume 68. Springer, Berlin; 1985:x+398.Google Scholar
  10. Kuang H-W: Minimum time function for differential inclusion with state constraints. Mathematica Applicata 2000,13(2):31–36.MathSciNetMATHGoogle Scholar
  11. Lions J-L: Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Dunod, Gauthier-Villars, Paris; 1968:xiii+426.MATHGoogle Scholar
  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)Google Scholar
  13. Mahmudov EN: Optimization of discrete inclusions with distributed parameters. Optimization 1990,21(2):197–207. 10.1080/02331939008843535MathSciNetView ArticleMATHGoogle Scholar
  14. Mahmudov EN: Mathematical Analysis and Applications. Papatya, Istanbul; 2002:392.Google Scholar
  15. Makarov VL, Rubinov AM: Mathematical Theory of Economic Dynamics and Equilibria. Nauka, Moscow; 1973:335. English translation in Springer, Berlin, 1977Google Scholar
  16. Mordukhovich BS: Optimal Control of Nonconvex Discrete and Differential Inclusions. Sociedad Matematica Mexicana, Mexico; 1998:vi + 324.Google Scholar
  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/BF02672708MathSciNetView ArticleGoogle Scholar
  18. Pšeničnyĭ BN: Convex Analysis and Extremal Problems, Series in Nonlinear Analysis and Its Applications. Nauka, Moscow; 1980:320.Google Scholar
  19. Rockafellar RT: Convex Analysis. Princeton University Press, New Jersey; 1972.MATHGoogle Scholar
  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, 1967MATHGoogle Scholar


© Elimhan N. Mahmudov. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.