Skip to main content
SpringerLink
Log in

Optimal control of discrete and differential inclusions with distributed parameters in the gradient form

  • Published:
Journal of Dynamical and Control Systems Aims and scope Submit manuscript

This paper is devoted to optimization of so-called first-order differential (P C ) inclusions in the gradient form on a square domain. As a supplementary problem, discrete-approximation problem (P A ) is considered. In the Euler–Lagrange form, necessary and sufficient conditions are derived for the problems (P A ) and partial differential inclusions (P C ), respectively. The results obtained are based on a new concept of locally adjoint mappings. The duality theorems are proved and duality relation is established.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. Amann and P. Quittner. Elliptic boundary-value problems involving measures: existence, regularity, and multiplicity. Adv. Differ. Equations 3 (1998), No. 6, 753–813.

    MathSciNet  MATH  Google Scholar 

  2. J.-P. Aubin and A. Cellina. Differential inclusions: Set-valued maps and viability theory. Springer-Verlag (1984).

  3. J.-P. Aubin and G. Da Prato. Stochastic viability and invariance. Ann. Scu. Norm. Super. Pisa 17 (1990), No. 4, 595613.

    Google Scholar 

  4. A. G. Butkovskii. Theory of optimal control by systems with distributed parameters [in Russian]. Nauka, Moscow (1965).

    Google Scholar 

  5. A. Cernea. Some second-order necessary conditions for nonconvex hyperbolic differential inclusion problems. J. Math. Anal. Appl. 253 (2001), No. 2, 616–639.

    Article  MathSciNet  MATH  Google Scholar 

  6. F. Clarke, Yu. Ledyaev, and M. Radulescu. Approximate invariance and differential inclusions in Hilbert spaces. J. Dynam. Control Systems 3 (1997), 493–518.

    MathSciNet  MATH  Google Scholar 

  7. V. F. Demiyanov and L. V. Vasiliev. Nondifferentiable optimization. Springer-Verlag (1986).

  8. P. Diamond and V. Opoitsev. Stability of a class of differential inclusions. Dynam. Control 11 (2001), 229–242.

    Article  MathSciNet  MATH  Google Scholar 

  9. I. Ekeland and R. Temam. Convex analysis and variational problems. SIAM (1987).

  10. E. Fornasini and G. Marchesini. Doubly-indexed dynamical systems: State-space models and structural properties. Theory Comput. Systems 12 (1978), 59–72.

    MathSciNet  MATH  Google Scholar 

  11. S. R. Grace, R. P. Agarwal, and D. O’Regan. A selection of oscillation criteria for second-order differential inclusions. Appl. Math. Lett. 22 (2009), No. 2, 153–158.

    Article  MathSciNet  MATH  Google Scholar 

  12. T. X. D. Ha. Some variants of the ekeland variational principle for a set-valued map. J. Optim. Theory Appl. 124 (2005), 187–206.

    Article  MathSciNet  MATH  Google Scholar 

  13. A. D. Ioffe and V. M. Tikhomirov. Theory of extremal problems. North-Holland Publ., Amsterdam–New York (1979).

  14. T. Kaczorek. Two-dimensional linear systems. Springer-Verlag (1985).

  15. J. L. Lions. Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Études math. Dunod (1968).

  16. P.-L. Lions. Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, 2: Viscosity solutions and uniqueness. Commun. Partial Differ. Equations 8 (1983), 1229–1276.

    Article  MATH  Google Scholar 

  17. E. N. Mahmudov. On duality in problems of optimal control described by convex differential inclusions of Goursat–Darboux type. J. Math. Anal. Appl. 307 (2005), No. 2, 628–640.

    Article  MathSciNet  MATH  Google Scholar 

  18. _______, The optimality principle for discrete and first-order partial differential inclusions. J. Math. Anal. Appl. 308 (2005), No. 2, 605–619.

    Article  MathSciNet  MATH  Google Scholar 

  19. _______, Necessary and sufficient conditions for discrete and differential inclusions of elliptic type. J. Math. Anal. Appl. 323 (2006), No. 2, 768–789.

    Article  MathSciNet  MATH  Google Scholar 

  20. _______, Locally adjoint mappings and optimization of the first boundary value problem for hyperbolic type discrete and differential inclusions. Nonlin. Anal. Theory Methods Appl. 67 (2007), No. 10, 2966–2981.

    Article  MathSciNet  MATH  Google Scholar 

  21. _______, Sufficient conditions for optimality for differential inclusions of parabolic type and duality. J. Global Optim. 41 (2008), 31–42.

    Article  MathSciNet  MATH  Google Scholar 

  22. V. L. Makarov and A. M. Rubinov. Mathematical theory of economic dynamics and equilibria. Springer-Verlag (1977).

  23. B. S. Mordukhovich. Variational analysis and generalized differentiation. I: Basic theory. II: Applications. Springer-Verlag (2006).

  24. C. Olech. Control theory and topics in functional analysis. In: Existence theory in optimal control 1, Int. Atomic Energy Agency, Vienna (1976), 291–328.

  25. A. Orpel. On solutions of the dirichlet problem for a class of partial differential inclusions with superlinear nonlinearities. Num. Funct. Anal. Optim. 23 (2002), No. 15, 367–381.

    Article  MathSciNet  MATH  Google Scholar 

  26. N. S. Papageorgiou. Existence of solutions for hyperbolic differential inclusions in Banach spaces. Arch. Math. 28 (1992), 205–213.

    MATH  Google Scholar 

  27. B. N. Pshenichnyi. Convex analysis and extremal problems. Nauka, Moscow (1980).

    MATH  Google Scholar 

  28. R. T. Rockafellar. Convex analysis. Princeton Univ. Press (1997).

  29. A. N. Tikhonov and A. A. Samarsky. Equations of mathematical physics. Dover Publ. (1990).

  30. H. D. Tuan. On solution sets of nonconvex darboux problems and applications to optimal control with endpoint constraints. ANZIAM J. 37 (1996), No. 3, 354–391.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    E. N. Mahmudov & M. E. Unal

Authors
  1. E. N. Mahmudov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. E. Unal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. N. Mahmudov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mahmudov, E.N., Unal, M.E. Optimal control of discrete and differential inclusions with distributed parameters in the gradient form. J Dyn Control Syst 18, 83–101 (2012). https://doi.org/10.1007/s10883-012-9135-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10883-012-9135-6

Key words and phrases

2000 Mathematics Subject Classification

Search

Navigation