Skip to main content
SpringerLink
Log in

Optimization of Higher-Order Differential Inclusions with Special Boundary Value Conditions

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The paper is devoted to Lagrange problem of optimal control theory with higher-order differential inclusions (HODI) and special boundary conditions. Optimality conditions are derived for HODIs, as well as for their discrete analogy. In this case, discretization method of the second-order differential inclusion is used to form sufficient optimality conditions for HODIs and periodic boundary conditions, the so-called transversality conditions. And to construct an Euler–Lagrange-type inclusion, a locally adjoint mapping is used, which is closely related to the coderivative concept of Mordukhovich. In turn, this approach requires several important equivalence results concerning LAMs to the discrete and discrete-approximate problems. The results obtained are demonstrated by the optimization of some “linear” optimal control problems, for which the Weierstrass–Pontryagin maximum principle and transversality conditions are formulated.

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. Alber, Y.I., Burachik, R.S., Iusem, A.N.: A proximal point method for nonsmooth convex optimization problems in Banach spaces. Abstract Appl. Anal. 2, 97–120 (1997)

    Article  MathSciNet  Google Scholar 

  2. Arthi, G., Balachandran, K.: Controllability of damped second-order impulsive neutral functional differential systems with infinite delay. J. Optim. Theory Appl. 152, 799–813 (2012)

    Article  MathSciNet  Google Scholar 

  3. Auslender, A., Mechler, J.: Second order viability problems for differential inclusions. J. Math. Anal. Appl. 181, 205–218 (1994)

    Article  MathSciNet  Google Scholar 

  4. Azzam, D.L., Castaing, C., Thibault, L.: Three boundary value problems for second order differential inclusions in Banach spaces. Contr. Cyber. 31(3), 659–693 (2002)

    MathSciNet  MATH  Google Scholar 

  5. Azzam, D.L., Makhlouf, A., Thibault, L.: Existence and relaxation theorem for a second order differential inclusion. Numer. Funct. Anal. Optim. 31, 1103–1119 (2010)

    Article  MathSciNet  Google Scholar 

  6. Benchohra, M., Graef, J.R., Henderson, J., Ntouyas, S.K.: Nonresonance impulsive higher order functional nonconvex-valued differential inclusions. Electr. J. Qual. Theory Differ. Equ. 13, 1–13 (2002)

    MathSciNet  MATH  Google Scholar 

  7. Benchohra, M., Ntouyas, S.K.: Controllability for an infinite-time horizon of second-order differential inclusions in Banach spaces with nonlocal conditions. J. Optim. Theory Appl. 109, 85–98 (2001)

    Article  MathSciNet  Google Scholar 

  8. Cannarsa, P., Frankowska, H., Scarinci, T.: Sensitivity relations for the Mayer problem with differential inclusions. ESAIM: COCV 21, 789–814 (2015)

    MathSciNet  MATH  Google Scholar 

  9. Cernea, A.: On the existence of solutions for a higher order differential inclusion without convexity. Electr. J. Qual. Theory Differ. Equ. 8, 1–8 (2007)

    MathSciNet  MATH  Google Scholar 

  10. Chalishajar, D.N.: Controllability of second order Impulsive neutral functional differential inclusions with infinite delay. J. Optim. Theory Appl. 154, 672–684 (2012)

    Article  MathSciNet  Google Scholar 

  11. Chang, Y.K., Li, W.T.: Controllability of second-order differential and integro differential Inclusions in Banach spaces. J. Optim. Theory Appl. 129, 77–87 (2006)

    Article  MathSciNet  Google Scholar 

  12. Cibulka, R., Dontchev, A.L., Veliov, V.M.: Graves-type theorems for the sum of a Lipschitz function and a set-valued mapping. SIAM J. Control Optim. 54(6), 3273–3296 (2016)

    Article  MathSciNet  Google Scholar 

  13. Conway, B.A., Larson, K.M.: Collocation versus differential inclusion in direct optimization. J. Guidance Contr. Dyn. 21(5), 780–785 (1998)

    Article  Google Scholar 

  14. An, D.T.V., Yen, N.D.: Differential stability of convex optimization problems under inclusion constraints. Appl. Anal. 94, 108–128 (2015)

    Article  MathSciNet  Google Scholar 

  15. Giannessi, F., Maugeri, A.: Variational Analysis and Applications. Springer, Berlin (2005)

    Book  Google Scholar 

  16. Ioffe, A.D., Tikhomirov, V.: Theory of extremal problems. “Nauka”. Moscow (1974) English transl. North-Holland. Amsterdam (1978)

  17. Jeyakumar, V., Wu, Z.Y.: A qualification free sequential Pshenichnyi-Rockafellar lemma and convex semidefinite. J. Convex Anal. 13(3), 773–784 (2006)

    MathSciNet  MATH  Google Scholar 

  18. Khan, A.A., Tammer, C., Zalinescu, C.: Set-Valued Optimization. Springer, Berlin (2016)

    MATH  Google Scholar 

  19. Kourogenis, N.C.: Strongly nonlinear second order differential inclusions with generalized boundary conditions. J. Math. Anal. Appl. 287, 348–364 (2003)

    Article  MathSciNet  Google Scholar 

  20. Loewen, P.D., Rockafellar, R.T.: Bolza problems with general time constraits. SIAM J. Contr. Optim. 35, 2050–2069 (1997)

    Article  Google Scholar 

  21. Mahmudov, E.N.: Optimization of Mayer problem with Sturm-Liouville-type differential inclusions. J. Optim. Theory Appl. 177(2), 345–375 (2018)

    Article  MathSciNet  Google Scholar 

  22. Mahmudov, E.N.: Single Variable Differential and Integral Calculus. Springer, Paris (2013)

    Book  Google Scholar 

  23. Mahmudov, E.N., Pshenichnyi, B.N.: The optimality principle for discrete and differential inclusions of parabolic type with distributed parameters, and duality. Russ. Acad. Sci. Izvestiya Math. 42(2), 299 (1994)

    Article  Google Scholar 

  24. Mahmudov, E.N.: Approximation and optimization of Darboux type differential inclusions with set-valued boundary conditions. Optim. Lett. 7, 871–891 (2013)

    Article  MathSciNet  Google Scholar 

  25. Mahmudov, E.N.: Approximation and Optimization of Discrete and Differential Inclusions. Elsevier, Boston (2011)

    MATH  Google Scholar 

  26. Mahmudov, E.N.: Approximation and optimization of higher order discrete and differential inclusions. Nonlin. Differ. Equ. Appl. NoDEA 21, 1–26 (2014)

    Article  MathSciNet  Google Scholar 

  27. Mahmudov, E.N.: Convex optimization of second order discrete and differential inclusions with inequality constraints. J. Convex Anal. 25, 1–26 (2018)

    Article  MathSciNet  Google Scholar 

  28. Mahmudov, E.N.: Optimal control of second order delay-discrete and delay-differential inclusions with state constraints. Evol. Equ. Contr. Theory 7(3), 501–529 (2018)

    Article  MathSciNet  Google Scholar 

  29. Mahmudov, E.N.: Locally adjoint mappings and optimization of the first boundary value problem for hyperbolic type discrete and differential inclusions. Nonlinear Anal. 67, 2966–2981 (2007)

    Article  MathSciNet  Google Scholar 

  30. Mahmudov, E.N.: Necessary and sufficient conditions for discrete and differential inclusions of elliptic type. J. Math. Anal. Appl. 323, 768–789 (2006)

    Article  MathSciNet  Google Scholar 

  31. Mahmudov, E.N., Mardanov, M.J.: On duality in optimal control problems with second-order differential inclusions and initial-point constraints. Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 46, 115–128 (2020)

    MathSciNet  MATH  Google Scholar 

  32. Marco, L., Murillo, J.A.: Lyapunov functions for second order differential inclusions: a viability approach. J. Math. Anal. Appl. 262, 339–354 (2001)

    Article  MathSciNet  Google Scholar 

  33. Martinez-Legaz, J.E., Thera, M.: A convex representation of maximal monotone operators. Spec. Issue Prof. Ky Fan. J. Nonlin. Convex Anal. 2, 243–247 (2001)

    MathSciNet  MATH  Google Scholar 

  34. Mordukhovich, B.S., Nam, N.M.: An Easy Path to Convex Analysis and Applications. Morgan & Claypool Publishers, San Rafael (2014)

    MATH  Google Scholar 

  35. Mordukhovich, B.S., Outrata, J.V., Sarabi, M.E.: Full stability of locally optimal solutions in second-order cone programs. SIAM J. Optim. 24(4), 1581–1613 (2014)

    Article  MathSciNet  Google Scholar 

  36. Mordukhovich, B.S.: Discrete approximations and refined Euler-Lagrange conditions for nonconvex differential inclusions. SIAM J. Contr. Optim. 33, 882–915 (1995)

    Article  MathSciNet  Google Scholar 

  37. Ying, G., Xinmin, Y., Jin, Y., Hong, Y.: Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps. J. Ind. Manag. Optim. 11, 673–683 (2014)

    MathSciNet  MATH  Google Scholar 

  38. Zhang, Q., Li, G.: Nonlinear boundary value problems for second order differential inclusions. Inter. J. Nonlinear Sci. 9, 84–103 (2010)

    MATH  Google Scholar 

Download references

Acknowledgements

The author would like to express his sincere thanks to Prof. Lionel Thibault, editor of JOTA, and anonymous reviewers for their valuable suggestions that improved the final manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics Engineering, Istanbul Technical University, Istanbul, Turkey

    Elimhan N. Mahmudov

  2. Azerbaijan National Academy of Sciences, Institute of Control Systems, Baku, Azerbaijan

    Elimhan N. Mahmudov

Authors
  1. Elimhan N. Mahmudov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elimhan N. Mahmudov.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Lionel Thibault.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mahmudov, E.N. Optimization of Higher-Order Differential Inclusions with Special Boundary Value Conditions. J Optim Theory Appl 192, 36–55 (2022). https://doi.org/10.1007/s10957-021-01936-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-021-01936-6

Keywords

Mathematical Subject Classification

Search

Navigation