Abstract

This paper first deals with the Neumann problem for discrete, discrete-approximate problems and elliptic differential inclusions. Then using the obtained results for discrete inclusions in the form of Euler-Lagrange inclusions necessary and sufficient conditions for optimality are derived for the problems under consideration. Further, these problems are generalized to mixed problems, where the Neumann and Dirichlet conditions are satisfied separately in different parts of the set-valued mapping domain. As an example, a linear optimal control problem is considered, from which the Weierstrass-Pontryagin maximum condition follows. Thus, relying only on the previous auxiliary results, we can obtain the main results of this paper for mixed problems. In turn, the Neumann problem is generalized to the multidimensional case with a second order elliptic operator.

Author information

Contact details are reproduced from the original publication and may be historical.

Elimhan N. Mahmudov

(1) Department of Mathematics, Istanbul Technical University, Istanbul, Turkey
(2) Azerbaijan National Aviation Academy, Baku, Azerbaijan
and: Research Center for Mathematical Modeling and Optimization, UNEC, Azerbaijan State University of Economics, Baku, Azerbaijan

elimhan22@yahoo.com

Dilara I. Mastaliyeva

Institute of Control Systems of the Ministry of Science and Education of the
Republic of Azerbaijan, Baku, Azerbaijan

dilara.mestelieva@gmail.com

Keywords

  • Discrete-approximate
  • elliptic differential inclusion
  • Neumann
  • Dirichlet
  • mixed problems
  • necessary and sufficient conditions

Mathematics Subject Classification

35J15, 47J22, 49K20

References

  1. S. Adly, A. Hantoute, M. Théra: Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions, Nonlin. Analysis Theory Methods Appl. 75 (2012), art.no. 9851008.https://doi.org/10.1016/j.na.2010.11.009Google Scholar
  2. F. Alvarez, S. Flores: Remarks on Lipschitz solutions to measurable differential inclusions and an existence result for some nonconvex variational problems, J. Convex Analysis 15 (2008) 349–363.https://doi.org/10.68381/jca15025Google Scholar
  3. H. Attouch, M. Marques Alves, B. F. Svaiter: A dynamic approach to a proximal Newton method for monotone inclusions in Hilbert spaces, with complexity O(1/n2), J. Convex Analysis 23 (2016) 139–180.https://doi.org/10.68381/jca23006Google Scholar
  4. J. P. Aubin, A. Cellina: Differential Inclusions. Set-Valued Maps and Viability Theory, Grundlehren der Mathematischen Wissenschaften 264, Springer, Berlin (1984).https://doi.org/10.1007/978-3-642-69512-4Google Scholar
  5. A. Auslender, J. Mechler: Second order viability problems for differential inclusions, J. Math. Analysis Appl. 181 (1994) 205–218.https://doi.org/10.1006/jmaa.1994.1015Google Scholar
  6. D. L. Azzam, S. Lounis, L. Thibault: Existence solutions for second-order differential inclusions with nonconvex perturbations, Appl. Analysis 86 (2007) 1199–1210.https://doi.org/10.1080/00036810701460511Google Scholar
  7. D. L. Azzam, A. Makhlouf, L. Thibault: On perturbed sweeping process, Appl. Analysis 95 (2016) 303–322.https://doi.org/10.1080/00036811.2014.1002482Google Scholar
  8. G. Barletta, P. Candito, D. Motreanu: Constant sign and sign-changing solutions for quasilinear elliptic equations with Neumann boundary condition, J. Convex Analysis 21 (2014) 53–66.https://doi.org/10.68381/jca21003Google Scholar
  9. F. Bernicot, J. Venel: Differential inclusions with proximal normal cones in Banach spaces, J. Convex Analysis 17 (2010) 451–484.https://doi.org/10.68381/jca17032Google Scholar
  10. O. Carja, M. D. P. Monteiro Marques: Viability results for nonautonomous differential inclusions, J. Convex Analysis 7 (2000) 437–444.https://doi.org/10.68381/jca07023Google Scholar
  11. S. Carl: Extremal solutions for quasilinear elliptic inclusions with state-dependent subdifferentials, J. Optim. Theory Appl. 104 (2000) 323–342.https://doi.org/10.1023/A:1004609713176Google Scholar
  12. A. Cernea: Some second-order necessary conditions for nonconvex hyperbolic differential inclusion problem, J. Math. Analysis Appl. 253 (2001) 616–639.https://doi.org/10.1006/jmaa.2000.7170Google Scholar
  13. F. H. Clarke: Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics 264, Springer, New York (2013).https://doi.org/10.1007/978-1-4471-4820-3Google Scholar
  14. T. Haddad, J. Noel, L. Thibault: Perturbed sweeping process with a subsmooth set depending on the state, Lin. Nonlin. Analysis 2/1 (2016) 155–174.Google Scholar
  15. A. Ioffe: Existence and relaxation theorems for unbounded differential inclusions, J. Convex Analysis 13 (2006) 353–362.https://doi.org/10.68381/jca13029Google Scholar
  16. A. Jourani, L. Thibault, D. Zagrodny: Differential properties of the Moreau envelope, J. Funct. Analysis 266 (2014) 1185–1237.https://doi.org/10.1016/j.jfa.2013.11.008Google Scholar
  17. A. A. Khan, C. Tammer, C. Zălinescu: Set-valued Optimization. An Introduction with Applications, Springer, Berlin (2015).https://doi.org/10.1007/978-3-642-54265-7Google Scholar
  18. E. Lehrer, S. Sorin: Minmax via differential inclusion, J. Convex Analysis 14 (2007) 271–273.https://doi.org/10.68381/jca14019Google Scholar
  19. P. D. Loewen, R. T. Rockafellar: Bolza problems with general time constraints, SIAM J. Control Optim. 35 (1997) 2050–2069.https://doi.org/10.1137/S0363012996298801Google Scholar
  20. E. N. Mahmudov: On duality in optimal control problems described by convex discrete and differential inclusions, Avtom.Telemekh. 2 (1987) 13–25.Google Scholar
  21. E. N. Mahmudov: Approximation and Optimization of Discrete and Differential Inclusions, Elsevier, Boston (2011).https://doi.org/10.1007/978-3-642-31724-8_38Google Scholar
  22. E. N. Mahmudov: Approximation and optimization of higher order discrete and differential inclusions, Nonlin. Diff. Equations Appl. 21 (2014) 1–26.https://doi.org/10.1007/s00030-013-0234-1Google Scholar
  23. E. N. Mahmudov: Optimization of Mayer problem with Sturm-Liouville-type differential inclusions, J. Optim. Theory Appl. 177 (2018) 345–375.https://doi.org/10.1007/s10957-018-1260-2Google Scholar
  24. E. N. Mahmudov: Optimal control of hyperbolic type discrete and differential inclusions described by the Laplace operator, ESAIM Control Optim. Calc. Var. 28 (2022) 1–21.https://doi.org/10.1051/cocv/2022061Google Scholar
  25. E. N. Mahmudov: Optimal control of second order sweeping processes with discrete and differential inclusions, J. Convex Analysis 29 (2022) 269–290.https://doi.org/10.68381/jca29017Google Scholar
  26. E. N. Mahmudov, Ö. Değer: Optimal control of elliptic differential inclusions with Dirichlet and Neumann boundary conditions, J. Dyn. Control Syst. 17 (2011) 163–185.https://doi.org/10.1007/s10883-011-9114-3Google Scholar
  27. E. N. Mahmudov, M. J. Mardanov: On duality in optimal control problems with second-order differential inclusions and initial-point constraints, Proc. Inst. Math. Mech. Nat. Acad. Sci. Azerb. 46 (2020) 115–128.https://doi.org/10.29228/proc.22Google Scholar
  28. E. N. Mahmudov, D. Mastaliyeva: Optimization of neutral functional-differential inclusions, J. Dyn. Control Syst. 21 (2015) 25–46.https://doi.org/10.1007/s10883-014-9215-xGoogle Scholar
  29. E. N. Mahmudov, D. I. Mastaliyeva: Optimal control of higher-order retarded differential inclusions, J. Indust. Manage. Optim. 1-14; doi: 10.3934/jimo. 2022226.https://doi.org/10.3934/jimoGoogle Scholar
  30. S. A. Marano: Elliptic equations and differential inclusions, Proc. 2nd World Congress Nonlin. Anal. Nonlin. Anal. 30 (1997) 1763–1770.https://doi.org/10.1016/S0362-546X(97)00252-6Google Scholar
  31. B. S. Mordukhovich: Discrete approximations and refined Euler-Lagrange conditions for nonconvex differential inclusions, SIAM J. Control Optim. 33 (1995) 882–915.https://doi.org/10.1137/S0363012993245665Google Scholar
  32. B. S. Mordukhovich: Variational Analysis and Generalized Differentiation. I: Basic Theory; II: Applications, Grundlehren der Mathematischen Wissenschaften Vol. 330 und 331, Springer, Berlin (2006).https://doi.org/10.1007/3-540-31246-3Google Scholar
  33. W. Pompe: Sufficient conditions for an existence of a solution to a differential inclusion, J. Convex Analysis 21 (2014) 715–726.https://doi.org/10.68381/jca21038Google Scholar
  34. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko: The Mathematical Theory of Optimal Processes, John Wiley, New York (1965).Google Scholar
  35. A. A. Tolstonogov: Compactness of BV solutions of a convex sweeping process of measurable differential inclusion, J. Convex Analysis 27 (2020) 673–695.https://doi.org/10.68381/jca27034Google Scholar
  36. S. S. Yusubov: Necessary optimality conditions for singular controls, Comput. Math. Math. Physics 47 (2007) 1446–1451.https://doi.org/10.1134/S0965542507090060Google Scholar

Suggested citation

E. N. Mahmudov, D. I. Mastaliyeva. “Optimization of Elliptic Type Differential Inclusions with Mixed Boundary Conditions.” Journal of Convex Analysis 32 (2025), No. 4, 1045–1064. https://doi.org/10.68381/jca32055

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