Optimization of the Lagrange Problem with second order discrete and differential inclusions and spatial boundary conditions

Gülseren Çiçek; Elimhan Mahmudov

doi:10.15672/hujms.1601668

Optimization of the Lagrange Problem with second order discrete and differential inclusions and spatial boundary conditions

Abstract

In this paper, we derive optimality conditions for the Lagrange problem with second-order differential inclusions (DFIs) and spatial boundary conditions. The Lagrangian and the set-valued mapping here also depend on the derivative of the sought trajectory. The difficulties that arise here are related to the construction of the adjoint discrete and differential inclusions. The novelty here lies in using the discretized method to establish the optimality condition for both discrete and DFIs. Optimality conditions for the discrete problem are generated by applying the concept locally adjoint mapping(LAM). Equivalence theorems are used to obtain the so-called Mahmudov's adjoint conditions for the discrete-approximat problem. Moreover passing to the limit, we get sufficient optimality conditions for the continuous problem. Unlike the Euler-Lagrange DFI, which only provides first-order optimality conditions, Mahmudov's adjoint inclusion is a powerful tool for establishing optimality conditions for higher-order problems. The findings are reinforced with an example. We also obtain similar results for the non-convex problem by using the concept of local tents.

Keywords

References

[1] N.U. Ahmed and K.L. Teo, Optimal Control of Distributed Parameter Systems, Elsevier Sci. Inc., New York, 1981.

Details

Primary Language

English

Subjects

Calculus of Variations, Mathematical Aspects of Systems Theory and Control Theory

Journal Section

Research Article

Authors

Gülseren Çiçek ^*
0000-0002-3012-3939
Türkiye

Elimhan Mahmudov
0000-0003-2879-6154
Azerbaijan

Early Pub Date

December 30, 2025

Publication Date

December 30, 2025

Submission Date

December 15, 2024

Acceptance Date

October 20, 2025

Published in Issue

Year 2026 Number: Advanced Online Publication

DOI

https://doi.org/10.15672/hujms.1601668

IZ

https://izlik.org/JA55XW52SJ

Cite

RIS / Bibtex

APA

Çiçek, G., & Mahmudov, E. (2025). Optimization of the Lagrange Problem with second order discrete and differential inclusions and spatial boundary conditions. Hacettepe Journal of Mathematics and Statistics, Advanced Online Publication. https://doi.org/10.15672/hujms.1601668

AMA

1.Çiçek G, Mahmudov E. Optimization of the Lagrange Problem with second order discrete and differential inclusions and spatial boundary conditions. Hacettepe Journal of Mathematics and Statistics. 2025;(Advanced Online Publication). doi:10.15672/hujms.1601668

Chicago

Çiçek, Gülseren, and Elimhan Mahmudov. 2025. “Optimization of the Lagrange Problem With Second Order Discrete and Differential Inclusions and Spatial Boundary Conditions”. Hacettepe Journal of Mathematics and Statistics, no. Advanced Online Publication. https://doi.org/10.15672/hujms.1601668.

EndNote

Çiçek G, Mahmudov E (December 1, 2025) Optimization of the Lagrange Problem with second order discrete and differential inclusions and spatial boundary conditions. Hacettepe Journal of Mathematics and Statistics Advanced Online Publication

IEEE

[1]G. Çiçek and E. Mahmudov, “Optimization of the Lagrange Problem with second order discrete and differential inclusions and spatial boundary conditions”, Hacettepe Journal of Mathematics and Statistics, no. Advanced Online Publication, Dec. 2025, doi: 10.15672/hujms.1601668.

ISNAD

Çiçek, Gülseren - Mahmudov, Elimhan. “Optimization of the Lagrange Problem With Second Order Discrete and Differential Inclusions and Spatial Boundary Conditions”. Hacettepe Journal of Mathematics and Statistics. Advanced Online Publication (December 1, 2025). https://doi.org/10.15672/hujms.1601668.

JAMA

1.Çiçek G, Mahmudov E. Optimization of the Lagrange Problem with second order discrete and differential inclusions and spatial boundary conditions. Hacettepe Journal of Mathematics and Statistics. 2025. doi:10.15672/hujms.1601668.

MLA

Çiçek, Gülseren, and Elimhan Mahmudov. “Optimization of the Lagrange Problem With Second Order Discrete and Differential Inclusions and Spatial Boundary Conditions”. Hacettepe Journal of Mathematics and Statistics, no. Advanced Online Publication, Dec. 2025, doi:10.15672/hujms.1601668.

Vancouver

1.Gülseren Çiçek, Elimhan Mahmudov. Optimization of the Lagrange Problem with second order discrete and differential inclusions and spatial boundary conditions. Hacettepe Journal of Mathematics and Statistics. 2025 Dec. 1;(Advanced Online Publication). doi:10.15672/hujms.1601668