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Journals / Istanbul Journal of Mathematics / 2025 / Volume: 3 Issue: 2

Optimization of discrete and differential inclusions of Lagrange Problem involving different delays

Gülseren Çiçek(ISTANBUL UNIVERSITY)0000-0002-3012-3939
Pages
—
Publication Date
2025-12-16
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

In optimization problems done so far, as far as we know, conditions for derivatives depending on a single delay are considered, while in our study, the velocity depends on both the present and two different instants in the past of the state.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper, optimality conditions for the Lagrange Problem with differential inclusions(DFIs) involving different delays and boundary conditions are obtained.
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Key Findings & Data:The velocity in the handled problem depends not only on the state at this instant, but also upon the history of the trajectory until this instant.
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Conclusion & Impact:In optimization problems done so far, as far as we know, conditions for derivatives depending on a single delay are considered, while in our study, the velocity depends on both the present and two different instants in the past of the state.
🔬Design:Modeling & Simulation
📍Scope:Comprehensive Sample
⚙️Variables:functional differential inclusions · Lagrange problem · delayed discrete inclusion

Abstract

In this paper, optimality conditions for the Lagrange Problem with differential inclusions(DFIs) involving different delays and boundary conditions are obtained. The velocity in the handled problem depends not only on the state at this instant, but also upon the history of the trajectory until this instant. We use the term Locally adjoint mapping(LAM) to determine adjoint discrete inclusions. The difficulty in the paper appears in the discretization method since the velocity depends also on two different states in the past. Some equivalence relations are applied to derive necessary and sufficient conditions for the discrete approximation problem. Passing to the limit, sufficient optimality conditions are established for the Lagrange optimal problem described by DFIs involving different delays. Similar results for the non-convex delayed Lagrange problem are obtained. We hope that with this study we will contribute to the development of optimal theory with delayed differential inclusions. In optimization problems done so far, as far as we know, conditions for derivatives depending on a single delay are considered, while in our study, the velocity depends on both the present and two different instants in the past of the state.

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Cite This Article

Çiçek, G. (2025). Optimization of discrete and differential inclusions of Lagrange Problem involving different delays. Istanbul Journal of Mathematics, Volume: 3 Issue: 2. https://doi.org/10.26650/ijmath.2025.00027

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References (13)

  1. Aubin, J.P., Cellina, A., 1984, Differential Inclusions, Springer-Verlag: Berlin. google scholar
  2. Cicek, G., Mahmudov, E.N., 2021, Optimization of Mayer functional in problems with discrete and differential inclusions and viability constraints, Turkish J.of Math., 45(5), 2084-2102. google scholar
  3. Çiçek, G., Mahmudov, E.N., 2016, The Problem of Mayer for Discrete and Differential Inclusions with Initial Boundary Constraints, Applied Mathematics and Information Sciences, 10(4), 1719-1728. google scholar
  4. Haddad, G., 1981, Monotone viable trajectories for functional-differential inclusions, Journal Diff. Equations, 42, 1-24. google scholar
  5. Kisielewicz, M., 1991, Differential Inclusions and Optimal Control, Kluwer Acad.Pub.: London, 0-7923-0675-9. google scholar
  6. Mahmudov, E.N., 2011, Approximation and Optimization of Discrete and Differential Inclusions, Elsevier: Boston, USA, 978-0-12-388428-2. google scholar
  7. Mahmudov, E.N., Mastaliyeva, D., 2015, Optimization of Neutral Functional-DifferentialInclusions, J Dyn Control Syst, 21, 25-46. google scholar
  8. Mahmudov, E.N., Mastaliyeva, D., 2023, Optimal control of higher-order retarded differential inclusions, Journal of Industrial and Management Optimization, 19(9), 6544-6557. google scholar
  9. Mordukhovich, B.S., 2006, Variational Analysis and Generalized Differentiation, I: Basic Theory; II. Applications, Grundlehren Series (Funde-mental Principles of Mathematical Sciences), Vol. 330 and 331, Springer. google scholar
  10. Mordukhovich, B., Nguyen, D., Nguyen, T. et al. 2025, Well-Posedness and Stability of Discrete Approximations for Controlled Sweeping Processes with Time Delay. J Optim Theory Appl., 206(70), 1-28. google scholar
  11. Rockafellar, R.T., 1972, Convex Analysis, Princeton University Press, New Jersey. 0-691-08069-0. google scholar
  12. Aubin, J.P., Cellina, A., 1984, Differential Inclusions, Springer-Verlag: Berlin. google scholar
  13. Mordukhovich, B., Nguyen, D., Nguyen, T. et al. 2025, Well-Posedness and Stability of Discrete Approximations for Controlled Sweeping Processes with Time Delay. J Optim Theory Appl., 206(70), 1-28. google scholar