SpringerLink
Log in

Sufficient conditions for interval-valued optimal control problems in admissible orders

  • Fuzzy systems and their mathematics
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

This paper addresses the optimal control problems with an interval-valued objective function. We consider a type of total order relationships \(\le _{adm}\) between two intervals. For each total order relationship, Kuhn–Tucker sufficient conditions for the optimal control problems with an interval-valued objective function are obtained. A numerical example is considered and solved. Kuhn–Tucker sufficient conditions provided under the total order relationship \(\le _{adm}\) are sufficient conditions under the partial order relationship \(\preceq _{LU}\) for interval-valued optimal control problems. The results of this paper could help construct evolutionary algorithms to solve interval-valued optimal control problems.

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

Access this article

Log in via an institution

Price includes VAT (France)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Data availability

All relevant data are within the paper.

References

  • Abu Arqub O (2017) Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm–Volterra integrodifferential equations. Neural Comput Appl 28:1591–1610

    Article  Google Scholar 

  • Abu Arqub O, Singh J, Maayah B et al (2023a) Reproducing kernel approach for numerical solutions of fuzzy fractional initial value problems under the Mittag–Leffler kernel differential operator. Math Methods Appl Sci 46(7):7965–7986

  • Abu Arqub O, Singh J, Alhodaly M (2023b) Adaptation of kernel functions-based approach with Atangana–Baleanu–Caputo distributed order derivative for solutions of fuzzy fractional Volterra and Fredholm integrodifferential equations. Math Methods Appl Sci 46(7):7807–7834

  • Abualigah L, Yousri D, Abd Elaziz M et al (2021a) Aquila optimizer: a novel meta-heuristic optimization algorithm. Comput Ind Eng 157:107250

  • Abualigah L, Diabat A, Mirjalili S et al (2021b) The arithmetic optimization algorithm. Comput Methods Appl Mech Eng 376:113609

  • Abualigah L, Abd Elaziz M, Sumari P et al (2022) Reptile search algorithm (RSA): a nature-inspired meta-heuristic optimizer. Expert Syst Appl 191:116158

    Article  Google Scholar 

  • Agarwal D, Singh P, El Sayed MA (2023) The Karush–Kuhn–Tucker (KKT) optimality conditions for fuzzy-valued fractional optimization problems. Math Comput Simul 205:861–877

    Article  MathSciNet  Google Scholar 

  • Agushaka JO, Ezugwu AE, Abualigah L (2022) Dwarf mongoose optimization algorithm. Comput Methods Appl Mech Eng 391:114570

    Article  ADS  MathSciNet  Google Scholar 

  • Alshammari M, Al-Smadi M, Arqub OA et al (2020) Residual series representation algorithm for solving fuzzy duffing oscillator equations. Symmetry 12(4):572

    Article  ADS  Google Scholar 

  • Aubin JP, Cellina A (1984) Differential inclusions. Springer, New York

    Book  Google Scholar 

  • Bhunia AK, Samanta SS (2014) A study of interval metric and its application in multi-objective optimization with interval objectives. Comput Ind Eng 74:169–178

    Article  Google Scholar 

  • Bustince H, Fernandez J, Kolesárová A et al (2013a) Generation of linear orders for intervals by means of aggregation functions. Fuzzy Sets Syst 220:69–77

  • Bustince H, Galar M, Bedregal B et al (2013b) A new approach to interval-valued Choquet integrals and the problem of ordering in interval-valued fuzzy set applications. IEEE Trans Fuzzy Syst 21:1150–1162

  • Chai R, Savvaris A, Tsourdos A et al (2020) Solving multiobjective constrained trajectory optimization problem by an extended evolutionary algorithm. IEEE Trans Cybern 50(4):1630–1643

    Article  PubMed  Google Scholar 

  • Chakraverty S, Mahato NR, Jeswal SK (2022) Sign function and ANN based pole placement for computing interval controls. ISA Trans 119:17–24

    Article  CAS  PubMed  Google Scholar 

  • Chalco-Cano Y, Lodwick WA, Rufian-Lizana A (2013) Optimality conditions of type KKT for optimization problem with interval-valued objective function via generalized derivative. Fuzzy Optim Decis Mak 12:305–322

    Article  MathSciNet  Google Scholar 

  • Cheng J, Liu Z, Wu Z et al (2016) Direct optimization of uncertain structures based on degree of interval constraint violation. Comput Struct 164:83–94

    Article  Google Scholar 

  • Chiu C, Hung Y (2020) One wheel vehicle real world control based on interval type 2 fuzzy controller. Mechatronics 70:102387

    Article  Google Scholar 

  • Cui Y, Pang JS (2021) Modern nonconvex nondifferentiable optimization, Society for Industrial and Applied Mathematics

  • Das S, Mondal R, Shaikh AA et al (2022) An application of control theory for imperfect production problem with carbon emission investment policy in interval environment. J Frankl Inst 359(5):1925–1970

    Article  MathSciNet  Google Scholar 

  • Das S, Mandal G, Manna AK et al (2023) Effects of emission reduction and rework policy in a production system of green products: an interval valued optimal control theoretic approach. Comput Ind Eng 179:109212

    Article  Google Scholar 

  • EL Sayed MA, Abo-Sinna MA (2021) A novel approach for full intuitionistic fuzzy multi-objective fractional transportation problem. Alex Eng J 60:1447–1463

    Article  Google Scholar 

  • El Sayed MA, Baky IA, Singh P (2020) A modified TOPSIS approach for solving stochastic fuzzy multi-level multi-objective fractional decision making problem. Opsearch 57:1374–1403

    Article  MathSciNet  Google Scholar 

  • El Sayed MA, Farahat FA, Elsisy MA (2022) A novel interactive approach for solving uncertain bi-level multi-objective supply chain model. Comput Ind Eng 169:108225

    Article  Google Scholar 

  • Elsisy MA, Elsaadany AS, El Sayed MA (2020) Using interval operations in the Hungarian method to solve the fuzzy assignment problem and its application in the rehabilitation problem of valuable buildings in Egypt. Complexity 2020:1–11

    Google Scholar 

  • Elsisy MA, El Sayed MA, Abo-Elnaga Y (2021) A novel algorithm for generating Pareto frontier of bi-level multi-objective rough nonlinear programming problem. Ain Shams Eng J 12(2):2125–2133

    Article  Google Scholar 

  • Ezugwu AE, Agushaka JO, Abualigah L et al (2022) Prairie dog optimization algorithm. Neural Comput Appl 34:20017–20065

    Article  Google Scholar 

  • Farhadinia B (2014) Pontryagin’s minimum principle for fuzzy optimal control problems. Iran J Fuzzy Syst 11:27–43

    MathSciNet  Google Scholar 

  • Fu C, Cao L (2019) An uncertain optimization method based on interval differential evolution and adaptive subinterval decomposition analysis. Adv Eng Softw 134:1–9

    Article  Google Scholar 

  • Ge X, Zhu Y (2013) A necessary condition of optimality for uncertain optimal control problem. Fuzzy Optim Decis Mak 12:41–51

    Article  MathSciNet  Google Scholar 

  • Gong DW, Ji XF, Sun J et al (2014) Interactive evolutionary algorithms with decision-makers preferences for solving interval multi-objective optimization problems. Neurocomputing 137:241–251

    Article  Google Scholar 

  • Ishibuchi H, Tanaka H (1990) Multiobjective programming in optimization of the interval objective function. Eur J Oper Res 48:219–225

    Article  Google Scholar 

  • Kushner HJ (1972) Necessary conditions for continuous parameter stochastic optimization problems. SIAM J Control 10:550–565

    Article  MathSciNet  Google Scholar 

  • Leal U, Lodwick W, Silva G et al (2021) Interval optimal control for uncertain problems. Fuzzy Sets Syst 402:142–154

    Article  MathSciNet  Google Scholar 

  • Lian Z, Shi P, Lim C (2021) Hybrid-triggered interval type-2 fuzzy control for networked systems under attacks. Inf Sci 567:332–347

    Article  MathSciNet  Google Scholar 

  • Mangasaria OL (1966) Sufficient conditions for the optimal control of nonlinear systems. SIAM J Control 4:139–152

    Article  MathSciNet  Google Scholar 

  • Manna AK, Bhunia AK (2022) Investigation of green production inventory problem with selling price and green level sensitive interval-valued demand via different metaheuristic algorithms. Soft Comput 26(19):10409–10421

    Article  Google Scholar 

  • Moore RE, Kearfott RB, Cloud MJ (2009) Introduction to interval analysis. SIAM, Philadelphia

    Book  Google Scholar 

  • Neumaier A (1990) Interval methods for systems of equations. Cambridge University Press, Cambridge

    Google Scholar 

  • Oyelade ON, Ezugwu AE, Mohamed TIA et al (2022) Ebola optimization search algorithm: a new nature-inspired metaheuristic optimization algorithm. IEEE Access 10:16150–16177

    Article  Google Scholar 

  • Ruidas S, Seikh MR, Nayak PK (2021) A production inventory model with interval-valued carbon emission parameters under price-sensitive demand. Comput Ind Eng 154:107154

    Article  Google Scholar 

  • Ruidas S, Seikh MR, Nayak PK et al (2023) An interval-valued green production inventory model under controllable carbon emissions and green subsidy via particle swarm optimization. Soft Comput 27:9709–9733

    Article  Google Scholar 

  • Sahoo L, Bhunia AK, Kapur PK (2012) Genetic algorithm based multi-objective reliability optimization in interval environment. Comput Ind Eng 62:152–160

    Article  Google Scholar 

  • Sampath Dakshina Murthy A, DrS KR, DrK SN et al (2015) Tracking of a manoeuvering target ship using radar measurements. Indian J Sci Technol 8:0974–5645

    Article  Google Scholar 

  • Sampath Dakshina Murthy A, Pavani T, Lakshmi K (2016) An application of firefly hybrid extended Kalman filter tracking a reentry object. Indian J Sci Technol 9:28

    Google Scholar 

  • Treanţă S (2021) On a new class of interval-valued variational control problems. In: Debnath P, Konwar N, Radenović S (eds) Metric fixed point theory. Forum for interdisciplinary mathematics. Springer, Singapore

    Google Scholar 

  • Treanţă S (2022) Characterization results of solutions in interval-valued optimization problems with mixed constraints. J Glob Optim 82:951–964

    Article  MathSciNet  Google Scholar 

  • Van Su T, Dinh DH (2020) Duality results for interval-valued pseudoconvex optimization problem with equilibrium constraints with applications. Comput Appl Math 39:127

    Article  MathSciNet  Google Scholar 

  • Wang L, Liu J, Yang C et al (2021) A novel interval dynamic reliability computation approach for the risk evaluation of vibration active control systems based on PID controllers. Appl Math Model 92:422–446

    Article  MathSciNet  Google Scholar 

  • Wu HC (2007) The Karush–Kuhn–Tucker optimality conditions in an optimization problem with interval-valued objective function. Eur J Oper Res 176:46–59

    Article  MathSciNet  Google Scholar 

  • Xu Z (2020) Stochastic recursive optimal control problem with obstacle constraint involving diffusion type control. Adv Differ Equ 2020:381

    Article  MathSciNet  Google Scholar 

  • Xu ZS, Yager RR (2006) Some geometric aggregation operators based on intuitionistic fuzzy sets. Int J Gen Syst 35:417–433

    Article  MathSciNet  Google Scholar 

  • Yuan Z, Wang W, He S (2020) Interval-based LQR strategy for optimal control of proton exchange membrane fuel cell system with interval uncertainties. ISA Trans 100:334–345

    Article  Google Scholar 

  • Zhang CL, Huang NJ (2022) On Ekeland’s variational principle for interval-valued functions with applications. Fuzzy Sets Syst 436:152–174

    Article  MathSciNet  Google Scholar 

  • Zhao Y, Zhu Y (2010) Fuzzy optimal control of linear quadratic models. Comput Math Appl 60:67–73

    Article  MathSciNet  Google Scholar 

  • Zhou XY (1996) Sufficient conditions of optimality for stochastic systems with controllable diffusions. IEEE Autom Control 41:1176–1179

    Article  MathSciNet  Google Scholar 

  • Zhou XY (1998) Stochastic near-optimal controls: necessary and sufficient conditions for near-optimality. SIAM J Control Optim 36:929–947

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11401469, F0118) and the Natural Science Basic Research Program of Shaanxi (Program No. 2023-JC-YB-623).

Author information

Author notes

  1. L. Li and J. Zhang have contributed equally to this work.

Authors and Affiliations

  1. School of Science, **’an University of Posts and Telecommunications, **’an, 710121, China

    Lifeng Li & Jianke Zhang

  2. Shaanxi Key Laboratory of Network Data Analysis and Intelligent Processing, **’an University of Posts and Telecommunications, **’an, 710121, China

    Lifeng Li & Jianke Zhang

Authors
  1. Lifeng Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianke Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors have equal contributions. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Lifeng Li.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, L., Zhang, J. Sufficient conditions for interval-valued optimal control problems in admissible orders. Soft Comput 28, 2843–2850 (2024). https://doi.org/10.1007/s00500-023-09563-1

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-023-09563-1

Keywords

Search

Navigation