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Optimal control of higher-order Hilfer fractional non-instantaneous impulsive stochastic integro-differential systems

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Abstract

Nowadays, engineers and biochemical industries have benefited greatly from optimal control analysis and its computational methods. Furthermore, the optimal control theory is a powerful instrument in infectious disease modeling and control of vibration in civil engineering structures under random loadings. In this paper, a new solution representation and optimal control of second-order Hilfer fractional stochastic integro-differential systems (HFSIDSs) with non-instantaneous impulsive (NI) are studied. Existence and uniqueness of solutions are proved in the finite-dimensional space by using Schaefer’s type fixed-point theorem with low conservative conditions on nonlinear part. Further, Lagrange problem is considered to establish optimal control results for HFSIDSs with NI. Finally, a pharmacotherapy type Hilfer fractional model is discussed in the example section.

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References

  1. Agarwal R, Hristova S, O’Regan D (2017) Non-instantaneous impulses in differential equations, non-instantaneous impulses in differential equations. Springer, Cham, pp 1–72

    Google Scholar 

  2. Hernandez E, O’Regan D (2013) On a new class of abstract impulsive differential equations. Proc Am Math Soc 141:1641–1649

    Article  MathSciNet  Google Scholar 

  3. Feckan M, Wang J, Zhou Y (2014) Periodic solutions for nonlinear evolution equations with non-instantaneous impulses. Nonauton Dyn Syst 1:93–101

    MathSciNet  Google Scholar 

  4. Gautam GR, Dabas J (2015) Mild solutions for class of neutral fractional functional differential equations with not instantaneous impulses. Appl Math Comput 259:480–489

    MathSciNet  Google Scholar 

  5. Lakshmikantham V (1995) Theory of integro-differential equations, vol 1. CRC Press, Boca Raton

    Google Scholar 

  6. Mahmudov NI (2003) Approximate controllability of semilinear deterministic and stochastic evolutions equations in abstract space. SIAM J Control Optim 42:1604–1622

    Article  MathSciNet  Google Scholar 

  7. Sathiyaraj T, Wang JR, Balasubramaniam P (2019) Ulam’s stability of Hilfer fractional stochastic differential systems. Eur Phys J Plus 134:1–14

    Article  Google Scholar 

  8. Sathiyaraj T, Balasubramaniam P (2019) Controllability of Hilfer fractional stochastic system with multiple delays and Poisson jumps. Eur Phys J Special Top 228:245–260

    Article  Google Scholar 

  9. Hilfer R (2000) Applications of fractional calculus in physics. World Scientific, Singapore

    Book  Google Scholar 

  10. Hilfer R (2002) Experimental evidence for fractional time evolution in glass forming materials. J Chem Phys 284:399–408

    Google Scholar 

  11. Miller KS, Ross B (1993) An introduction to the fractional calculus and differential equations. Wiley, New York

    Google Scholar 

  12. Podlubny I (1999) Fractional differential equations. Academic Press, New York

    Google Scholar 

  13. Furati KM, Kassim ND, Tatar NE (2012) Existence and uniqueness for a problem involving Hilfer fractional derivative. Comput Math Appl 64:1616–1626

    Article  MathSciNet  Google Scholar 

  14. Gu H, Trujillo JJ (2015) Existence of mild solution for evolution equation with Hilfer fractional derivative. Appl Math Comput 257:344–354

    MathSciNet  Google Scholar 

  15. Rafal K (2016) A new representation formula for the Hilfer fractional derivative and its application. J Comput Appl Math 308:39–45

    Article  MathSciNet  Google Scholar 

  16. Asawasamrit S, Kijjathanakorn A, Ntouyas SK, Tariboon J (2018) Nonlocal boundary value problems for Hilfer fractional differential equations. Bull Korean Math Soc 55:1639–1657

    MathSciNet  Google Scholar 

  17. Wang J, Zhang Y (2015) Nonlocal initial value problems for differential equations with Hilfer fractional derivative. Appl Math Comput 266:850–859

    MathSciNet  Google Scholar 

  18. Wongcharoen A, Ntouyas SK, Tariboon J (2020) Nonlocal boundary value problems for Hilfer type pantograph fractional differential equations and inclusions. Adv Differ Equ 2020:279

    Article  MathSciNet  Google Scholar 

  19. Abdelhedi W (2021) Fractional differential equations with a \(\Psi \)-Hilfer fractional derivative. Comput Appl Math 40:1–19

    Article  MathSciNet  Google Scholar 

  20. Wongcharoen A, Ntouyas SK, Tariboon J (2020) On coupled systems for Hilfer fractional differential equations with nonlocal integral boundary conditions. J Math 2020(2875152):1–12

    Article  MathSciNet  Google Scholar 

  21. Wongcharoen A, Ntouyas SK, Jessada T (2020) Boundary value problems for Hilfer fractional differential inclusions with nonlocal integral Boundary conditions. Mathematics 8:1905

    Article  Google Scholar 

  22. Agarwal OP (2004) A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn 38:323–337

    Article  MathSciNet  Google Scholar 

  23. d’Albis H, Augeraud-V’eron E (2021) Optimal prevention and elimination of infectious diseases. J Math Econ 93:102487

    Article  MathSciNet  Google Scholar 

  24. Khajji B, Kouidere A, Elhia M, Balatif O, Rachik M (2021) Fractional optimal control problem for an age-structured model of COVID-19 transmission. Chaos Solitons Fract 143:110625

    Article  MathSciNet  Google Scholar 

  25. Sweilam NH, Al-Mekhlafi SM, Baleanu D (2019) Optimal control for a fractional tuberculosis infection model including the impact of diabetes and resistant strains. J Adv Res 17:125–137

    Article  Google Scholar 

  26. Dhayal R, Malik M, Abbas S, Debbouche A (2020) Optimal controls for second-order stochastic differential equations driven by mixed-fractional Brownian motion with impulses. Math Methods Appl Sci 43:4107–4124

    MathSciNet  Google Scholar 

  27. Dhayal R, Gómez-Aguilar JF, Fernandez-Anaya G (2022) Optimal controls for fractional stochastic differential systems driven by Rosenblatt process with impulses. Optim Control Appl Methods 43:386–401

    Article  MathSciNet  Google Scholar 

  28. Zhou Y (2014) Basic theory of fractional differential equations. World Scientific, Singapore

    Book  Google Scholar 

  29. Tomovski Z (2012) Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional operator. Nonlinear Anal Theory Methods Appl 75:3364–3384

    Article  MathSciNet  Google Scholar 

  30. Zhang S, Sun J (2016) On existence and uniqueness of random impulsive differential equations. J Syst Sci Complex 29:300–314

    Article  MathSciNet  Google Scholar 

  31. Shen J, Liu X (2006) Global existence results for impulsive differential equations. J Math Anal Appl 314:546–557

    Article  MathSciNet  Google Scholar 

  32. Deimling K (1985) Nonlinear functional analysis. Springer, Berlin

    Book  Google Scholar 

  33. Dhage BC (2004) A random version of Schaefer’s fixed point theorem with applications to functional random integral equations. Tamkang J Math 35:197–205

    Article  MathSciNet  Google Scholar 

  34. Mahto L, Abbas S (2013) Approximate controllability and optimal control of impulsive fractional semilinear delay differential equations with non-local conditions. J Abstr Differ Equ Appl 4:44–59

    MathSciNet  Google Scholar 

  35. Balder E (1987) Necessary and sufficient conditions for L1-strong–weak lower semicontinuity of integral functional. Nonlinear Anal 11:1399–1404

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work is supported by Fundamental Research Grant Scheme, Ministry of Higher Education, Malaysia, under Grant No. FRGS/1/2020/STG06/UCSI/02/1, and UCSI University Grants, Malaysia, under Grant No. REIG-IASDA-2023/045.

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Authors and Affiliations

  1. Institute of Actuarial Science and Data Analytics, UCSI University, 56000, Cheras, Kuala Lumpur, Malaysia

    T. Sathiyaraj & Seng Huat Ong

  2. Department of Mathematics, The Gandhigram Rural Institute - Deemed to be University, Gandhigram, Tamil Nadu, 624 302, India

    P. Balasubramaniam

  3. Division of Mathematics, Saveetha School of Engineering, Saveetha University, Chennai, 602 105, Tamil Nadu, India

    T. Sathiyaraj

  4. School of Physics and Electronic Science, Zunyi Normal University, Zunyi, 563006, China

    Hao Chen

Authors
  1. T. Sathiyaraj
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  2. P. Balasubramaniam
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  3. Hao Chen
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  4. Seng Huat Ong
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Contributions

Conceptualization, O.S.H.; methodology, T.S.; software, P.B.; validation, O.S.H.; formal analysis, T.S.; investigation, T.S. and P.B.; resources, O.S.H. and H.C.; data curation, T.S.; writing-original draft preparation, T.S.; writing-review and editing, O.S.H. and H.C; visualization, T.S.; supervision, P.B.; project administration, T.S.; funding acquisition, O.S.H. and T.S. All authors have read and agreed to the published version of the manuscript.

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Correspondence to T. Sathiyaraj.

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Sathiyaraj, T., Balasubramaniam, P., Chen, H. et al. Optimal control of higher-order Hilfer fractional non-instantaneous impulsive stochastic integro-differential systems. J Eng Math 146, 3 (2024). https://doi.org/10.1007/s10665-024-10358-y

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  • DOI: https://doi.org/10.1007/s10665-024-10358-y

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