Skip to main content
SpringerLink
Log in

An Investigation on Existence and Optimal Feedback Control for Fractional Neutral Stochastic Evolution Hemivariational Inequalities

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

This paper addresses some interesting results of optimal feedback control for neutral stochastic fractional systems in Hilbert spaces. We study the primary results by utilizing the theoretical concepts related to fractional calculus, fixed point theory of multivalued maps, and properties of generalized Clarke’s subdifferential type. Initially, we prove the existence of a mild solution by applying a fixed point technique. Then, we calculate the existence of feasible pair by employing the Filippov theorem and the Cesari property. Also, optimal feedback control results is developed under sufficient conditions. In addition, involving nonlocal conditions for the given control systems is discussed in a separate section. Further, we extend the given control systems with Sobolev-type and the existence of mild solution for the obtained system is evaluated by using the same fixed point theorem. In the end, a simple example is given for the effectiveness of the discussion.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Abbas, S.: Pseudo almost automorphic solutions of fractional order neutral differential equation. Semigroup Forum 81(3), 393–404 (2010)

    MathSciNet  Google Scholar 

  2. Abbas, S.: Pseudo almost periodic solution of stochastic functional differential equations. Int. J. Evol. Equ. 5(4), 1–13 (2011)

    MathSciNet  Google Scholar 

  3. Ahmed, H.M., El-Owaidy, H.M., Al-Nahhas, M.A.: Neutral fractional stochastic partial differential equations with Clarke subdifferential. Appl. Anal. 100(15), 3220–3232 (2021)

    MathSciNet  Google Scholar 

  4. Aubin, J.P., Frankowska, H.: Set Valued Analysis. Berkhauser, Boston (1990)

    Google Scholar 

  5. Baishya, C., Premakumari, R.N., Samei, M.E., Naik, M.K.: Chaos control of fractional order nonlinear Bloch equation by utilizing sliding mode controller. Chaos, Solitons Fractals 174, 1–18 (2023). (113773)

    MathSciNet  Google Scholar 

  6. Balder, E.J.: Necessary and sufficient conditions for L1-strong-weak lower semicontinuity of integral functionals. Nonlinear Anal. Theory Methods Appl. 11(12), 1399–1404 (1987)

    Google Scholar 

  7. Belhadji, B., Alzabut, J., Samei, M.E., Fatima, N.: On the global behaviour of solutions for a delayed viscoelastic-type Petrovesky wave equation with p-Laplacian operator and logarithmic source. Mathematics 10(22), 1–39 (2022). (4194)

    Google Scholar 

  8. Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162, 494–505 (1991)

    MathSciNet  Google Scholar 

  9. Chen, J., Liu, Z.H., Lomovtsev, F.E., Obukhovskii, V.: Optimal feedback control for a class of second-order evolution differential inclusions with Clarke’s subdifferential. J. Nonlinear Var. Anal. 6(5), 551–565 (2022)

    Google Scholar 

  10. Chen, P., Yongxiang, L., Xuping, Z.: Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families. Discret. Contin. Dyn. Syst.-Ser. B 26(3), 1531–1547 (2021)

    MathSciNet  Google Scholar 

  11. Chikh, S.B., Amara, A., Etemad, S., Rezapour, S.: On Hyers–Ulam stability of a multi-order boundary value problems via Riemann–Liouville derivatives and integrals. Adv. Differ. Equ. 547, 1–20 (2020)

    MathSciNet  Google Scholar 

  12. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    Google Scholar 

  13. Denkowski, Z., Migorski, S., Papageorgiou, N.S.: An Introduction to non-linear analysis: Theory. Kluwer Academic Plenum Publishers, Boston, Dordrecht, London, New York (2003)

    Google Scholar 

  14. Etemad, S., Rezapour, S.: On the existence of solutions for fractional boundary value problems on the ethane graph. Adv. Differ. Equ. 276, 1–20 (2020)

    MathSciNet  Google Scholar 

  15. Franklin, G.F., Powell, J.D., Emami-Naeini, A.: Feedback Control of Dynamic Systems. Addison-Wesley, Boston (1986)

    Google Scholar 

  16. Haslinger, J., Panagiotopoulos, P.D.: Optimal control of systems governed by hemivariational inequalities. Existence and approximation results. Nonlinear Anal. Theory Methods Appl. 24(1), 105–119 (1995)

    MathSciNet  Google Scholar 

  17. Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis (Theory). Kluwer Academic Publishers, Dordrecht Boston, London (1997)

    Google Scholar 

  18. Huang, Y., Liu, Z.H., Zeng, B.: Optimal control of feedback control systems governed by hemivariational inequalities. Comput. Math. Appl. 70(8), 2125–2136 (2015)

    MathSciNet  Google Scholar 

  19. Gou, H., Li, Y.: Extremal mild solutions to Hilfer evolution equations with non-instantaneous impulses and nonlocal conditions. Fract. Calc. Appl. Anal. 26, 1145–1185 (2023)

    MathSciNet  Google Scholar 

  20. Gou, H.: Study on Sobolev-type Hilfer evolution equations with non-instantaneous impulses. Int. J. Comput. Math. 100(5), 1153–1170 (2023)

    MathSciNet  Google Scholar 

  21. Gou, H., Li, Y.: A study on approximate controllability of non-autonomous evolution system with nonlocal conditions using sequence method. Optimization 71(16), 4763–4783 (2022)

    MathSciNet  Google Scholar 

  22. Jiang, Y., Zhang, Q., Huang, N.: Fractional stochastic evolution hemivariational inequalities and optimal controls. Topol. Methods Nonlinear Anal. 55, 493–515 (2020)

    MathSciNet  Google Scholar 

  23. Johnson, M., Vijayakumar, V.: Optimal control results for Sobolev-type fractional stochastic Volterra-Fredholm integrodifferential systems of order \(\vartheta \in (1, 2)\) via Sectorial operators. Numer. Funct. Anal. Optim. 44(1), 1–22 (2023)

    MathSciNet  Google Scholar 

  24. Kamenskii, M.I., Nistri, P., Obukhovskii, V.V., Zecca, P.: Optimal feedback control for a semilinear evolution equation. J. Optim. Theory Appl. 82, 503–517 (1994)

    MathSciNet  Google Scholar 

  25. Khan, A., Khan, H., Gómez-Aguilar, J.F., Abdeljawad, T.: Existence and Hyers–Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel. Chaos, Solitons Fractals 127, 422–427 (2019)

    MathSciNet  Google Scholar 

  26. Khan, A., Syam, M.I., Zada, A., Khan, H.: Stability analysis of nonlinear fractional differential equations with Caputo and Riemann–Liouville derivatives. Eur. Phys. J. Plus 133, 1–9 (2018)

    Google Scholar 

  27. Khan, A., Khan, Z.A., Abdeljawad, T., Khan, H.: Analytical analysis of fractional-order sequential hybrid system with numerical application. Adv. Contin. Discret. Models 1, 1–19 (2022)

    MathSciNet  Google Scholar 

  28. Khan, A., Gómez-Aguilar, J.F., Khan, T.S., Khan, H.: Stability analysis and numerical solutions of fractional order HIV\(/\)AIDS model. Chaos, Solitons Fractals 122, 119–128 (2019)

    MathSciNet  Google Scholar 

  29. Khan, H., Gómez-Aguilar, J.F., Alkhazzan, A., Khan, A.: A fractional order HIV-TB coinfection model with nonsingular Mittag-Leffler Law. Math. Methods Appl. Sci. 43(6), 3786–3806 (2020)

    MathSciNet  Google Scholar 

  30. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, p. 204. Elsevier, Amsterdam (2006)

    Google Scholar 

  31. Li, X., Yong, J.: Optimal Control Theory for Infinite Dimensional Systems. Birkhauser, Boston (1995)

    Google Scholar 

  32. Li, X., Liu, Z.H., Papageorgiou, N.S.: Solvability and pullback attractor for a class of differential hemivariational inequalities with its applications. Nonlinearity 36, 1323–1348 (2023)

    MathSciNet  Google Scholar 

  33. Lu, L., Liu, Z.H., Zhao, J.: A class of delay evolution hemivariational inequalities and optimal feedback controls. Methods Nonlinear Anal. 51, 1–22 (2018)

    MathSciNet  Google Scholar 

  34. Liu, Y., Liu, Z., Papageorgiou, N.S.: Sensitivity analysis of optimal control problems driven by dynamic history-dependent variational-hemivariational inequalities. J. Differ. Equ. 342, 559–595 (2023)

    MathSciNet  Google Scholar 

  35. Liu, Y., Liu, Z.H., Peng, S., Wen, C.F.: Optimal feedback control for a class of fractional evolution equations with history-dependent operators. Fract. Calc. Appl. Anal. 25, 1108–1130 (2022)

    MathSciNet  Google Scholar 

  36. Liu, Z.H., Papageorgiou, N.S.: Double phase Dirichlet problems with unilateral constraints. J. Differ. Equ. 316(15), 249–269 (2022)

    MathSciNet  Google Scholar 

  37. Liu, Z.H., Migórski, S., Zeng, B.: Optimal feedback control and controllability for hyperbolic evolution inclusions of Clarke’s subdifferential type. Comput. Math. Appl. 74, 3183–3194 (2017)

    MathSciNet  Google Scholar 

  38. Liu, Z.H.: Existence results for Quasilinear parabolic hemivariational inequalities. J. Differ. Equ. 244, 1395–1409 (2008)

    MathSciNet  Google Scholar 

  39. Mees, A.I.: Dynamics of Feedback Systems. Wiley, New York (1981)

    Google Scholar 

  40. Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities, Models and Analysis of Contact Problems. Springer, New York (2013)

    Google Scholar 

  41. Migórski, S., Ochal, A.: Optimal control of parabolic hemivariational inequalities. J. Glob. Optim. 17, 285–300 (2000)

    MathSciNet  Google Scholar 

  42. Mahmudov, N.I., Udhayakumar, R., Vijayakumar, V.: On the approximate controllability of second-order evolution hemivariational inequalities. Results Math. 75, 1–20 (2020)

    MathSciNet  Google Scholar 

  43. Mahmudov, N.I., Murugesu, R., Ravichandran, C., Vijayakumar, V.: Approximate controllability results for fractional semilinear integro-differential inclusions in Hilbert spaces. Results Math. 71(1–2), 45–61 (2017)

    MathSciNet  Google Scholar 

  44. Panagiotopoulos, P.D.: Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer, Berlin (1993)

    Google Scholar 

  45. Panagiotopoulos, P.D.: Nonconvex superpotentials in sense of F. H. Clarke and Applications. Mech. Res. Commun. 8, 335–340 (1981)

    MathSciNet  Google Scholar 

  46. Park, J.Y., Park, S.H.: Optimal control problems for anti-periodic quasi-linear hemivariational inequalities. Optim. Control Appl. Methods 28, 275–287 (2007)

    MathSciNet  Google Scholar 

  47. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983)

    Google Scholar 

  48. Phuong, N.D., Sakar, F.M., Etemad, S., Rezapour, S.: A novel fractional structure of a multi-order quantum multi-integro-differential problem. Adv. Differ. Equ. 633, 1–23 (2020)

    MathSciNet  Google Scholar 

  49. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    Google Scholar 

  50. Prato, G.D., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  51. Sakthivel, R., Anandhi, E.R.: Approximate controllability of impulsive differential equations with state-dependent delay. Int. J. Control 83(2), 387–393 (2010)

    MathSciNet  Google Scholar 

  52. Santra, S.S., Mondal, P., Samei, M.E., Alotaibi, H., Altanji, M., Botmart, T.: Study on the oscillation of solution to second-order impulsive systems. AIMS Math. 8(9), 22237–22255 (2023)

    MathSciNet  Google Scholar 

  53. Shukla, A., Sukavanam, N., Pandey, D.N.: Controllability of semilinear stochastic control system with finite delay. IMA J. Math. Control. Inf. 35(2), 427–449 (2018)

    MathSciNet  Google Scholar 

  54. Shukla, A., Sukavanam, N., Pandey, D.N.: Complete controllability of semilinear stochastic systems with delay in both state and control. Math. Rep. 18(2), 247–259 (2016)

    MathSciNet  Google Scholar 

  55. Vijayakumar, V.: Approximate controllability for a class of second-order stochastic evolution inclusions of Clarke’s subdifferential type. Results Math. 73(42), 1–17 (2018)

    MathSciNet  Google Scholar 

  56. Wang, J., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal. Real World Appl. 12, 262–272 (2011)

    MathSciNet  Google Scholar 

  57. Yan, Z.: Sensitivity analysis for a fractional stochastic differential equation with \(S^p\)-weighted pseudo almost periodic coefficients and infinite delay. Fract. Calc. Appl. Anal. 25, 2356–2399 (2022)

    MathSciNet  Google Scholar 

  58. Yan, Z., Zhou, Y.H.: Optimization of exact controllability for fractional impulsive partial stochastic differential systems via analytic sectorial operators. Int. J. Nonlinear Sci. Numer. Simul. 22(5), 559–579 (2021)

    MathSciNet  Google Scholar 

  59. Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)

    Google Scholar 

Download references

Funding

There are no funders to report for this submission.

Author information

Authors and Affiliations

  1. Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tamil Nadu, 632 014, India

    S. Vivek & V. Vijayakumar

Authors
  1. S. Vivek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Vijayakumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

SV: Conceptualization, Methodology, Validation, Visualization, Writing—original draft. VV: Conceptualization, Formal analysis, Resources, Supervision, Writing—original draft, Writing—Review & Editing.

Corresponding author

Correspondence to V. Vijayakumar.

Ethics declarations

Conflict of interest

This work does not have any 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

Vivek, S., Vijayakumar, V. An Investigation on Existence and Optimal Feedback Control for Fractional Neutral Stochastic Evolution Hemivariational Inequalities. Qual. Theory Dyn. Syst. 23, 25 (2024). https://doi.org/10.1007/s12346-023-00881-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-023-00881-1

Keywords

Mathematics Subject Classification

Search

Navigation