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Research Article

Discussion on optimal feedback control for stochastic fractional differential system by hemivariational inequalities

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Received 27 Dec 2023, Accepted 26 Apr 2024, Published online: 07 May 2024

References

  • Arjunan, M. M. (2021). On fractional neutral Volterra–Fredholm integro-differential systems with non-dense domain and non-instantaneous impulses. Malaya Journal of Matematik, 9(1), 212–216. https://doi.org/10.26637/MJM
  • Aubin, J. P., & Frankowska, H. (1990). Set valued analysis. Berkhauser.
  • Balachandran, K., & Park, J. Y. (2009). Controllability of fractional integrodifferential systems in Banach spaces. Nonlinear Analysis: Hybrid Systems, 3(4), 363–367.
  • Balasubramaniam, P., & Tamilalagan, P. (2015). Approximate controllability of a class of fractional neutral stochastic integro-differential inclusions with infinite delay by using Mainardi's function. Applied Mathematics and Computation, 256, 232–246. https://doi.org/10.1016/j.amc.2015.01.035
  • Balder, E. J. (1987). Necessary and sufficient conditions for L1-strong-weak lower semicontinuity of integral functionals. Nonlinear Analysis: Theory, Methods and Applications, 11(12), 1399–1404. https://doi.org/10.1016/0362-546X(87)90092-7
  • Benchohra, M., Henderson, J., Ntouyas, S. K., & Ouahab, A. (2008). Existence results for fractional order functional differential equations with infinite delay. Journal of Mathematical Analysis and Applications, 338(2), 1340–1350. https://doi.org/10.1016/j.jmaa.2007.06.021
  • Clarke, F. H. (1983). Optimization and nonsmooth analysis. Wiley.
  • Denkowski, Z., Migorski, S., & Papageorgiou, N. S. (2003). An introduction to non-linear analysis: Theory. Kluwer Academic Plenum Publishers.
  • Dineshkumar, C., Vijayakumar, V., Udhayakumar, R., Shukla, A., & K. S. Nisar (2023). Controllability discussion for fractional stochastic Volterra–Fredholm integro-differential systems of order 1<r<2. International Journal of Nonlinear Sciences and Numerical Simulation, 24(5), 1947–1979. https://doi.org/10.1515/ijnsns-2021-0479
  • Franklin, G. F., Powell, J. D., & Emami-Naeini, A. (1986). Feedback control of dynamic systems. Addison-Wesley.
  • Guendouzi, T., & Bousmaha, L. (2014). Approximate controllability of fractional neutral stochastic functional integro-differential inclusions with infinite delay. Qualitative Theory of Dynamical Systems, 13(1), 89–119. https://doi.org/10.1007/s12346-014-0107-y
  • Guendouzi, T., & Hamada, I. (2013). Relative controllability of fractional stochastic dynamical systems with multiple delays in control. Malaya Journal of Matematik, 1(1), 86–97. https://doi.org/10.26637/mjm101/009
  • Haslinger, J., & Panagiotopoulos, P. D. (1995). Optimal control of systems governed by hemivariational inequalities. Existence and approximation results. Nonlinear Analysis: Theory, Methods and Applications, 24(1), 105–119. https://doi.org/10.1016/0362-546X(93)E0022-U
  • Hu, S., & Papageorgiou, N. S. (1997). Handbook of multivalued analysis (theory). Kluwer Academic Publishers.
  • Huang, Y., Liu, Z., & Zeng, B. (2015). Optimal control of feedback control systems governed by hemivariational inequalities. Computers and Mathematics with Applications, 70(8), 2125–2136. https://doi.org/10.1016/j.camwa.2015.08.029
  • Johnson, M., & Vijayakumar, V. (2023a). An analysis on the optimal control for fractional stochastic delay integrodifferential systems of order 1<γ<2. Fractal and Fractional, 7(4), 1–22. https://doi.org/10.3390/fractalfract7040284
  • Johnson, M., & Vijayakumar, V. (2023b). Optimal control results for Sobolev-type fractional stochastic Volterra–Fredholm integrodifferential systems of order v∈(1,2) via sectorial operators. Numerical Functional Analysis and Optimization, 44(6), 439–460. https://doi.org/10.1080/01630563.2023.2180645
  • Johnson, M., & Vijayakumar, V. (2024). An analysis on the optimal control results for second-order Sobolev-type delay differential inclusions of Clarke's subdifferential type. Communications in Nonlinear Science and Numerical Simulation, 128, Article 107649. https://doi.org/10.1016/j.cnsns.2023.107649
  • Kamenskii, M. I., Nistri, P., Obukhovskii, V. V., & Zecca, P. (1994). Optimal feedback control for a semilinear evolution equation. Journal of Optimization Theory and Applications, 82(3), 503–517. https://doi.org/10.1007/BF02192215
  • Kavitha, K., & Vijayakumar, V. (2022). An analysis regarding to approximate controllability for Hilfer fractional neutral evolution hemivariational inequality. Qualitative Theory of Dynamical Systems, 21(3), 1–22. https://doi.org/10.1007/s12346-022-00611-z
  • Kavitha, K., & Vijayakumar, V. (2023). Optimal control for Hilfer fractional neutral integrodifferential evolution equations with infinite delay. Optimal Control Applications and Methods, 44(1), 130–147. https://doi.org/10.1002/oca.v44.1
  • Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and applications of fractional differential equations. In North-Holland mathematics studies (Vol. 204). Elsevier Science B. V.
  • Kumar, A., & Pandey, D. N. (2019). Approximate controllability of multi-term time-fractional stochastic differential inclusions with nonlocal conditions. Malaya Journal of Matematik, 7(4), 687–699. https://doi.org/10.26637/MJM
  • Li, X., Liu, Z. H., & Papageorgiou, N. S. (2023). Solvability and pullback attractor for a class of differential hemivariational inequalities with its applications. Nonlinearity, 36(2), 1323–1348. https://doi.org/10.1088/1361-6544/acb191
  • Li, X., & Yong, J. (1995). Optimal control theory for infinite dimensional systems. Birkhauser.
  • Liu, Y., Liu, Z. H., & Papageorgiou, N. S. (2023). Sensitivity analysis of optimal control problems driven by dynamic history-dependent variational-hemivariational inequalities. Journal of Differential Equations, 342, 559–595. https://doi.org/10.1016/j.jde.2022.10.009
  • Liu, Y., Liu, Z. H., Peng, S., & Wen, C. (2022). Optimal feedback control for a class of fractional evolution equations with history-dependent operators. Fractional Calculus and Applied Analysis, 25(3), 1108–1130. https://doi.org/10.1007/s13540-022-00054-y
  • Liu, Z. H. (2008). Existence results for quasilinear parabolic hemivariational inequalities. Journal of Differential Equations, 244(6), 1395–1409. https://doi.org/10.1016/j.jde.2007.09.001
  • Liu, Z. H., Migórski, S., & Zeng, B. (2017). Optimal feedback control and controllability for hyperbolic evolution inclusions of Clarke's subdifferential type. Computers and Mathematics with Applications, 74(12), 3183–3194. https://doi.org/10.1016/j.camwa.2017.08.024
  • Liu, Z. H., & Papageorgiou, N. S. (2022). Double phase Dirichlet problems with unilateral constraints. Journal of Differential Equations, 316(15), 249–269. https://doi.org/10.1016/j.jde.2022.01.040
  • Lu, L., Liu, Z. H., & Zhao, J. (2018). A class of delay evolution hemivariational inequalities and optimal feedback controls. Topological Methods in Nonlinear Analysis, 51, 1–22. https://doi.org/10.12775/TMNA.2017.061
  • Ma, Y. K., Vijayakumar, V., Shukla, A., Nisar, K. S., Thilagavathi, K., Nashine, H. K., Singh, A. K., & Zakarya, M. (2023). Discussion on the existence of mild solution for fractional derivative by Mittag–Leffler kernel to fractional stochastic neutral differential inclusions. Alexandria Engineering Journal, 63, 271–282. https://doi.org/10.1016/j.aej.2022.08.006
  • Mees, A. I. (1981). Dynamics of feedback systems. Wiley.
  • Migórski, S., Ochal, A., & Sofonea, M. (2013). Nonlinear inclusions and hemivariational inequalities, models and analysis of contact problems. In Advances in mechanics and mathematics. Springer.
  • Moustapha, D., Diop, M. A., & Ezzinbi, K. (2017). Optimal feedback control law for some stochastic integrodifferential equations on Hilbert spaces. European Journal of Control, 37, 54–62. https://doi.org/10.1016/j.ejcon.2017.05.006
  • Panagiotopoulos, P. D. (1981). Nonconvex superpotentials in sense of F. H. Clarke and applications. Mechanics Research Communications, 8(6), 335–340. https://doi.org/10.1016/0093-6413(81)90064-1
  • Panagiotopoulos, P. D. (1993). Hemivariational inequalities: Applications in mechanics and engineering. Springer.
  • Park, J. Y., & Park, S. H. (2007). Optimal control problems for anti-periodic quasi-linear hemivariational inequalities. Optimal Control Applications and Methods, 28(4), 275–287. https://doi.org/10.1002/oca.v28:4
  • Pazy, A. (1983). Semigroups of linear operators and applications to partial differential equations. Springer-Verlag.
  • Podlubny, I. (1999). Fractional differential equations. Academic Press.
  • Pradeesh, J., & Vijayakumar, V. (2024). A new approach on the approximate controllability results for Hilfer fractional stochastic hemivariational inequalities of order 1<μ<2. Qualitative Theory of Dynamical Systems, 23(4), 1–37.
  • Prato, G. D., & Zabczyk, J. (1992). Stochastic equations in infinite dimensions. Cambridge University Press.
  • Raja, M. M., Shukla, A., Nieto, J. J., Vijayakumar, V., & Nisar, K. S. (2022). A note on the existence and controllability results for fractional integrodifferential inclusions of order r∈(1,2] with impulses. Qualitative Theory of Dynamical Systems, 21(150), 1–38.
  • Raja, M. M., Vijayakumar, V., Shukla, A., Nisar, K. S., Sakthivel, N., & Kaliraj, K. (2022). Optimal control and approximate controllability for fractional integrodifferential evolution equations with infinite delay of order r∈(1,2). Optimal Control Applications and Methods, 43(4), 996–1019. https://doi.org/10.1002/oca.v43.4
  • Selvarasu, S., Kalamani, P., & Arjunan, M. M. (2016). Approximate controllability of nonlocal impulsive fractional neutral stochastic integro-differential equations with state-dependent delay in Hilbert spaces. Malaya Journal of Matematik, 4(4), 571–598. https://doi.org/10.26637/mjm404/006
  • Shukla, A., Sukavanam, N., & Pandey, D. N. (2016). Complete controllability of semilinear stochastic systems with delay in both state and control. Mathematical Reports, 18(2), 247–259.
  • Shukla, A., Sukavanam, N., & Pandey, D. N. (2018). Controllability of semilinear stochastic control system with finite delay. IMA Journal of Mathematical Control and Information, 35(2), 427–449.
  • Shukla, A., Vijayakumar, V., & Nisar, K. S. (2022). A new exploration on the existence and approximate controllability for fractional semilinear impulsive control systems of order r∈(1,2). Chaos, Solitans and Fractals, 154, 1–8.
  • Sobczyk, K. (1991). Stochastic differential equations with applications to physics and engineering. Kluwer Academic Publishers.
  • Vijayakumar, V., Nisar, K. S., Chalishajar, D. N., Shukla, A., Malik, M., Alsaadi, A., & Aldosary, S. F. (2022). A note on approximate controllability of fractional semilinear integro-differential control systems via resolvent operators. Fractal and Fractional, 6(2), 1–15.
  • Vivek, S., & Vijayakumar, V. (2023a). A note concerning to optimal feedback control for Caputo fractional neutral stochastic evolution systems. Qualitative Theory of Dynamical Systems, 22(4), 1–20. https://doi.org/10.1007/s12346-023-00855-3
  • Vivek, S., & Vijayakumar, V. (2023b). An analysis on the approximate controllability of neutral functional hemivariational inequalities with impulses. Optimization, 1–24. https://doi.org/10.1080/02331934.2023.2239851
  • Vivek, S., & Vijayakumar, V. (2024). A class of time-optimal feedback control for fractional neutral evolution hemivariational inequalities with fixed delay. Optimization, 1–29. https://doi.org/10.1080/02331934.2023.2295468
  • Vivek, S., & Vijayakumar, V. (2024). New discussion on optimal feedback control for Caputo fractional neutral evolution systems governed by hemivariational inequalities. Mathematical Methods in the Applied Sciences, 47(6), 3903–3920. https://doi.org/10.1002/mma.9794
  • Wang, J., & Zhou, Y. (2011a). A class of fractional evolution equations and optimal controls. Nonlinear Analysis: Real World Applications, 12(1), 262–272.
  • Wang, J., & Zhou, Y. (2011b). Study of an approximation process of time optimal control for fractional evolution systems in Banach spaces. Advances in Difference Equations, 2011, Article 385324.
  • Williams, W. K., Vijayakumar, V., Nisar, K. S., & Shukla, A. (2023). Atangana–Baleanu semilinear fractional differential inclusions with infinite delay: Existence and approximate controllability. Journal of Computational and Nonlinear Dynamics, 18(2), 1–18. https://doi.org/10.1115/1.4056357
  • Zhao, J., Liu, Z. H., Vilches, E., Wen, C., & Yao, J. C. (2021). Optimal control of an evolution hemivariational inequality involving history-dependent operators. Communications in Nonlinear Science and Numerical Simulation, 103, Article 105992.
  • Zhou, Y. (2014). Basic theory of fractional differential equations. World Scientific.

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