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Optimal control for fractional stochastic differential system driven by fractional Brownian motion with Poisson jumps

Year 2022, Volume: 4 Issue: 1, 1 - 14, 30.08.2022

Abstract

The objective of this article is to investigate the optimal controls for a class of fractional stochastic di erential system driven by fractional Brownian motion with Poisson jumps in Hilbert space setting. The sucient conditions for the existence of mild solution results are formulated and proved by virtue of fractional calculus, solution operator and stochastic analysis techniques. Furthermore, the existence of optimal control of the proposed problem is presented by using Balder's theorem. Finally, stochastic integrodi erential equations are provided to validate the applicability of the derived theoretical results.

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Project Number

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References

  • [1] P. Balasubramniam, P. Tamilalagan, The solvability and optimal controls for impulsive fractional stochastic integrodi erential equations via resolvent operators, Journal of Optimization Theory and Applications, 174, 139-155, 2017.
  • [2] G. Da prato, J. Zabczyk, Stochastic Equations in In nite Dimensions, Cambridge University Press, Cambridge, 1992.
  • [3] S. Das, Functional Fractional Calculus, Springer-Verlag, Berlin, Heidelberg, 2011.
  • [4] A. D. Fitt, A. R. H. Goodwin, K. A. Ronaldson, W. A. Wakeham, A fractional di erential equation for a MEMS viscometer used in the oil industry, Journal of Computational and Applied Mathematics, 229, 373-381, 2009.
  • [5] W. G. Glockle, T. F. Nonnenmacher, A fractional calculus approach of self-similar protein dynamics, Biophysical Journal, 68(1), 46-53, 1995.
  • [6] H. Rudolf, Applications of fractional calculus in physics, World Scienti c, 2000.
  • [7] J. Luo, T. Taniguchi, The existence and uniqueness for non-Lipschitz stochastic neutral delay evolution equations driven by Poisson jumps, Stochastics and Dynamics, 9(1), 135-152, 2009.
  • [8] A. Anguraj, K. Ravikumar, Existence and stability results for impulsive stochastic functional integrodi erential equations with Poisson jumps, Journal of Applied Nonlinear Dynamics, 8(3),407-417, 2019.
  • [9] P. Muthukumar, K. Thiagu, Existence of solutions and approximate controllability of fractional nonlocal stochastic di erential equations of order 1 < q  2, with in nite delay and Poisson jumps, Di erential Equations and Dynamical Systems, 26(1-3), 15-36, 2018
  • [10] F. A. Rihan, C. Rajivganthi, P. Muthukumar, Fractional stochastic di erential equations with Hilfer fractional derivative: Poisson jumps and optimal control, Discrete Dynamics in Nature and Society, 1-11, 2017.
  • [11] F. Biagini, Y. Hu, B. Oksendal, T. Zhang, Stochastic calculus for fractional Brownian motion and applications, Springer Science & Business Media, 2008.
  • [12] B. Maslowski, B. Schmalfuss, Random dynamical systems and stationary solutions of di erential equations driven by the fractional Brownian motion, Stochastic analysis and applications, 22(6),1577-1607, 2004.
  • [13] J. Han, L. Yan, Controllability of a stochastic functional di erential equation driven by a fractional Brownian motion, Advances in Di erence Equations, 104(1), 2018.
  • [14] P. Tamilalagan, P. Balasubramaniam, Approximate controllability of fractional stochastic di erential equations driven by mixed fractional Brownian motion via resolvent operator, International Journal of Control, 90(8), 1713-1727, 2017.
  • [15] C. A. Tudor, Analysis of the Rosenblatt process. ESAIM: Probability and Statistics, 12, 230-257, 2018.
  • [16] M. Maejima, C. A. Tudor, On the distribution of the Rosenblatt process, Statistics & Probability Letters, 83(6), 1490-1495, 2013.
  • [17] G. J. Shen, Y. Ren, Neutral stochastic partial di erential equations with delay driven by Rosenblatt process in a Hilbert space, Journal of the Korean Statistical Society, 44(1), 123-133, 2015.
  • [18] R. Sakthivel, P. Revathi, Y. Ren, G. Shen, Retared stochastic di erential equations with in nite delay driven by Rosenblatt process, Stochastic Analysis and Applications, 36(2), 304-323, 2018.
  • [19] L. Urszula, H. Schattler, Antiangiogenic therapy in cancer treatment as an optimal control proble, SIAM Journal on Control and Optimization, 46(3), 1052-1079, 2007.
  • [20] A. Ivan, J. J. Nieto, C. J. Silva, D. F. Torres, Ebola model and optiimal control with vaccination constraints, J Ind Manag Optim., 14(2), 427-446, 2018.
  • [21] H. Aicha, J. J. Nieto, D. Amar, Solvability and optimal controls of impulsive Hilfer fractional delay evolution inclusions with Clarke subdi erential, Journal of Computational and Applied Mathematics, 344, 725-737,2018.
  • [22] P. Tamilalagan, P. Balasubramniam, The solvability and optimal controls for fractional stochastic di erential equations driven by Poisson jumps via resolvent operators, Applied mathematics and Optimization, 77(3), 443-462, 2018.
  • [23] O. P. Agrawal, A general formulation and solution scheme for fractional optimal control probalems, Nonlinear Dynamics, 38(1-4), 323-337, 2004.
  • [24] X. B. Shu, Y. Lai, Y. Chen, The existence of mild solutions for impulsive fractional partial di erential equations, Nonlinear Analysis: Theory, Methods & Applications, 74(5), 2003-2011, 2011.
  • [25] E. Balder, Necessary and sucient conditions for L1-strong-weak lower semicontinuity of integral functional, Nonlinear Anal. TMA 11, 1399-1404, 1987.
Year 2022, Volume: 4 Issue: 1, 1 - 14, 30.08.2022

Abstract

Project Number

-

References

  • [1] P. Balasubramniam, P. Tamilalagan, The solvability and optimal controls for impulsive fractional stochastic integrodi erential equations via resolvent operators, Journal of Optimization Theory and Applications, 174, 139-155, 2017.
  • [2] G. Da prato, J. Zabczyk, Stochastic Equations in In nite Dimensions, Cambridge University Press, Cambridge, 1992.
  • [3] S. Das, Functional Fractional Calculus, Springer-Verlag, Berlin, Heidelberg, 2011.
  • [4] A. D. Fitt, A. R. H. Goodwin, K. A. Ronaldson, W. A. Wakeham, A fractional di erential equation for a MEMS viscometer used in the oil industry, Journal of Computational and Applied Mathematics, 229, 373-381, 2009.
  • [5] W. G. Glockle, T. F. Nonnenmacher, A fractional calculus approach of self-similar protein dynamics, Biophysical Journal, 68(1), 46-53, 1995.
  • [6] H. Rudolf, Applications of fractional calculus in physics, World Scienti c, 2000.
  • [7] J. Luo, T. Taniguchi, The existence and uniqueness for non-Lipschitz stochastic neutral delay evolution equations driven by Poisson jumps, Stochastics and Dynamics, 9(1), 135-152, 2009.
  • [8] A. Anguraj, K. Ravikumar, Existence and stability results for impulsive stochastic functional integrodi erential equations with Poisson jumps, Journal of Applied Nonlinear Dynamics, 8(3),407-417, 2019.
  • [9] P. Muthukumar, K. Thiagu, Existence of solutions and approximate controllability of fractional nonlocal stochastic di erential equations of order 1 < q  2, with in nite delay and Poisson jumps, Di erential Equations and Dynamical Systems, 26(1-3), 15-36, 2018
  • [10] F. A. Rihan, C. Rajivganthi, P. Muthukumar, Fractional stochastic di erential equations with Hilfer fractional derivative: Poisson jumps and optimal control, Discrete Dynamics in Nature and Society, 1-11, 2017.
  • [11] F. Biagini, Y. Hu, B. Oksendal, T. Zhang, Stochastic calculus for fractional Brownian motion and applications, Springer Science & Business Media, 2008.
  • [12] B. Maslowski, B. Schmalfuss, Random dynamical systems and stationary solutions of di erential equations driven by the fractional Brownian motion, Stochastic analysis and applications, 22(6),1577-1607, 2004.
  • [13] J. Han, L. Yan, Controllability of a stochastic functional di erential equation driven by a fractional Brownian motion, Advances in Di erence Equations, 104(1), 2018.
  • [14] P. Tamilalagan, P. Balasubramaniam, Approximate controllability of fractional stochastic di erential equations driven by mixed fractional Brownian motion via resolvent operator, International Journal of Control, 90(8), 1713-1727, 2017.
  • [15] C. A. Tudor, Analysis of the Rosenblatt process. ESAIM: Probability and Statistics, 12, 230-257, 2018.
  • [16] M. Maejima, C. A. Tudor, On the distribution of the Rosenblatt process, Statistics & Probability Letters, 83(6), 1490-1495, 2013.
  • [17] G. J. Shen, Y. Ren, Neutral stochastic partial di erential equations with delay driven by Rosenblatt process in a Hilbert space, Journal of the Korean Statistical Society, 44(1), 123-133, 2015.
  • [18] R. Sakthivel, P. Revathi, Y. Ren, G. Shen, Retared stochastic di erential equations with in nite delay driven by Rosenblatt process, Stochastic Analysis and Applications, 36(2), 304-323, 2018.
  • [19] L. Urszula, H. Schattler, Antiangiogenic therapy in cancer treatment as an optimal control proble, SIAM Journal on Control and Optimization, 46(3), 1052-1079, 2007.
  • [20] A. Ivan, J. J. Nieto, C. J. Silva, D. F. Torres, Ebola model and optiimal control with vaccination constraints, J Ind Manag Optim., 14(2), 427-446, 2018.
  • [21] H. Aicha, J. J. Nieto, D. Amar, Solvability and optimal controls of impulsive Hilfer fractional delay evolution inclusions with Clarke subdi erential, Journal of Computational and Applied Mathematics, 344, 725-737,2018.
  • [22] P. Tamilalagan, P. Balasubramniam, The solvability and optimal controls for fractional stochastic di erential equations driven by Poisson jumps via resolvent operators, Applied mathematics and Optimization, 77(3), 443-462, 2018.
  • [23] O. P. Agrawal, A general formulation and solution scheme for fractional optimal control probalems, Nonlinear Dynamics, 38(1-4), 323-337, 2004.
  • [24] X. B. Shu, Y. Lai, Y. Chen, The existence of mild solutions for impulsive fractional partial di erential equations, Nonlinear Analysis: Theory, Methods & Applications, 74(5), 2003-2011, 2011.
  • [25] E. Balder, Necessary and sucient conditions for L1-strong-weak lower semicontinuity of integral functional, Nonlinear Anal. TMA 11, 1399-1404, 1987.
There are 25 citations in total.

Details

Primary Language English
Subjects Software Engineering (Other)
Journal Section Articles
Authors

K. Ravikumar

Ramkumar Kumark

Elsayed Elsayed 0000-0003-0894-8472

Project Number -
Publication Date August 30, 2022
Acceptance Date March 14, 2022
Published in Issue Year 2022 Volume: 4 Issue: 1

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