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

Abstract

The objective of this article is to investigate the optimal controls for a class of fractional stochastic dierential 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 integrodierential equations are provided to validate the applicability of the derived theoretical results.

Keywords

• Fractional calculus
• Stochastic dierential system
• Mild solution
• Optimal control
• Fractional Brownian motion
• Poisson jumps.

References

1. [1] P. Balasubramniam, P. Tamilalagan, The solvability and optimal controls for impulsive fractional stochastic integrodierential equations via resolvent operators, Journal of Optimization Theory and Applications, 174, 139-155, 2017.
2. [2] G. Da prato, J. Zabczyk, Stochastic Equations in Innite Dimensions, Cambridge University Press, Cambridge, 1992.
3. [3] S. Das, Functional Fractional Calculus, Springer-Verlag, Berlin, Heidelberg, 2011.
4. [4] A. D. Fitt, A. R. H. Goodwin, K. A. Ronaldson, W. A. Wakeham, A fractional dierential equation for a MEMS viscometer used in the oil industry, Journal of Computational and Applied Mathematics, 229, 373-381, 2009.
5. [5] W. G. Glockle, T. F. Nonnenmacher, A fractional calculus approach of self-similar protein dynamics, Biophysical Journal, 68(1), 46-53, 1995.
6. [6] H. Rudolf, Applications of fractional calculus in physics, World Scientic, 2000.
7. [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. [8] A. Anguraj, K. Ravikumar, Existence and stability results for impulsive stochastic functional integrodierential equations with Poisson jumps, Journal of Applied Nonlinear Dynamics, 8(3),407-417, 2019.

9. [9] P. Muthukumar, K. Thiagu, Existence of solutions and approximate controllability of fractional nonlocal stochastic dierential equations of order 1 < q  2, with innite delay and Poisson jumps, Dierential Equations and Dynamical Systems, 26(1-3), 15-36, 2018
10. [10] F. A. Rihan, C. Rajivganthi, P. Muthukumar, Fractional stochastic dierential equations with Hilfer fractional derivative: Poisson jumps and optimal control, Discrete Dynamics in Nature and Society, 1-11, 2017.
11. [11] F. Biagini, Y. Hu, B. Oksendal, T. Zhang, Stochastic calculus for fractional Brownian motion and applications, Springer Science & Business Media, 2008.
12. [12] B. Maslowski, B. Schmalfuss, Random dynamical systems and stationary solutions of dierential equations driven by the fractional Brownian motion, Stochastic analysis and applications, 22(6),1577-1607, 2004.
13. [13] J. Han, L. Yan, Controllability of a stochastic functional dierential equation driven by a fractional Brownian motion, Advances in Dierence Equations, 104(1), 2018.
14. [14] P. Tamilalagan, P. Balasubramaniam, Approximate controllability of fractional stochastic dierential equations driven by mixed fractional Brownian motion via resolvent operator, International Journal of Control, 90(8), 1713-1727, 2017.
15. [15] C. A. Tudor, Analysis of the Rosenblatt process. ESAIM: Probability and Statistics, 12, 230-257, 2018.
16. [16] M. Maejima, C. A. Tudor, On the distribution of the Rosenblatt process, Statistics & Probability Letters, 83(6), 1490-1495, 2013.
17. [17] G. J. Shen, Y. Ren, Neutral stochastic partial dierential equations with delay driven by Rosenblatt process in a Hilbert space, Journal of the Korean Statistical Society, 44(1), 123-133, 2015.
18. [18] R. Sakthivel, P. Revathi, Y. Ren, G. Shen, Retared stochastic dierential equations with innite delay driven by Rosenblatt process, Stochastic Analysis and Applications, 36(2), 304-323, 2018.
19. [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. [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. [21] H. Aicha, J. J. Nieto, D. Amar, Solvability and optimal controls of impulsive Hilfer fractional delay evolution inclusions with Clarke subdierential, Journal of Computational and Applied Mathematics, 344, 725-737,2018.
22. [22] P. Tamilalagan, P. Balasubramniam, The solvability and optimal controls for fractional stochastic dierential equations driven by Poisson jumps via resolvent operators, Applied mathematics and Optimization, 77(3), 443-462, 2018.
23. [23] O. P. Agrawal, A general formulation and solution scheme for fractional optimal control probalems, Nonlinear Dynamics, 38(1-4), 323-337, 2004.
24. [24] X. B. Shu, Y. Lai, Y. Chen, The existence of mild solutions for impulsive fractional partial dierential equations, Nonlinear Analysis: Theory, Methods & Applications, 74(5), 2003-2011, 2011.
25. [25] E. Balder, Necessary and sucient conditions for L1-strong-weak lower semicontinuity of integral functional, Nonlinear Anal. TMA 11, 1399-1404, 1987.