# Article

MSC: 34K50, 93B05, 93E03
Full entry | PDF   (0.3 MB)
Keywords:
approximate controllability; fixed-point theorem; fractional stochastic differential system; Hilbert space, Poisson jumps
Summary:
The paper is motivated by the study of interesting models from economics and the natural sciences where the underlying randomness contains jumps. Stochastic differential equations with Poisson jumps have become very popular in modeling the phenomena arising in the field of financial mathematics, where the jump processes are widely used to describe the asset and commodity price dynamics. This paper addresses the issue of approximate controllability of impulsive fractional stochastic differential systems with infinite delay and Poisson jumps in Hilbert spaces under the assumption that the corresponding linear system is approximately controllable. The existence of mild solutions of the fractional dynamical system is proved by using the Banach contraction principle and Krasnoselskii's fixed-point theorem. More precisely, sufficient conditions for the controllability results are established by using fractional calculations, sectorial operator theory and stochastic analysis techniques. Finally, examples are provided to illustrate the applications of the main results.
References:
[1] Balasubramaniam, P., Vembarasan, V., Senthilkumar, T.: Approximate controllability of impulsive fractional integro-differential systems with nonlocal conditions in Hilbert space. Numer. Funct. Anal. Optim. 35 (2014), 177-197. DOI 10.1080/01630563.2013.811420 | MR 3175636 | Zbl 1288.34074
[2] Caputo, M.: Elasticità e dissipazione. Zanichelli Publisher, Bologna Italian (1969).
[3] Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman & Hall/CRC Financial Mathematics Series Chapman & Hall/CRC, Boca Raton (2004). MR 2042661 | Zbl 1052.91043
[4] Cui, J., Yan, L.: Existence result for fractional neutral stochastic integro-differential equations with infinite delay. J. Phys. A, Math. Theor. 44 (2011), Article ID 335201, 16 pages. MR 2822114 | Zbl 1232.34107
[5] Cui, J., Yan, L.: Successive approximation of neutral stochastic evolution equations with infinite delay and Poisson jumps. Appl. Math. Comput. 218 (2012), 6776-6784. DOI 10.1016/j.amc.2011.12.045 | MR 2880333 | Zbl 1248.34120
[6] Prato, G. Da, Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44 Cambridge University Press, Cambridge (1992). MR 1207136 | Zbl 0761.60052
[7] Dabas, J., Chauhan, A., Kumar, M.: Existence of the mild solutions for impulsive fractional equations with infinite delay. Int. J. Differ. Equ. 2011 (2011), Article ID 793023, 20 pages. MR 2843512 | Zbl 1239.34094
[8] El-Borai, M. M., El-Nadi, K. E.-S., Fouad, H. A.: On some fractional stochastic delay differential equations. Comput. Math. Appl. 59 (2010), 1165-1170. DOI 10.1016/j.camwa.2009.05.004 | MR 2579480 | Zbl 1189.60117
[9] Hasse, M.: The Functional Calculus for Sectorial Operators. Operator theory: Advances and Applications. Vol. 196 Birkhäuser, Basel (2006). MR 2244037
[10] Hausenblas, E., Marchis, I.: A numerical approximation of parabolic stochastic partial differential equations driven by a Poisson random measure. BIT 46 (2006), 773-811. DOI 10.1007/s10543-006-0099-3 | MR 2285208 | Zbl 1112.65004
[11] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204 Elsevier, Amsterdam (2006). MR 2218073 | Zbl 1092.45003
[12] Kolmanovskii, V., Myshkis, A.: Applied Theory of Functional Differential Equations. Mathematics and Its Applications. Soviet Series 85 Kluwer Academic Publishers, Dordrecht (1992). MR 1256486
[13] Lakshmikantham, V., Bainov, D. D., Simeonov, P. S.: Theory of Impulsive Differential Equations. Series in Modern Applied Mathematics 6 World Scientific, Singapore (1989). MR 1082551 | Zbl 0719.34002
[14] Liu, J., Yan, L., Cang, Y.: On a jump-type stochastic fractional partial differential equation with fractional noises. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75 (2012), 6060-6070. DOI 10.1016/j.na.2012.06.012 | MR 2956125 | Zbl 1246.35215
[15] Long, H., Hu, J., Li, Y.: Approximate controllability of stochastic PDE with infinite delays driven by Poisson jumps. IEEE International Conference on Information Science and Technology. Wuhan, Hubei, China (2012), 23-25.
[16] Mahmudov, N. I.: Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. SIAM J. Control Optim. 42 (2003), 1604-1622. DOI 10.1137/S0363012901391688 | MR 2046377 | Zbl 1084.93006
[17] Miller, K. S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication John Wiley & Sons, New York (1993). MR 1219954 | Zbl 0789.26002
[18] Muthukumar, P., Rajivganthi, C.: Approximate controllability of fractional order stochastic variational inequalities driven by Poisson jumps. Taiwanese J. Math. 18 (2014), 1721-1738. DOI 10.11650/tjm.18.2014.3885 | MR 3284028
[19] Ren, Y., Zhou, Q., Chen, L.: Existence, uniqueness and stability of mild solutions for time-dependent stochastic evolution equations with Poisson jumps and infinite delay. J. Optim. Theory Appl. 149 (2011), 315-331. DOI 10.1007/s10957-010-9792-0 | MR 2787714 | Zbl 1241.34089
[20] Sakthivel, R., Ganesh, R., Ren, Y., Anthoni, S. M.: Approximate controllability of nonlinear fractional dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 18 (2013), 3498-3508. DOI 10.1016/j.cnsns.2013.05.015 | MR 3081379
[21] Sakthivel, R., Ren, Y.: Complete controllability of stochastic evolution equations with jumps. Rep. Math. Phys. 68 (2011), 163-174. DOI 10.1016/S0034-4877(12)60003-2 | MR 2900843 | Zbl 1244.93028
[22] Sakthivel, R., Revathi, P., Ren, Y.: Existence of solutions for nonlinear fractional stochastic differential equations. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 81 (2013), 70-86. DOI 10.1016/j.na.2012.10.009 | MR 3016441 | Zbl 1261.34063
[23] Sakthivel, R., Suganya, S., Anthoni, S. M.: Approximate controllability of fractional stochastic evolution equations. Comput. Math. Appl. 63 (2012), 660-668. DOI 10.1016/j.camwa.2011.11.024 | MR 2871665 | Zbl 1238.93099
[24] Shu, X.-B., Wang, Q.: The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order $1<\alpha<2$. Comput. Math. Appl. 64 (2012), 2100-2110. DOI 10.1016/j.camwa.2012.04.006 | MR 2960829 | Zbl 1268.34155
[25] Sukavanam, N., Kumar, S.: Approximate controllability of fractional order semilinear delay systems. J. Optim. Theory Appl. 151 (2011), 373-384. DOI 10.1007/s10957-011-9905-4 | MR 2852407 | Zbl 1251.93039
[26] Tai, Z., Wang, X.: Controllability of fractional-order impulsive neutral functional infinite delay integrodifferential systems in Banach spaces. Appl. Math. Lett. 22 (2009), 1760-1765. DOI 10.1016/j.aml.2009.06.017 | MR 2560992 | Zbl 1181.34078
[27] Taniguchi, T., Luo, J.: The existence and asymptotic behaviour of mild solutions to stochastic evolution equations with infinite delays driven by Poisson jumps. Stoch. Dyn. 9 (2009), 217-229. DOI 10.1142/S0219493709002646 | MR 2531628 | Zbl 1181.60102
[28] Triggiani, R.: A note on the lack of exact controllability for mild solutions in Banach spaces. SIAM J. Control Optim. 15 (1977), 407-411. DOI 10.1137/0315028 | MR 0435991 | Zbl 0354.93014
[29] Zhao, H.: On existence and uniqueness of stochastic evolution equation with Poisson jumps. Stat. Probab. Lett. 79 (2009), 2367-2373. DOI 10.1016/j.spl.2009.08.006 | MR 2556370 | Zbl 1182.60018

Partner of