×

zbMATH — the first resource for mathematics

Approximate controllability of fractional stochastic evolution equations. (English) Zbl 1238.93099
Summary: A class of dynamic control systems described by nonlinear fractional stochastic differential equations in Hilbert spaces is considered. Using fixed point technique, fractional calculations, stochastic analysis technique and methods adopted directly from deterministic control problems, a new set of sufficient conditions for approximate controllability of fractional stochastic differential equations is formulated and proved. In particular, we discuss the approximate controllability of nonlinear fractional stochastic control system under the assumptions that the corresponding linear system is approximately controllable. The results in this paper are generalization and continuation of the recent results on this issue. An example is provided to show the application of our result. Finally as a remark, the compactness of semigroup is not assumed and subsequently the conditions are obtained for exact controllability result.

MSC:
93E03 Stochastic systems in control theory (general)
93B05 Controllability
34A08 Fractional ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agarwal, R.P.; Baghli, S.; Benchohra, M., Controllability for semilinear functional and neutral functional evolution equations with infinite delay in Fréchet spaces, Applied mathematics and optimization, 60, 253-274, (2009) · Zbl 1179.93041
[2] Benchohra, M.; Ouahab, A., Controllability results for functional semilinear differential inclusions in Fréchet spaces, Nonlinear analysis: theory, methods & amp; applications, 61, 405-423, (2005) · Zbl 1086.34062
[3] Abada, N.; Benchohra, M.; Hammouche, H., Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions, Journal of differential equations, 246, 3834-3863, (2009) · Zbl 1171.34052
[4] Klamka, J., Constrained controllability of semilinear systems with delays, Nonlinear dynamics, 56, 169-177, (2009) · Zbl 1170.93009
[5] Klamka, J., Schauder’s fixed-point theorem in nonlinear controllability problems, Control and cybernetics, 29, 153-165, (2000) · Zbl 1011.93001
[6] Klamka, J., Constrained exact controllability of semilinear systems, Systems and control letters, 47, 139-147, (2002) · Zbl 1003.93005
[7] Mahmudov, N.I., Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM journal on control and optimization, 42, 1604-1622, (2003) · Zbl 1084.93006
[8] Sakthivel, R.; Anandhi, E.R., Approximate controllability of impulsive differential equations with state-dependent delay, International journal of control, 83, 387-393, (2010) · Zbl 1184.93021
[9] Klamka, J., Constrained approximate controllability, IEEE transactions on automatic control, 45, 1745-1749, (2000) · Zbl 0991.93013
[10] Fu, X.; Mei, K., Approximate controllability of semilinear partial functional differential systems, Journal of dynamical and control systems, 15, 425-443, (2009) · Zbl 1203.93022
[11] Sakthivel, R.; Ren, Y.; Mahmudov, N.I., On the approximate controllability of semilinear fractional differential systems, Computers and mathematics with applications, 62, 1451-1459, (2011) · Zbl 1228.34093
[12] Mahmudov, N.I.; Denker, A., On controllability of linear stochastic systems, International journal of control, 73, 144-151, (2000) · Zbl 1031.93033
[13] Klamka, J., Stochastic controllability of linear systems with delay in control, Bulletin of the Polish Academy of sciences: technical sciences, 55, 23-29, (2007) · Zbl 1203.93190
[14] Sakthivel, R.; Ren, Y.; Mahmudov, N.I., Approximate controllability of second-order stochastic differential equations with impulsive effects, Modern physics letters B, 24, 1559-1572, (2010) · Zbl 1211.93026
[15] Sakthivel, R.; Nieto, Juan J.; Mahmudov, N.I., Approximate controllability of nonlinear deterministic and stochastic systems with unbounded delay, Taiwanese journal of mathematics, 14, 1777-1797, (2010) · Zbl 1220.93011
[16] Klamka, J., Stochastic controllability of systems with multiple delays in control, International journal of applied mathematics and computer science, 19, 39-47, (2009) · Zbl 1169.93005
[17] Klamka, J., Stochastic controllability of systems with variable delay in control, Bulletin of the Polish Academy of sciences: technical sciences, 56, 279-284, (2008)
[18] Klamka, J., Stochastic controllability of linear systems with state delays, International journal of applied mathematics and computer science, 17, 5-13, (2007) · Zbl 1133.93307
[19] Podlubny, I., Fractional differential equations, (1999), San Diego Academic Press · Zbl 0918.34010
[20] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier Science B.V. Amsterdam · Zbl 1092.45003
[21] Agarwal, R.P.; Benchohra, M.; Ahmed, B., Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions, Computers & mathematics with applications, 62, 1200-1214, (2011) · Zbl 1228.34009
[22] Agarwal, R.P.; Benchohra, M.; Slimani, B.A., Existence results for differential equations with fractional order and impulses, Memoirs on differential equations and mathematical physics, 44, 1-21, (2008) · Zbl 1178.26006
[23] Mophou, G.M.; N’Guérékata, G.M., Existence of mild solution for some fractional differential equations with nonlocal conditions, Semigroup forum, 79, 2, 322-335, (2009) · Zbl 1180.34006
[24] N’Guérékata, G.M., A Cauchy problem for some fractional abstract differential equation with nonlocal conditions, Nonlinear analysis: theory, methods & amp; applications, 70, 5, 1873-1876, (2009) · Zbl 1166.34320
[25] Zhou, Y.; Jiao, F., Existence of mild solutions for fractional neutral evolution equations, Computers & mathematics with applications, 59, 1063-1077, (2010) · Zbl 1189.34154
[26] Wang, J.R.; Zhou, Y.; Wei, W.; Xua, H., Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls, Computers & mathematics with applications, 62, 1427-1441, (2011) · Zbl 1228.45015
[27] Wang, J.R.; Zhou, Y., A class of fractional evolution equations and optimal controls, Nonlinear analysis: real world applications, 12, 262-272, (2011) · Zbl 1214.34010
[28] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge University Press Cambridge · Zbl 0761.60052
[29] Sobczyk, K., Stochastic differential equations with applications to physics and engineering, (1991), Kluwer Academic Publishers London · Zbl 0762.60050
[30] El-Borai, M.M., Some probability densities and fundamental solutions of fractional evolution equations, Chaos, solitons and fractals, 14, 433-440, (2002) · Zbl 1005.34051
[31] Debbouche, A.; El-Borai, M.M., Weak almost periodic and optimal mild solutions of fractional evolution equations, Electronic journal of differential equations, 46, 1-8, (2009) · Zbl 1171.34331
[32] Dauer, J.P.; Mahmudov, N.I., Controllability of stochastic semilinear functional differential systems in Hilbert spaces, Journal of mathematical analysis and applications, 290, 373-394, (2004) · Zbl 1038.60056
[33] Mahmudov, N.I., Controllability of linear stochastic systems in Hilbert spaces, Journal of mathematical analysis and applications, 259, 64-82, (2001) · Zbl 1031.93032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.