Approximate controllability of fractional stochastic evolution equations.

*(English)*Zbl 1238.93099Summary: 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) |

##### Keywords:

approximate controllability; stochastic systems; fractional differential equation; fixed point principle
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\textit{R. Sakthivel} et al., Comput. Math. Appl. 63, No. 3, 660--668 (2012; Zbl 1238.93099)

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