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In this paper, we investigate the approximate controllability of Hilfer fractional neutral stochastic differential equations. Firstly, the existence and uniqueness of mild solutions for these equations are obtained by means of the Banach contraction mapping principle. Then, combining the techniques of stochastic analysis theory, fractional calculations and operator semigroup theory, a new set of sufficient conditions for approximate controllability of these equations is formulated. At last, an example is presented to illustrate the obtained results.

In this paper, the approximate controllability for a class of Hilfer fractional differential equations (FDEs) of order 1 < < 2 and type 0 ≤ ≤ 1 is considered.The existence and uniqueness of mild solutions for these equations are established by applying the Banach contraction principle. Further, we obtain a set of sufficient conditions for the approximate controllability of these equations.Finally, an example is presented to illustrate the obtained results. KEYWORDSapproximate controllability, fractional differential equations, Hilfer fractional derivative MSC CLASSIFICATION26A33; 34A08; 35R11; 93B05 k=0 (−z) k k!Γ(1− − k) is defined only when ∈ (0, 1). There are only few papers that deal with the FDEs of order 1 < < 2. By using sectorial operator and -resolvent family, Shu and Wang 16 investigated the existence of mild solutions for Caputo FDEs of order ∈ (1, 2). Li et al. 17 concerned the Cauchy problems for Riemann-Liouville FDEs of order ∈ (1, 2). Li et al. 18 researched the Caputo FDEs of order ∈ (1, 2). To the best of our knowledge, there is no results about Hilfer FDEs of order 1 < < 2 and type 0 ≤ ≤ 1.Math Meth Appl Sci. 2020;43:242-254. wileyonlinelibrary.com/journal/mma

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In this paper, we investigate a class of Hilfer fractional stochastic differential equations with nonlocal conditions. We first study the existence of mild solutions of these equations by means of stochastic analysis theory, fractional calculations, and operator semigroup theory. Further, the existence of optimal pairs for the corresponding Lagrange control systems is investigated. Finally, an example is presented to illustrate our obtained results.