Controllability of Fractional Stochastic Delay Equations
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چکیده
0 g(s, x(s), x(s− τ(s))) (t− s)1−α dω(s), t ∈ J = [0, T ], x(t) = ψ(t), t ∈ [−r, 0], (1.1) where 0 < α ≤ 1, T > 0 and A is a linear closed operator , defined on a given Hilbert space X . It is assumed that A generates an analytic semigroup S(t), t ≥ 0. The state x(.) takes its values in the Hilbert spaceX , and the control function u(.) is in L2(J, U), the Hilbert space of admissible control functions with Ua Hilbert space. B is a bounded linear operator from U into X . Let K be a separable Hilbert space, and let (Ω, F, Ft, P ) be a complete probability space furnished with a complete family of right continuous increasing sigma algebras{Ft} satisfying Ft ⊂ F for t ≥ 0. The process {ω(t), t ≥ 0} is a K-valued, Ft-adapted Brownian motion with P{ω(0) = 0} = 1, and ψ(.) is an X-valued F0-measurable random variable independent of the Brownian motion ω(.). For any Banach space Y , let L2(Ω, Y ) denote the space of strongly measurable, Y -valued, square integrable random variables equipped with the norm topology ∥ x ∥L2(Ω,Y )= {E ∥ x ∥Y } 1 2 , where E is defined as integration with respect to the probability measure P. Then L2(Ω, Y ) is also Hilbert space since Y is a Hilbert space. Let τ(.) be a continuous nonnegative function on R+ and define r = sup{τ(t) − t : t ≥ 0} < ∞. Let ψ ∈ L2([−r, 0], Xγ), the family of all continuous square integrable stochastic processes ψ(.) such that sup{E ∥ ψ ∥γ} < ∞, for −r ≤ t ≤ 0. Let I = [−r, T ] and M(I, Y ) denote the space of Ft-adapted stochastic processes defined on I, taking values in Y , having square integrable norms, that are continuous in t on I in the mean square sense. This is a Banach space with respect to the norm topology
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