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Cao, Nan; Fu, Xianlong

Controllability of semilinear neutral stochastic integrodifferential evolution systems with fractional Brownian motion. (English) Zbl 1525.45008

J. Integral Equations Appl. 34, No. 4, 409-432 (2022).
Consider the following semilinear neutral integro-differential equations with state-dependent delay: \[ d \bigg[ x(t) + \int_0^t N(t-s) x(s) ds \bigg] = \bigg[ - A x(t) + \int_0^t \gamma(t-s) x(s) ds + f(t,x_t) + B u(t) \bigg] dt \] \[ + g(t,x_t) dW(t) + h(t,x_t) d B^H(t), \quad t \in [0,T]. \] Here \(-A\) is the generator of an analytic semigroup. Moreover, \(W\) is a \(Q\)-Wiener process and \(B^H\) is a fractional Brownian motion with Hurst parameter \(H \in (\frac{1}{2},1)\).
The authors show that, under suitable conditions, the system admits a unique mild solution (see Theorem 3.4). Furthermore, it is shown that under appropriate conditions the system is approximately controllable on \([0, T ]\) (see Theorem 3.8). These results are proven by using the theory of analytic resolvent operators and the Banach fixed point theorem. An example is presented as well.
Reviewer: Stefan Tappe (Freiburg)

Cited in 1 Document

MSC:

45J05 Integro-ordinary differential equations
45R05 Random integral equations
47G20 Integro-differential operators
47N20 Applications of operator theory to differential and integral equations
60G22 Fractional processes, including fractional Brownian motion
60H20 Stochastic integral equations
93B05 Controllability

Keywords:

approximate controllability; fractional Brownian motion; neutral integro-differential system; resolvent operator
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References:

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