Controllability of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with delay and Poisson jumps

Abstract

The current paper is concerned with the controllability of nonlocal secondorder impulsive neutral stochastic functional integro-differential equations with infinite delay and Poisson jumps in Hilbert spaces. Using the theory of a strongly continuous cosine family of bounded linear operators, stochastic analysis theory and with the help of the Banach fixed point theorem, we derive a new set of sufficient conditions for the controllability of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with infinite delay and Poisson jumps. Finally, an application to the stochastic nonlinear wave equation with infinite delay and Poisson jumps is given.

Approximate controllability of retarded impulsive stochastic integro-differential equations driven by fractional Brownian motion

“…Now, define a continuous linear mapping B from U into H as Bu = 2u 2 w 1 + ∞ n=2 u n w n for u = ∞ n=2 u n w n ∈ U. We assume that the following conditions hold: ]dsdt + f (t, y(t − r 2 (t)))dt +σ(t)dB H (t), t ∈ [0, T], t t k , ∆y(t k ) = y(t + k ) − y(t k ) = I k (y(t k )), k = 1, ..., m, y(t) = ϕ(t), −τ ≤ t ≤ 0, (19) Moreover, if b is bounded and C 1 such that b is bounded and uniformly continuous, then (H.2) is satisfied, hence equation 18has a resolvent operator (R(t)) t≥0 on H. Besides, the continuity ofF andĜ and assumption (ii) it ensues that f and are continuous. In accordance with assumption (vi) we obtain…”

“…Controllability generally means that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls [39]. The controllability of nonlinear stochastic systems in infinite dimensional spaces has recently received a lot of attentions (see [2,3,8,9,19,24,25,31,39] and the references therein). Moreover, the approximate controllability means that the system can be steered to arbitrary small neighborhood of final state.…”

“…The impulsive neutral integro-differential equations with delay properties has been used for modeling the evolution of physical systems, in which the response of the system depends not only on the current state, but also on the past history of the system. Several authors have investigated the impulsive neutral integrodifferential equations with delay (see [19,20,28,35,43]) and references therein.…”

Approximate controllability for time-dependent impulsive neutral stochastic partial differential equations with memory

“…In [10], Huan derived a set of sufficient conditions for the controllability results of nonlocal second-order impulsive neutral stochastic integro-differential equations with infinite delay in Hilbert spaces by means of the Banach fixed point theorem combined with theories of a strongly continuous cosine families of bounded linear operators. Recently, Huan and Gao [11] have extended the results of the paper [10] for a class of nonlocal second-order impulsive neutral stochastic integro-differential equations with infnite delay and Poisson jumps. For more detail on the well-posedness and controllability of stochastic systems with impulsive effect, we refer the reader to [1,2,5,12,23] and the references therein.…”

APPROXIMATE CONTROLLABILITY OF DAMPED SECOND-ORDER IMPULSIVE NEUTRAL STOCHASTIC INTEGRO-DIFFERENTIAL SYSTEM WITH STATE-DEPENDENT DELAY Vol. 5(2) July 2017, No. 21, pp. 221-228 DIEM DANG HUAN

“…Secondorder equations have been examined in [44]. The deterministic version for the existence and the controllability of second-order differential equations have been thoroughly studied by several authors (see [1,2,3,6,10,13,16,42,43,44] and the references therein) while the controllability for stochastic version are not yet sufficiently investigated, and there are only few works on it [8,23,30,31,32,33,36,35,38].…”