On the Controllability of Nonlocal Second‐Order Impulsive Neutral Stochastic Integro‐Differential Equations with Infinite Delay

Abstract

In this paper, we investigate the controllability for a class of nonlocal second-order impulsive neutral stochastic integro-differential equations with infinite delay in Hilbert spaces. More precisely, a set of sufficient conditions for the controllability results of nonlocal second-order impulsive neutral stochastic integro-differential equations with infinite delay are derived by means of the Banach fixed point theorem combined with theories of a strongly continuous cosine family of bounded linear operators. As an application, an example is provided to illustrate the obtained theory.

The Controllability of Fractional Damped Stochastic Integrodifferential Systems

“…Remark III.8. Several authors in [14,16,19,20,26,29] have studied the existence of solutions and extended the theory of controllability results for integer order systems. Compared to the above, in this paper, we investigate the complete controllability of fractional nonlinear stochastic integrodifferential systems.…”

Optimal controls for second‐order stochastic differential equations driven by mixed‐fractional Brownian motion with impulses

“…Many physical phenomena, for example, the vibration of hinged bars, the transverse motion of an extensible beam, and charge on a capacitor, can be modeled by second‐order differential equations. Recently, based on the theory of strongly continuous cosine families and fixed point approach, the existence, uniqueness, stability, and controllability results for various classes of second‐order differential equations and SDEs with impulses have been investigated extensively by many authors . The general definition of an optimal control problem requires the minimization of a criterion function of the states and control inputs of the system over a set of admissible control functions.…”

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

“…Therefore, there is a real need to discuss impulsive differential control systems with memory (delay). 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.…”

“…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.…”

“…) = A(t)v(t)dt, t ≥ s, v(s) = z ∈ H, has an associated evolution family {U(t, s)} t≥s as U(t, s)z(y) = S(t − s)e t s a(ρ,y)dρ z (y).From the above expression, it follows that U(t, s) is a compact operator and every t, s ∈ J with t > sU(t, s) ≤ e −(1+λ)(t−s) .Thus, (H1) is true.Now, we define the linear continuous mapping B fromU = u = Put x(t)(•) = v(t, •) and u(t) = µ(t, y) where µ(t, y) : J × [0, π] → [0, π] is continuous.We choose B = I the identity operator and define the operatorsG(t, v)(•) = (t, v(•)), F(t, v, v)(•) = f (t, v(•), v(•)), σ(t, v, v)(•) = σ(t, v(•), v(•)).Then, under the above conditions, we can represent the stochastic control system(10) in the abstract form(1).Assume that the linear operator LT 0 be defined by (L T 0 u)(y) = T 0 S(T − s)e t s a(ρ,y)dρ µ(s, y)ds.…”