Existence of Optimal Mild Solutions and Controllability of Fractional Impulsive Stochastic Partial Integro‐Differential Equations with Infinite Delay

Abstract

In this paper, we investigate a new class of fractional impulsive stochastic partial integro-differential equations with infinite delay in Hilbert spaces. By using the stochastic analysis theory, fractional calculus, analytic -resolvent operator and the fixed point technique combined with fractional powers of closed operators, we firstly give the existence of of mild solutions and optimal mild solutions for the these equations. Next, the controllability of the controlled fractional impulsive stochastic partial integro-differential systems with not instantaneous impulses is presented. Finally, examples are also given to illustrate our results. Key Words:Fractional impulsive stochastic partial integro-differential equations, optimal mild solutions, controllability, analytic -resolvent operators, fixed point. 1 p . Then ((H), ∥ ⋅ ∥  ) is a Banach space. Now, we give knowledge on the -resolvent operator which appeared in [4]. Definition 2.1. A one-parameter family of bounded linear operators ( (t)) t≥0 on H is called an -resolvent operator for c D t x(t) = Ax(t) + ∫ t 0 R(t − s)x(s)ds, x 0 = ∈ H, x ′ (0) = 0, (II.

Total controllability results for a class of time‐varying switched dynamical systems with impulses on time scales

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“…The relevance of this model emerges from the way that impulses can be viewed as a suitable model for describing a process, which at specific moments changes its state quickly and that cannot be described by ordinary differential equations. Many authors studied the instantaneous impulsive differential equations, one can see, for example, Yan and Jia [10] and Zhou et al [11].…”

Group controllability of discrete‐time second‐order multi‐agent systems with two‐time‐scale feature

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Approximate and trajectory controllability of fractional stochastic differential equation with non‐instantaneous impulses and Poisson jumps

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