The optimal behavior of solutions to fractional impulsive stochastic integro-differential equations and its control problems

Abstract

In this paper, we study a new class of fractional nonlinear impulsive stochastic integro-differential equations with infinite delay in separable Hilbert spaces. Firstly, by using the measure of noncompactness, stochastic analysis theory, solution operators and suitable fixed point theorems, we prove the existence of mild solutions for these systems. Secondly, the existence of optimal mild solutions is obtained. Thirdly, we establish the controllability of the controlled fractional impulsive stochastic functional integro-differential systems with not instantaneous impulses. Finally, an example is provided to show the application of our results.