Solvability and optimal controls of neutral stochastic integro-differential equations driven by fractional Brownian motion

Abstract

In this article, the authors set up an optimal control of neutral stochastic integro-differential equations (NSIDEs) driven by fractional Brownian motion (fBm) in a Hilbert space by using Grimmer resolvent operators. Sufficient conditions for mild solutions are formulated and proved by using the Banach contraction mapping principle and stochastic analytic techniques. We have extended the problem in [Issaka et al. (2020) Results on nonlocal stochastic integro-differential equations are driven by a fractional Brownian motion. Open Mathematics, 18(1), 1097–1112] to NSIDEs driven by fBm and have used modified techniques to make them compatible with optimal controls of stochastic integro-differential systems. In addition, the optimal control of the proposed problem is presented using Balder's theorem. Such optimal control of NSIDEs with fBm is widely used in automatic control, aircraft and air traffic control, electrical networks, wavelet expansions, etc. Finally, an example illustrates the potential of the main results.