Optimal control for neutral stochastic systems with infinite time delay and deviated argument driven by Rosenblatt process

Abstract

In this article, the authors set up an optimal control for a class of neutral Stochastic Integro-Differential Equations (SIDEs) with infinite delay and deviated arguments driven by Rosenblatt process in Hilbert space. Sufficient conditions for the existence of mild solutions are formulated and proved by using fixed point theorem and stochastic analysis techniques. We have proved the axiomatic definition of new phase space for infinite time delay stochastic process. We have extended the problem in Das et al. (2014) to neutral SIDEs with infinite delay and have used modified techniques to make them compatible with integro-differential systems. In addition, the existence of optimal control of the proposed problem is presented by using Balder’s theorem. We dropped down the supplemented boundedness and range conditions of Das et al. (2014) and also the growth condition utilized in Das et al. (2017). Thus, our result extends the work of Das et al. (2014, 2017). Such delay systems with Rossenblatt switching are widely used in the fields such as automatic control, aircraft and air traffic control, electrical networks, wavelet expansions, etc. Finally, an example illustrates the potential of the main results.