The purpose of this paper is to find the time optimal control to a target set for semilinear stochastic functional differential equations involving time delays or memories under general conditions on a target set and nonlinear terms even though the equations contain unbounded principal operators. Our research approach is construct a fundamental solution for corresponding linear systems and establish variations of constant formula of solutions for given stochastic equations. The existence result of time optimal controls for one point target set governed by the given semilinear stochastic equation is also established.

In this paper, we prove some random fixed point theorems for generalized random $\alpha-\psi-$contractive mappings in a Polish space and, as some applications, we show the existence of random solutions of second order random differential equation.

Here, we propose a prescriptive simulation model of a process operator’s decision-making assisted by artificial intelligence (AI) algorithm in a technical system control loop. We analyze situations fraught with a catastrophic threat that may cause unacceptable damage. Operators’ decision-making is interpreted in terms of a subjectively admissible probability of disaster and subjectively necessary reliability of its assessment. We distinguish four extreme decision-making strategies corresponding to different ratios between the above variables. An experiment simulating a process facility, an AI algorithm and operator's decision-making strategy was held. It showed that depending on the properties of a controlled process (the speed of hazard onset) and the AI algorithm characteristics (Type I and II error rate), each of such strategies or some intermediate strategy may prove to be more beneficial than others. The same approach is applicable to the identification and analysis of sustainability of strategies applied in real-life operating conditions, as well as to the development of a computer simulator to train operators to control hazardous technological processes using AI-generated advice.