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The problem of stabilizing an unstable highly nonlinear hybrid neutral stochastic functional differential equation with infinite delay (HNSFDEwID) is addressed in this article. The objective of this study is to propose a discrete‐time feedback control (DTFC) approach for stabilizing the original system. Sufficient conditions are derived for the existence and uniqueness of the system's solution, as well as its boundedness, prerequisites for studying the system's dynamic behavior. Four distinct criteria for stabilizing the controlled system are provided, including H∞$$ {H}_{\infty } $$ stabilization, asymptotic stabilization, exponential stabilization, and almost surely exponentially stabilization. Additionally, the upper limit of the time interval for discrete observations is calculated based on these criteria. Finally, the proposed theory is validated through an illustrative example. [ABSTRACT FROM AUTHOR]

This paper addresses the existence of stability results for Ulam–Hyers (UHS) and Ulam–Hyers–Rassias (UHRS) in the setting of Caputo–Hadamard fractional functional stochastic differential equations with delay (FFSDEwD). We first prove existence and uniqueness using Banach fixed point theorem coupled with standard stochastic analysis techniques. Then, we deal with UHS and UHRS results of Caputo–Hadamard FFSDEwD through an application of Gronwall inequality. Our theoretical findings are corroborated with two numerical examples. [ABSTRACT FROM AUTHOR]

In this article, we investigate the existence and uniqueness of solution of controlled hybrid neutral stochastic differential equations with infinite delay (HNSFDEswID). It is known that the time lag generated by the controller in each discrete observation must be different. The controlled HNSFDEswID are affected by the variable delay induced by the control function, the infinite time delay, and the highly nonlinear coefficients of the systems itself, which makes our problem more sophisticated. Different from the classical Khasminskii‐type conditions in the literature, we provide some sufficient conditions for the mean square and almost sure exponential stability for controlled HNSFDEswID by using the M$$ M $$‐matrix technique and a suitable Lyapunov functional. In this sense, the main contributions of our results, compared with those in the literature, are the infinite time delay, the neutral term, and the new Lyapunov functions. Finally, we illustrate our results by a numerical example. [ABSTRACT FROM AUTHOR]