Mean square stability for controlled hybrid neutral stochastic differential equations with in nite delay

Abstract

In this article, we investigate the existence and uniqueness of solution of controlled hybrid neutral stochastic di erential equations with in nite delay (HNSFDEswID). It is known that the time lag generated by the controller in each discrete observation must be di erent. The controlled HNSFDEswID are a ected by the variable delay induced by the control function, the in nite time delay, and the highly nonlinear coe cients of the systems itself, which makes our problem more sophisticated. Di erent from the classical Khasminskii-types conditions in the literature, we provide some su cient conditions for 1 the mean square and almost sure exponential stability for controlled HNSFDEswID by using the 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 in nite time delay, the neutral term and the new Lyapunov functions. Finally, we illustrate our results by a numerical example.