Stabilization of highly nonlinear hybrid systems driven by Lévy noise and delay feedback control based on discrete-time state observations.

• Our paper shows how to stabilize an unstable nonlinear delay hybrid SDEs with Lévy noise. • We design a feedback control in both drift and diffusion parts based on discrete-time observations. • We prove the existence and boundedness of the unique global solution under a new set of suitable conditions. • We discuss several kinds of stability for the controlled system using Lyapunov functions. • Our findings improve and generalize some works done in the stabilization theory. There are many hybrid stochastic differential equations (SDEs) in the real-world that don't satisfy the linear growth condition (namely, SDEs are highly nonlinear), but they have highly nonlinear characteristics. Based on some existing results, the main difficulties here are to deal with those equations if they are driven by Lévy noise and delay terms, then to investigate their stability in this case. The present paper aims to show how to stabilize a given unstable nonlinear hybrid SDEs with Lévy noise by designing delay feedback controls in the both drift and diffusion parts of the given SDEs. The controllers are based on discrete-time state observations which are more realistic and make the cost less in practice. By using the Lyapunov functional method under a set of appropriate assumptions, stability results of the controlled hybrid SDEs are discussed in the sense of p th moment asymptotic stability and exponential stability. As an application, an illustrative example is provided to show the feasibility of our theorem. The results obtained in this paper can be considered as an extension of some conclusions in the stabilization theory. [ABSTRACT FROM AUTHOR]