Aperiodically delay intermittent control for Levy noise driven highly nonlinear stochastic hybrid systems based on discrete-time observations

To tackle the challenge of effectively stabilizing an inherently unstable highly nonlinear stochastic hybrid system while simultaneously minimizing costs, this paper introduces a novel control strategy termed aperiodically delayed intermittent control (ADIC), grounded in discrete-time observations. Diverging from the conventional stochastic hybrid system driven by Brownian motion, we adopt Lévy noise to characterize the random perturbations, which possess the capacity to capture the discontinuous, heavy tail, and peak pulse features associated with random jumps. Leveraging the Lyapunov functional approach and the strong ergodicity theory of Markov chains, we establish the mean-square exponential stability of a controlled highly nonlinear stochastic hybrid system subjected to Lévy noise. Furthermore, the range of the average time rate for ADIC, and the range of duration separating two consecutive observations are given. One particular note is our introduction of a novel technique for estimating the difference between the current state (or mode) and the discrete-time state (or mode). This innovation leads to less stringent conditions imposed on both the underlying system and the control function. Finally, we apply our theoretical findings to the modified FitzHugh-Nagumo models and the stochastic Gilpin-Ayala model.