Asynchronous Switching Fuzzy Control for Stochastic Highly Nonlinear Systems With Markov Jump and Periodic Time-Varying Delay: By Dupire’s Functional Itô Formula

This paper predominantly studies how to achieve the mean-square exponential stability (MSES) of stochastic highly nonlinear fuzzy systems with Markov jump and periodic time-varying delay (SHNFSMPTD), using asynchronous switching fuzzy control. Considering the time-varying delay and the drift and diffusion terms satisfying polynomial growth conditions, we construct two different functionals that explicitly depend on time t in the monotonically increasing and decreasing intervals, respectively. Combined with Dupire’s functional Itô formula, graph theory, and the properties of periodic time-varying delay, we can show that the negative definiteness of the functionals holds in both intervals, which leads to the stability of systems. In addition, by combining the generator matrix of Markov jump and the conditional probability matrix of asynchronous switching, we provide additional MSES criteria for SHNFSMPTD under asynchronous switching fuzzy control. Ultimately, the theoretical results obtained are utilized in a class of van der Pol-Duffing oscillators (VDPDO), and the six-robot systems with center pattern generator networks, and the feasibility of the theoretical results is illustrated through numerical simulations. Note to Practitioners—Stochastic highly nonlinear fuzzy systems with Markov jump and periodic time-varying delays are widely found in many practical systems, such as communication networks, control systems, and biological rhythm systems. In this paper, we propose for the first time to deal with the exponential stability problem of such systems using Dupire’s functional Itô formula. By constructing suitable functionals and applying Dupire’s functional Itô formula, we obtain the negative definiteness of the operator in both intervals. This paper provides a new method for the construction of Lyapunov functional and offers some new ideas for the stability study of this kind of systems. In the future, we attempt to study the noise stabilization of such systems with various attacks and intermittent communication under intermittent control.