Almost Surely Exponential Convergence Analysis of Time Delayed Uncertain Cellular Neural Networks Driven by Liu Process via Lyapunov–Krasovskii Functional Approach

Abstract

As with probability theory, uncertainty theory has been developed, in recent years, to portray indeterminacy phenomena in various application scenarios. We are concerned, in this paper, with the convergence property of state trajectories to equilibrium states (or fixed points) of time delayed uncertain cellular neural networks driven by the Liu process. By applying the classical Banach’s fixed-point theorem, we prove, under certain conditions, that the delayed uncertain cellular neural networks, concerned in this paper, have unique equilibrium states (or fixed points). By carefully designing a certain Lyapunov–Krasovskii functional, we provide a convergence criterion, for state trajectories of our concerned uncertain cellular neural networks, based on our developed Lyapunov–Krasovskii functional. We demonstrate under our proposed convergence criterion that the existing equilibrium states (or fixed points) are exponentially stable almost surely, or equivalently that state trajectories converge exponentially to equilibrium states (or fixed points) almost surely. We also provide an example to illustrate graphically and numerically that our theoretical results are all valid. There seem to be rare results concerning the stability of equilibrium states (or fixed points) of neural networks driven by uncertain processes, and our study in this paper would provide some new research clues in this direction. The conservatism of the main criterion obtained in this paper is reduced by introducing quite general positive definite matrices in our designed Lyapunov–Krasovskii functional.

Bifurcation Analysis of Time-Delayed Non-Commensurate Caputo Fractional Bi-Directional Associative Memory Neural Networks Composed of Three Neurons

“…As is the case with research on the stability problem, a huge number of results concerning the synchronization problem for BAMNNs have also been reported publicly in recent years (see [11][12][13]). Stability and synchronizability are closely related notions, and they both guarantee that NNs display outstanding application performance (see, for instance, references [3,10,[14][15][16]).…”

“…In order to attenuate or even remove such troubles, experts have developed several methods and have reported a great deal of interesting and inspirational results based on these methods. The interested readers can consult the studies of [6,8,10,11,13,14,16,[19][20][21] for more detailed explanations of the more meaningful results in this respect. Let us point out that, in these references, the method of Lyapunov-Krasovskii functionals has been widely utilized and quite impressive advances have been established via this approach.…”