Ito Differential Formula Research Articles

Pth moment exponential stability of stochastic fuzzy Cohen–Grossberg neural networks with discrete and distributed delays

In this paper, stochastic fuzzy Cohen–Grossberg neural networks with discrete and distributed delays are investigated. By using Lyapunov function and the Ito differential formula, some sufficient conditions for the pth moment exponential stability of such stochastic fuzzy Cohen–Grossberg neural networks with discrete and distributed delays are established. An example is given to illustrate the feasibility of our main theoretical findings. Finally, the paper ends with a brief conclusion. Methodology and achieved results is to be presented.

Approaches to T–S Fuzzy-Affine-Model-Based Reliable Output Feedback Control for Nonlinear Itô Stochastic Systems

This paper deals with the problem of reliable and robust $\mathscr {H}_{\infty }$ static output feedback (SOF) controller synthesis for continuous-time nonlinear stochastic systems with actuator faults. The nonlinear stochastic plant is expressed by an Ito-type Takagi–Sugeno fuzzy-affine model with parametric uncertainties, and a Markov process is employed to model the occurrence of actuator fault. The purpose is to design an admissible piecewise SOF controller, such that the resulting closed-loop system is stochastically stable with a prescribed disturbance attenuation level in an $\mathscr {H}_{\infty }$ sense. Specifically, based on a Markovian Lyapunov function combined with Ito differential formula, S-procedure, and some matrix inequality convexification procedures, two new approaches to the reliable SOF controller analysis and synthesis are proposed for the underlying stochastic fuzzy-affine systems. It is shown that the existence of desired reliable controllers is fully characterized in terms of strict linear matrix inequalities. Finally, simulation examples are presented to illustrate the effectiveness and advantages of the developed methods.

Dynamic analysis of stochastic fuzzy Cohen-Grossberg neural networks with time-varying delays

This paper is concerned with the problem of stochastic stability for a class of fuzzy Cohen-Grossberg neural networks, in which the interconnections and delays are time-varying. Based on a Lyapunov function and the Ito differential formula, a set of novel sufficient conditions on the pth moment exponential stability of the equilibrium of the system is derived. Moreover, an illustrative example is given to demonstrate the effectiveness of the results obtained.

Adaptive neural control of stochastic nonlinear systems with unmodeled dynamics and time-varying state delays

In this paper, a novel adaptive control scheme is investigated based on the backstepping design for a class of stochastic nonlinear systems with unmodeled dynamics and time-varying state delays. The radial basis function neural networks are used to approximate the unknown nonlinear functions obtained by using Ito differential formula and Young׳s inequality. The unknown time-varying delays and the unmodeled dynamics are dealt with by constructing appropriate Lyapunov–Krasovskii functions and introducing available dynamic signal. It is proved that all signals in the closed-loop system are bounded in probability and the error signals are semi-globally uniformly ultimately bounded (SGUUB) in mean square or the sense of four-moment. Simulation results illustrate the effectiveness of the proposed design.

Stability Analysis of Stochastic Fuzzy Neural Networks with Time-Varying Delays and Reaction–Diffusion Terms

This paper investigates the problem of the global exponential stability for a class of stochastic Takagi–Sugeno fuzzy neural networks (STSFNNs) with time-varying delays and reaction–diffusion terms. Based on the piecewise Lyapunov–Krasovskii functional, Poincare integral inequality, and Ito differential formula, we obtain some sufficient conditions ensuring the global exponential stability of an equilibrium point for STSFNNs with time-varying delays and reaction–diffusion terms. These sufficient conditions depend on the reaction–diffusion terms and time delays. Finally, some examples are given to show the effectiveness and superiority of the proposed approach.

Delay-dependent non-fragile robust dissipative filtering for uncertain nonlinear stochastic singular time-delay systems with Markovian jump parameters

The problem of delay-dependent non-fragile and robust dissipative filtering is investigated for a class of uncertain nonlinear stochastic singular time-delay Ito-type systems with Markovian jump parameters. With the system uncertainty and the filter gain perturbations, the closed-loop filtering error system is robust asymptotically stable and satisfies the dissipation performance. By constructing a Lyapunov-Krasovskii function and applying the Ito differential formula to compute the differential function along the system, a sufficient condition for the existence of a robust and non-fragile dissipative filter is derived in the form of linear matrix inequality (LMI). An illustrative numerical example is provided to demonstrate the effectiveness of the proposed approach.

Path-Following for a Cutting Tool Subject to Stochastic Disturbance

We extend the path-following problem of deterministic systems to stochastic nonlinear systems in this paper. By using backstepping design method and Ito differential formula, a smooth controller consist of a dynamic control law with a filtered-gradient update law has been designed for a cutting tool subject to stochastic disturbance. The controller can guarantee all the signals of closed-loop systems bounded in probability and the output signal track the given anticipant path with the speed assignment.

Robust Stability for Interval Stochastic Neural Networks with Time-Varying Discrete and Distributed Delays

This paper investigates the problem of robust stability for a class of stochastic interval neural networks with discrete and distributed time-varying delays. The discrete delays are assumed to be varying within a given interval, while the parameter uncertainties are assumed to be bounded in some given compact sets. Based on the Ito differential formula and stochastic stability theory, delay-range-dependent criteria for stochastic interval neural networks with time-varying delays are derived to guarantee robust global asymptotic stability in mean square. In this paper, when estimating the upper bound of the derivative of Lyapunov functionals, we are concerned with better handling of terms related to the discrete and distributed delays and establishing less conservative results. These robust stability conditions are presented in terms of linear matrix inequalities and can be efficiently solved via standard numerical software. An important feature of the results proposed in this paper is that the stability conditions are dependent on the upper and lower bounds of the discrete delays. Numerical examples are given to illustrate the effectiveness of the proposed method.

New Results on Robust Exponential Stability of Uncertain Stochastic Neural Networks with Mixed Time-Varying Delays

This letter considers the robust exponential stability of uncertain stochastic neural networks with mixed time-varying delays. By using Lyapunov---Krasovskii functional and Ito's differential formula, several new sufficient conditions guaranteeing the global robust exponential stability are derived in terms of linear matrix inequalities. Numerical examples are given to illustrate the effectiveness and less conservativeness of our results.

Improved exponential stability for stochastic Markovian jump systems with nonlinearity and time‐varying delay

AbstractThis paper is concerned with delay‐dependent exponential stability for stochastic Markovian jump systems with nonlinearity and time‐varying delay. An improved exponential stability criterion for stochastic Markovian jump systems with nonlinearity and time‐varying delay is proposed without ignoring any terms by considering the relationship among the time‐varying delay, its upper bound and their difference, and using both Itô's differential formula and Lyapunov stability theory. A numerical example is given to illustrate the effectiveness and the benefits of the proposed method. Copyright © 2009 John Wiley & Sons, Ltd.

Almost sure exponential stability of stochastic reaction diffusion systems

The Lyapunov direct method, as the most effective measure of studying stability theory for ordinary differential systems and stochastic ordinary differential systems, has not been generalized to research concerning stochastic partial differential systems owing to the emptiness of the corresponding Ito differential formula. The goal of this paper is just employing the Lyapunov direct method to investigate the stability of Ito stochastic reaction diffusion systems, including asymptotical stability in probability and almost sure exponential stability. The obtained results extend the conclusions of [X.X. Liao, X.R. Mao, Exponential stability and instability of stochastic neural networks, Stochastic Analysis and Applications 14 (2) (1996) 165–185; X.X. Liao, S.Z. Yang, S.J. Cheng, Y.L. Fu, Stability of general neural networks with reaction–diffusion, Science in China (F) 44 (5) (2001) 389–395].

Stability and stabilization on generalized delay stochastic neural network with distributed parameter

Inner stability and stabilization of Cohen-Grossberg generalized delay stochastic neural network with distributed parameter are discussed. The main method adopted is, combining inequality techniques, to apply Ito differential formula to the constructed average function with respect to spatial variables along the system considered under the integral operator. Some sufficient conditions are given.

Stochastic Transient Resonance of a Plate Under Rotating Machine

AbstractAn approach for obtaining the probabilistic characteristics of the plate response due to the excitation caused by rotating machine standing on it is presented. Some method for solving the problem of transient resonance applying the Ito differential formula and a set of Ito type differential equations is proposed.