Inequalities and pth moment exponential stability of impulsive delayed Hopfield neural networks

In this paper, the pth moment exponential stability for a class of impulsive delayed Hopfield neural networks is investigated. Some concise algebraic criteria are provided by a new method concerned with impulsive integral inequalities. Our discussion neither requires a complicated Lyapunov function nor the differentiability of the delay function. In addition, we also summarize a new result on the exponential stability of a class of impulsive integral inequalities. Finally, one example is given to illustrate the effectiveness of the obtained results.

Delayed impulive SDEs driven by multiplicative fBm noise and additive fBm noise

The stability and boundedness for delayed impulsive SDEs driven by fBm are studied in this paper. Two kinds of noises, i.e, additive fBm noise and mul-tiplicative fBm noise are both taken into consideration. By using stochastic Lyapunov technique and impulsive control theory, sufficient criteria for pth moment exponential stability and mean square ultimate boundedness are derived, for two kinds of fBm driven delayed impulsive SDEs, respectively. As application, the obtained results are used to do practical synchronization w.r.t. a class of chaotic systems, in which the response system is perturbed by additive fBm noises. Finally, A Chua chaotic oscillator is given to verify the validity and applicability of the derived results.

The pth moment exponential stability of uncertain differential equation with jumps

Under the axiom system of uncertainty theory, the paper mainly introduce the new definition of the pth moment exponential stability for uncertain differential equation with jumps. For illustrating the concept, some examples and counterexamples are given. Furthermore, we obtain a necessary and sufficient condition of stability in pth moment exponential for the linear uncertain differential equation with jumps. Also, the conclusion condition is illustrated very clearly by two examples.

Almost sure exponential stability of stochastic differential delay equations

This paper mainly studies whether the almost sure exponential stability of stochastic differential delay equations (SDDEs) is shared with that of the stochastic theta method. We show that under the global Lipschitz condition the SDDE is pth moment exponentially stable (for p 2 (0; 1)) if and only if the stochastic theta method of the SDDE is pth moment exponentially stable and pth moment exponential stability of the SDDE or the stochastic theta method implies the almost sure exponential stability of the SDDE or the stochastic theta method, respectively. We then replace the global Lipschitz condition with a finite-time convergence condition and establish the same results. Hence, our new theory enables us to consider the almost sure exponential stability of the SDDEs using the stochastic theta method, instead of the method of Lyapunov functions. That is, we can now perform careful numerical simulations using the stochastic theta method with a sufficiently small step size ?t. If the stochastic theta method is pth moment exponentially stable for a sufficiently small p ? (0,1), we can then deduce that the underlying SDDE is almost sure exponentially stable. Our new theory also enables us to show the pth moment exponential stability of the stochastic theta method to reproduce the almost sure exponential stability of the SDDEs.