| Reaction-diffusion systems are used to describe the states changes of fluid systems,neural networks and biological populations,and they play an important role in the appli-cations.The stability of reaction-diffusion systems and the synchronization of the coupled reaction-diffusion systems have attracted many concerns.In the real application,delay,stochastic disturbance and impulsive effects may enter the considered systems and study the reaction-diffusion systems with delay,stochastic disturbance and impulsive effects are necessary not only for the applications but also for the theory.This study will consider the synchronization and stability for the coupled reaction-diffusion systems,based on the theory of stability and using the Green formula,Poincar′e inequality and some techniques in the stochastic analysis.This study are mainly focused on the following topics.The first topic is to study the cluster synchronization.Using the matrix transfor-mation technique,we turn the cluster synchronization of the coupled reaction-diffusion systems into the asymptotical stability of the translated systems.In light of the Lyapunov functional method,by virtue of the Green formula and Poincar′e inequality,we provide some sufficient conditions to ensure the cluster synchronization.These sufficient con-ditions show the effect of spatial domain on the cluster synchronization.Moreover,the cluster synchronization of reaction-diffusion systems with time-varying delay is studied and sufficient conditions are obtained under some restrictions on the time-varying delay.The second topic is to study the mean square H_∞synchronization and adaptive H_∞synchronization for stochastic reaction-diffusion systems.First,we provide the sufficient conditions to ensure the H_∞performance for the synchronization error systems.Then we design an adaptive controller for the coupled stochastic reaction-diffusion systems and obtain the sufficient conditions for the H_∞synchronization.Moreover,the pinning control is studied when the number of the subsystem is larger,which is suitable control strategy for the complex coupled systems.The criterion for the mean square H_∞synchronization under the pinning control is obtained.The third topic is to study the finite-time stability for the impulsive reaction-diffusion systems.First,we adopt the impulsive expression by the average impulsive interval method.Based on this expression,we study the finite-time stability for the impulsive reaction-diffusion systems and criterion is obtained to ensure the finite-time stability.Since the expression of the impulsive interval via average impulsive interval method is not unique,and the non-uniqueness will bring out the conservative property,we adopt the bounded impulsive interval expression to study the finite-time stability of impulsive reaction-diffusion systems.Sufficient conditions are obtained to guarantee the finite-time stability of impulsive reaction-diffusion systems.The relationship between the system’s coefficients and impulsive constants on the finite-time stability is discussed in detail.The forth topic is to study the finite-time stability for the impulsive delay reaction-diffusion systems.Sufficient conditions are obtained to guarantee the finite-time stability of impulsive delay reaction-diffusion systems.These sufficient conditions show the effect of delay on the finite-time stability.At the same time,we obtain the criterion for the exponential stability of impulsive delay reaction-diffusion systems and the criterion of finite-time stability of delay reaction-diffusion systems without the impulsive effects.The fifth topic is to study the boundedness of stochastic delay differential systems.Two cases are considered:one is on a”good”impulse for the boundedness,that is,the considered system without impulse may not bounded,and we introduce an impulsive effect to make the system be bounded.The other is on the disturbed impulsive effect,that is,the considered system without impulse is bounded but the impulse is regarded as a disturbance.Under this situation,we study that the systems can be bounded with what kind of disturbed impulses.Sufficient conditions are obtained under these two cases for the p-moment unique boundedness of stochastic delay differential systems.From these sufficient conditions,we can see that,to get the boundedness,the system’s coefficients of non-impulsive systems and the impulsive coefficients,including the impulsive frequency and impulsive strength,should be balanced.When a considered system without impulse is unbounded,we should adopt a more powerful impulse with higher frequency to get the boundedness.When the considered system without impulse is bounded,the disturbed impulse should be weaker and with lower frequency to keep the boundedness. |
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