| In the past few decades, genetic regulatory networks has attracted a lot of attention for their wide application in the biological and engineering science. The study on the stability or the synchronization of the networks is essential for the understanding of living organisms at both molecular and cellular levels. Hence, it is very important to investigate the dynamical analysis problem of the genetic regulatory networksIn this thesis, the stochastic dynamical analysis problem for several classes of genetic regulatory networks are discussed. First, we investigate the stability analy-sis problem for the Markoviari jumping genetic regulatory networks with mixed time delays. Secondly, the stability analysis problem of genetic regulatory networks with random discrete time delays and distributed delays is discussed. Thirdly, we study the stability analysis problem for genetic regulatory networks with a finite set delay characterization. Fourth, the robust stability analysis problem for genetic regulatory networks with linear fractional uncertainties is investigated. Finally, we study the ex-ponential cluster synchronization problem of impulsive delayed genetic oscillators with external disturbances. The compendious frame and description of this thesis are listed as follows:(1)Stochastic stability of Markovian jumping genetic regulatory net-works with mixed time delays. Stability analysis problem is investigated for a class of Markovian jumping genetic regulatory networks with mixed time delays and stochastic perturbations. Compared with previous work, both the discrete time delay and distributed time delay are considered. Moreover, the stochastic noise is also taken into account. By utilizing a more general Lyapunov-Krasovskii functional based on the idea of"delay decomposing", we derive sufficient delay-dependent conditions ensuring the asymptotically stability of the genetic regulatory networks with mixed time delays and noise perturbations. Finally, simulation examples are exploited to illustrate the effectiveness of the developed theoretical results(2)Robust stability analysis for genetic regulatory networks with ran-dom discrete delays and distributed delays. Based on Bernoulli stochastic vari- ables, we investigate the delay-probability-distribution-dependent stability problem of stochastic genetic regulatory networks with random discrete time delays and distributed time delays, which exist, in both translation process and feedback regulation process. By introducing a new Lyapunov functional which takes into account the ranges of delays and employing some free-weighting matrices approach, some new delay-probability-distribution-dependent stability criteria are established to guarantee the genetic regu-latory networks to be stable. In addition, when estimating the upper bounds of the derivative of Lyapunov functionals. we carefully handle the additional useful terms about the distributed delays, which may lead to the less conservative results. Finally, numerical examples are given to illustrate the effectiveness of our theoretical results and less conservativeness of the proposed method.(3)St.ochastic stability of genetic regulatory networks with a finite set de-lay characterization. By introducing two indicator functions, the delay-distribution-dependent stability is derived for the stochastic genetic regulatory networks with a finite set delay characterization and interval parameter uncertainties. One important feature of the obtained results here is that the time-varying delays are assumed to be random and the sum of the occurrence probabilities of the delays are assumed to be1. By employing a new Lyapunov-Krasovskii functional dependent on auxiliary delay pa-rameters, which allow the time-varying delays to be not differentiable, less conservative mean-square stochastic stability criteria are obtained. Finally, numerical examples are given to illustrated the effectiveness and superiority of the derived results.(4)R.obust stability for genetic regulatory networks with linear fractional uncertainties. The asymptotic stability analysis problem for a class of delayed genetie regulatory networks with linear fractional uncertainties and stochastic perturbations is studied. By employing a more effective Lyapunov functional and using a lemma to estimate the derivative; of the Lyapunov functional, some new sufficient conditions for the stability problem of GRNs are derived in terms of linear matrix inequality (LMI). Finally, two numerical examples are used to demonstrate the usefulness of the main results and less conservatism of the derived conditions(5)Exponential cluster synchronization of impulsive delayed genetic os-cillators with external disturbances. The problem of the exponential cluster syn- chronization of coupled impulsive genetic oscillators with external disturbances and communication delay is investigated. Based on the Kronecker product, some new clus-ter synchronization criteria for coupled impulsive genetic oscillators with attenuation level are derived. The derived results are related to the impulsive strength, and the derived results also indicate that the maximal allowable bound of time delay is inversely proportional to the decay rate, the decay rate is proportional to the couple strength, the maximal allowable bound of time delay is proportional to attenuation level, and the attenuation level is inversely proportional to the couple strength. Moreover, the case when the feedback have different self-delay is also investigated. Finally, numerical examples are given to illustrate the effectiveness of the derived results... |
