Stability Of Several Stochastic Coupled Systems

In recent years,the stability of coupled systems has attracted many scholars' attention,and a lot of important results on the stability of deterministic coupled systems have been reported.However,coupled systems in the real world are always disturbed by a variety of environmental noises,and environmental noises can make the stability of a system change.From the viewpoint of control,it is of very important significance to study the stability of stochastic coupled systems.This thesis will focus on establishing several mathematical models of stochastic coupled systems and analyzing their stability by stochastic analysis techniques.The effects of environmental noises and coupling structure on the stability of the stochastic coupled systems established will be revealed.The main contents are as follows.Firstly,by introducing multiple dispersal and white noise into multi-group models,stochastic multi-group models with multi-dispersal are established.The Lyapunovtype criterion and the coefficient-type criterion that guarantee the exponential stability of the model are derived by combining graph theory and Lyapunov method.Moreover,coefficient-type criterion is applied to analyze the mean-square exponential stability of stochastic coupled oscillators.The results show that stochastic model is stable when the topology structure of digraphs satisfies some conditions and the intensity of white noises is limited in certain range.Secondly,the mathematical model of pantograph stochastic multi-group models with multi-dispersal is established.The exponential stability of the proposed model is analyzed and then the Lyapunov-type criterion and the coefficient-type criterion are both derived.Furthermore,the mean-square exponential stability for pantograph stochastic coupled oscillators is studied by applying the coefficient-type criterion.The results show that exponential stability of the proposed model has a close relationship to the intensity of white noises and the coefficient of pantograph delay as well as the topology structure of digraphs.Thirdly,by means of graph theory and Lyapunov method,input-to-state stability of stochastic multi-group models with multi-dispersal is investigated,and sufficient conditions on input-to-state stability are obtained.Furthermore,input-to-state stability of stochastic coupled oscillators is studied.The results show that input-to-state stability of the model relates to not only input-to-state stability of vertex systems but also the topology structure of digraphs when the intensity of white noises is appropriate.Fourthly,by combining Razumikhin method and graph theory,the Razumikhin-type criterion and the coefficient-type criterion on moment exponential stability and almost surely exponential stability criterion are all obtained.The numerical test is offered to verify the effectiveness of the theoretic results.The results show that the exponential stability of the system depends on the intensity of white noises and the strong connectedness of digraph.Finally,by means of Lyapunov method and M-matrix theory,stochastic stability,stochastic asymptotical stability,and global stochastic asymptotic stability of the model of stochastic neural networks with infinite delay and Markov switching are all analyzed,and sufficient conditions to the three kinds of stochastic stability are obtained.The results show that the three kinds of stochastic stability of the model relate closely to the generator matrix of Markov chain when the intensity of white noises is controlled in certain extent.