Asymptotic Analysis Of Differential Equations With Unbounded Delays And Its Applications

In the dynamic evolutionary processes of neural networks,ecosystems,epidemic models,power systems and economic operation systems,the time delays are ubiquitous and many delay effects are time-varying and long-term effective(i.e.unbounded delays).Furthermore,due to the interference of external environment,the state of the systems may also be subject to random noise and impulsive effect.Therefore,the mathematical model depicted by functional differential equation with unbounded delays and even including stochastic and impulsive effects should be more practical in some cases.Thus,this paper is concerned with asymptotic analysis of several kinds of functional differential equations with unbounded delays and impulsive stochastic effects,including invariant sets and attracting sets,boundedness,the existence of periodic attractor and its existence range of long time solution,Laypunov stability,input-to-state stability(ISS)and its applied to analyze the dynamic behaviors of several kinds of neural networks.The main contents are given as follows:In the first part,we give the invariant sets and attracting sets of differential systems with unbounded delays by constructing a differential inequality with unbounded delays and the method of M-matrix.Without the assumption on boundedness of time delays or system coefficients,we analyze a class of non-autonomous neural networks described by delay differential equations,and obtain the invariant sets and attracting sets,boundedness,Laypunov stability,the existence of periodic attractor and its existence range.Furthermore,we offer an appropriate weight learning algorithm to ensure input-to-state stability,and give the state estimation and attracting set for the networks.In the second part,we study the ISS properties of a class of stochastic delayed differential systems with unbounded delays and impulsive effects,in which stochastic disturbances involve white noise and Markov chain.We obtain some criteria on inputto-state stability(ISS)for impulsive delayed stochastic systems by establishing a novel differential inequality with unbounded delays and variable inputs,and investigate the impulsive stabilization of ISS for considered systems.The results not only indicate that the ISS properties still remain under certain impulsive perturbations for some continuous stable systems,but also show that an unstable system can be successfully stabilized to be input-to-state stable by impulses even if the corresponding continuous system is unstable.Moreover,we apply our criteria to analyze the ISS properties of impulsive stochastic neural networks with Markov switching and unbounded delays.