Stability And Sampled-data Control For Several Classes Of Stochastic Systems

As is known to all,stochastic systems play an important role in many branches of science,economy,physics and engineering and many scholars in the world have de-voted themselves to investigating various properties of stochastic systems.Especially,stability and sampled-data control theories have been interesting and important topics since they can describe significant characteristics of stochastic systems.Although a large number of works have appeared in the literature,a lot of problems remain to be solved.Especially for the sampled-data control theory,it is a new study in stabi-lization of stochastic systems,but there are few literature about sampled-data control theory for stabilization of stochastic retarded systems since quite a few difficulties arising from time delays.Therefore,throughout this thesis,we mainly consider stabil-ity and sampled-data control theories for four classes of stochastic systems,including nonlinear hybrid stochastic heat equations,stochastic neutral functional differential equations of Sobolev-type,stochastic systems with variable and distributed delays and stochastic retarded systems with controls.Now,we present our main results as follows:Firstly,we study a class of nonlinear hybrid stochastic heat equations.In this part,we mainly discuss the exponential stability problem of nonlinear hybrid stochas-tic heat equations(known as stochastic heat equations with Markovian switching)in an infinite state space.Different from traditional literature,we generalize Markov chain from a finite state space to an infinite state space.Moreover,we reduce some restrictions on Markov chain in the traditional literature,for instance,we do not as-sume that Markov chain is irreducible,such that,the analysis of Lyapunov exponents via stationary probability distribution in the traditional literature is invalid.There-fore,we utilize the fixed point theory to overcome these difficulties and we obtain the existence,uniqueness,and pth moment exponential stability of the mild solution.Also,we acquire the Lyapunov exponents by combining the fixed point theory and the Gronwall inequality,which generalize the application of nonlinear hybrid stochastic heat equation in engineering.Secondly,we are concerned with a class of stochastic neutral functional differ-ential equations of Sobolev-type with Poisson jumps.In this part,we present some non-Lipschitz conditions which are weaker than the traditional Lipschitz conditions.Combined with two different sets of assumptions,we establish the existence of the mild solution by applying Leray-Schauder alternative theory and Sadakovskii's fixed point theorem,respectively.Furthermore,we use Bihari's inequality to prove the Os-good type uniqueness.Also,the mean square exponential stability is investigated via Gronwall inequality.Hence,we generalize the study of continuous stochastic differ-ential equations to discontinuous stochastic differential equations and accelerate the development of stochastic neutral functional differential equations of Sobolev-type.Thirdly,we are interested in a class of stochastic systems with variable and dis-tributed delays.In the traditional works,there are lots of restricted conditions imposed on variable delays.For example,variable delays are required to be differential and the derivatives of variable delays are required to be bounded.To reduce such restricted conditions,we follow the well known Perron-Frobenius theorem and Ito formula and just assume the variable delays to be bounded.Then,we give a proof by contradiction to investigate the mean square exponential stability of stochastic delay systems and provide an optimal upper bound for variable and distributed delays.It greatly reduces the traditional restrictions imposed on variable and distributed delays.Furthermore,one example is presented to introduce the application in the neural networks.Then,we concentrate on a class of stochastic retarded systems with controls.In this part,we first design a linear feedback control to obtain the stability in probabil-ity and mean square of stochastic control systems with constant delays.In addition,we combine one example and a simulation figure to illustrate the obtained results.Then,for a class of stochastic retarded systems described by stochastic functional d-ifferential equations,different from the traditional feedback control which is based on the continuous observation of state,we put forward a novel sampled-data feedback control law depending on the discrete observation of state and time delay.It greatly enlarges the probability of finding discrete-time feedback controls for stochastic re-tarded systems and this novel sampled-data feedback control succeeds in handling the variable delays involved in stochastic functional differential equations.Compared to the continuous feedback control,the novel sampled-data feedback control we estab-lished can save more cost,manpower and material resources and it is more efficient in application of engineering.At last,we investigate a class of nonlinear stochas-tic retarded systems with sampled-data output.To reduce the error arising from the delays in the traditional literature,we introduce the observer,and based on it,we construct a corresponding predictor.Furthermore,to deal with the problem that in-put measurements do not coincide with sampled measurements,following the inter-sample-predictor,observer,predictor,delay-free controller(ISP-O-P-DFC)scheme,and combining sampling data and predictor,we design a novel sampled-data predic-tive feedback control to stabilize nonlinear stochastic retarded systems.This novel feedback control can provide a more accurate prediction for the future state value and make the error between the predictor and the real state value converge asymptotically.