Infinite Dimensional Stochastic Evolution Systems And Its Applications In Modeling Of Formation Mechanism In Coagulation Bath Of Carbon Fiber Precursor

In this thesis, we considered the theory of infinite dimensional stochastic evolution systems and its applications in modelling of formation mechanism in coagulation bath of carbon fiber precursor. The infinite dimensional evolution systems, defined in infinite dimensional function spaces, are described by functional differential equations, partial differential equations. The stochastic factors, considerd here, caused by modeling errors, missing measurements include random noise, impulsive effects, stochastic oscillators, time delays, system parameters random switching. As an important topic of stochastic dynamics, the stability for stochastic evolution systems in infinite dimension has gained great attention. The asymptotic behavior of stochastic evolution systems in infinite dimension, especially the convergence to a equilibrium as time goes to infinity attracts a lot of interests.Controllability, as a fundamental concept of control theory, plays an important role both in stochastic and deterministic control theory. The study of controllability of linear and nonlinear systems represented by infinite dimensional systems in Banach spaces has been raised by many authors.By using the theory of modern partial differential equations, stochastic analysis and the semigroup of operators, we discussed the stability and controllability of stochastic evolution systems in infinite dimension with stochastic noise, impulsive effects, stochastic perturbations, time delays and system parameters random switching.The content of this thesis is mainly divided into three parts. Firstly, we considered the stability and controllability for infinite dimensional stochastic evolution systems with stochastic oscillators, impulsive effects, Levy noise, time delays, system parameters random switching. Especially, by the concept of stability in distribution, we generalized the results derived by mean square stability and almost surely stability, then, we discussed the exponential stability and controllability for infinite dimensional stochastic evolution systems of high order or fractional order. By using the cosine families of operators and Caputo derivate, sufficient conditions for the stability and controllability of mild solutions are obtained. In the third part, we use the theory and technique developed in previous two parts to deal with some issues of formation mechanism in coagulation bath for carbon fiber precursor. By the stochastic evolution systems theory, we modeled the formation mechanism in coagulation bath of carbon fiber precursor. The stability and controllability conditions are derived by the stochastic analysis and semigroup of operators. At last, computer simulations are given to illustrate our results.The main contributions of the paper are as follows:(1) The stability of mild solution for the impulsive neutral nonlinear stochastic delay partial differential equations is investigated. Under the global Lipschitz condition, linear growth condition, contractive conditions, we considered the stability in pth moment of mild solutions to nonlinear impulsive stochastic delay partial differential equations (NISDPDEs). By employing a fixed point approach, sufficient conditions for the exponential stability in pth moment of mild solutions are derived.(2) The controllability of mild solution for a class of second order impulsive neutral stochastic functional systems is considered. We consider a class of impulsive neutral second order stochastic functional evolution equations. The Sadovskii fixed point theorem and the theory of strongly continuous cosine families of operators are used to investigate the sufficient conditions for the controllability of the system considered. An example is provided to illustrate our results. Furthermore, stability results of second order evolution equations are generalized.(3) The asymptotic stability of mild solution to the nonlinear fractional order stochastic partial differential equations with Poisson jumps is investigated. We consider a class of fractional stochastic partial differential equations with Poisson jumps. Sufficient conditions for the existence and asymptotic stability in pth moment of mild solutions are derived by employing the semigroup of operator method, Banach fixed point principle.. The stability results of linear model are generalized to cover a class of more general nonlinear ones.(4) In Hilbert space, we considered the asymptotic stability in distribution for stochastic delay reaction diffusion equations with Markov switching and Poisson jumps. With the help of operator in semi-group theory and stochastic systems in Hilbert space theory. Several criteria of asymptotic stability in distribution for stochastic reaction diffusion systems with delays and Markovian jumps driven by the Levy martingales in Hilbert spaces are presented. A proper approximating strong solution and a limiting type of argument are being constructed to pass on the stability of strong solutions to mild ones. Sufficient conditions are obtained to ensure the asymptotic stability in distribution for the mild solutions. In particular, stability results are generalized to cover a class of more general hybrid stochastic reaction diffusion systems with delays and jumps in infinite dimension..(5)The asymptotic properties of the model for formation mechanism in coagulation bath of carbon fiber precursor with random informations are studied. Combining with theory of semigroup operators and stochastic analysis in Hilbert space theory, the reaction diffusion model in coagulation bath of carbon fiber precursor is considered, by numerical simulation, the asymptotic properties for the reaction diffusion model with unknown informations are investigated.At the end, we summarize of content, advantage and the deficiency of the paper, narrate further development of the study.