Solutions And Control Problems Of Several Fractional Stochastic Evolution Equations

A stochastic partial differential equation is a class of partial differential equations including stochastic processes or random fields.The idea of linking partial differential equations with randomness can be traced back to the 1950 s.Fractional stochastic partial differential equations are new research subjects in recent years.Because of multi-scale,fractional calculus is more suitable for describing natural phenomena such as anomalous diffusions,memory effects and fractals.However,due to the nonlocality and strong singularity of fractional calculus,there are few results about fractional stochastic partial differential equations.Fractional Brownian motions were introduced by Kolmogorov in 1940.Fractional Brownian motions have been widely used in physical phenomena.A fractional Brownian motion is a generalization of a standard Brownian motion,but a fractional Brownian motion is neither a semi-martingale nor a Markov process.Poisson processes are important stochastic processes,which can be used to construct general independent increment processes.To sum up,there is great theoretical and practical significance to study fractional stochastic partial differential equations driven by fractional Brownian motions and Poisson processes.In this thesis,we study the existence and uniqueness of solutions,the existence of optimal controls and the approximate controllability of the corresponding control systems for several fractional stochastic partial differential equations.Firstly,a class of space fractional stochastic reaction-diffusion systems driven by Gaussian random fields is studied.The fractional Laplacian is a nonlocal operator,which is more complex than standard Laplacian operator.In this thesis,based on the properties of eigenvalues and eigenfunctions of fractional Laplacian operators,by using Gal�erkin method and Crandal-Liggett theorem,we obtain a uniform estimate of weak solutions under the conditions that the nonlinear term satisfies m-dissipative and certain growth,and then prove that the system has a unique weak solution.In addition,the existence of optimal control for a class of quadratic cost functional is discussed,and examples are given to illustrate the conclusions.Secondly,a class of time-space fractional stochastic anomalous diffusion equations driven by fractional Brownian motions and Poisson jumps is studied.The difficulty of this problem is that the equation is involved with fractional Brownian motions,Poisson jumps,Caputo time fractional derivatives and fractional Laplacian operators.In this thesis,the sufficient conditions for the existence and uniqueness of mild solutions are given by using iterative technique.Furthermore,the existence of optimal control for a class of nonconvex cost functional is studied,and two examples are given to illustrate the results.Finally,a class of time-space fractional stochastic anomalous diffusion equations driven by hybrid noises with delay is studied.The controllability of delay systems is more complex than that without delay ones.In this thesis,the linear fractional noises case and the nonlinear fractional noises case are discussed respectively.By using approximate solutions,the existence and uniqueness of mild solutions to linear noises case are proved.By using the fixed point theory,the existence and uniqueness of mild solutions to nonlinear noises case are proved.Then,by using the properties of mild solutions,the sufficient conditions for the approximate controllability of the corresponding control systems are given.As far as we know,there are a few of literatures on fractional stochastic partial differential equations and partial differential equations driven by fractional Brownian motions and Poisson jumps,and less on time-space fractional stochastic partial differential equations driven by fractional Brownian motions and Poisson jumps.The purpose of this thesis is to enrich the theory in this subject and promote the development of this research field.