Carleman Estimates For Some Stochastic Partial Differential Equations And Applications

This dissertation is mainly devoted to studying Carleman estimates for two classes of typical stochastic evolution equations: stochastic degenerate parabolic equations and refined stochastic beam equations.Also,these Carleman estimates are applied to solve the controllability,insensitizing controls problem and inverse problems,respectively.As a class of weighted energy estimates,Carleman-type estimate is one of important tools in studying the uniqueness,control and inverse problems of deterministic and stochastic partial differential equations.A weighted identity method is an important approach in establishing Carleman estimates.But it requires more regularity on coefficients and non-homogeneous terms for some stochastic partial differential equations.This will restrict applications of Carleman estimates to control problems.Later,a duality method twice is proposed to solve this problem.If we take stochastic parabolic equations as an example,the estimate derived by the duality method twice is more general and has more applications.The second chapter of this dissertation is devoted to establishing global Carleman estimates for a class of stochastic degenerate parabolic equations.In order to overcome difficulties brought by degeneracy of the equation,suitable weight functions are chosen to establish Carleman estimates for forward and backward stochastic degenerate parabolic equations,respectively,by the weighted identity method and duality method twice.By comparing these estimates,it is shown that two methods in establishing stochastic Carleman estimates have their own advantages when the equation has degeneracy.They have different requirements on the coefficients in some control problems,which will be the basis for the study of nonlinear problems.As applications of the Carleman estimates in chapter 2,the insensitizing control problem and controllability problem in the sense of Stackelberg-Nash equilibrium for stochastic degenerate parabolic equations are studied in chapter3,respectively.Moreover,a suitable Carleman estimate is established,when the weight functions depend only on the time variable.It may be applied to the study of inverse initial value problems for stochastic degenerate parabolic equations.The fourth chapter of this dissertation addresses a study of the controllability and inverse problem for refined stochastic beam equations.This paper proves that the classical stochastic beam equation is not exactly controllable,even if controls are effective everywhere in drift term,diffusion term and the whole boundary.However,by modifying this model and establishing an appropriate Carleman estimate,the refined stochastic beam equation is proved to be exactly controllable at any given time with controls in diffusion term and part of boundary.In addition,the uniqueness of inverse source problems for the refined stochastic beam equation is obtained by the Carleman estimate method.