Carleman Estimates Of A Class Of Stochastic Parabolic Equations Of Fourth Order And Application

This dissertation is devoted to establishing Carleman estimates for some forward and backward stochastic linear parabolic equations of fourth order.Also,an application of these estimates to insensitizing control problems is presented.As a class of important weighted energy estimates,Carleman estimates can be used to study deterministic and stochastic partial differential equations,and the related control and inverse problems.For forward stochastic parabolic equations of fourth order,in the known Carleman estimate,diffusion terms are required to have weak derivatives up to second order with respect to the space variable.For backward stochastic parabolic equations of fourth order,in the known Carleman estimate,drift terms are required to be square integrable with respect to the space variable.By a duality technique,these two Carleman estimates are improved in the dissertation.On one hand,for a forward equation,the diffusion term may have no weak derivative.Hence,the regularity requirement on some coefficients of the equation may be reduced.On the other hand,for a backward equation,an estimate is established with the drift term belonging to some Sobolev space of negative order.This result is applicable to a study of some control problems for coupled systems,while the known Carleman estimate fails.By virtue of two new Carleman estimates,this dissertation addresses a study of insensitizing control problems for stochastic parabolic equations of fourth order.The action of an insensitizing control is to guarantee that some energy of a system almost does not vary,when initial measurements have small errors.In the known results in this respect,energy functionals in general are related to the state variable or its derivative of first order.The more general case is studied in this dissertation.The energy may depend on the state variable and its derivatives up to second order.