Carleman Estimates For Stochastic Degenerate And Singular Partial Differential Equations And Its Applications

Degenerate and singular partial differential equations are widely used in physics,economics,biology and other disciplines,such as population genetic problem,option pricing problem,cardiac electrophysiological model and so on.In practical applications,the uncertainty of the environment is often involved.Therefore,it is very important to study the inverse problems and control problems involving stochastic degradation and singular partial differential equations.In this thesis,we mainly study the Carleman estimates of several stochastic degenerate and singular partial differential equations and their applications.The main contents are as follows:In the first chapter we review some developments of the related problems and summarize the main work of the pressertation.In Chapter 2,we study the Carleman estimates of the stochastic Grushin equation with singular potential and its applications to null controlllability and an inverse source problem.We construct two special weight functions to establish two Carleman estimates for backward/forward stochastic Grushin operator with singular potential by a weighted identity method.We apply the Carleman estimate of the backward stochastic Grushin equation to prove the observability inequality of the backward stochastic Grushin equation,and then we prove the null controllability for the forward stochastic Grushin equation for any time T.Finally,we apply the Carleman estimate of the forward stochastic Grushin equation to study the inverse source problem of determining two kinds of sources simultaneously,we obtain the uniqueness of the inverse source problem.In Chapter 3,we consider the Carleman estimate of the stochastic singular parabolic equations with first order term and its application to null controllability.We first establish the Carleman estimate for the backward stochastic singular parabolic equation without lower order terms.Based on this Carleman estimate and a duality technique,we obtain the Carleman estimate for the backward stochastic singular parabolic equation with thefirst order term,and then we use the Carleman estimation and the energy estimate of the stochastic singular parabolic equation to establish the observability inequality of the backward stochastic singular parabolic equation.Finally,we prove null controllability of the forward stochastic singular parabolic equations with the first order term.In Chapter 4,we mainly consider the Carleman estimate of stochastic degenerate wave equation and the inverse problem of determining three unknow coefficients in the stochastic degenerate wave equation.We first choose the appropriate weight function to establish the Carleman estimate of the forward stochastic degenerate wave equation,and then we apply the Carleman estimate to solve an inverse problem of simultaneously determining three unknowns,i.e.,a source term,an initial velocity,and an initial displacement.We prove the global uniqueness of the inverse problem.In Chapter 5,we study the Carleman estimate of the stochastic degenerate reaction-diffusion system in electrocardiology and its application to null controllability.Since the time derivatives of the electrical potentials coupled each other in the two equations,and the couplings lead to the degeneracy of the model.In this chaper,we first consider an approximate problem of the system.We first consider two equations in backward stochastic degenerate reaction-diffusion system as a whole to establish a uniform Carleman estimate.Based on this Carleman estimation,the observability inequality of the adjoint system corresponding to the approximate system is obtained,and then the null controllability of the approximate system is proved.Finally,by a limit process,we obtain the null controllability result of the forward stochastic degenerate reaction-diffusion system.