Two Carleman Estimates For A Bidomain Model In Electrocardiology And Its Applications

Recently,bidomain model in electrocardiology has become one of the hot research scholars at home and abroad,more and more scholars begin to study on this prob-lem.cardiovascular model mainly describes the electrical activity in the heart tissue.This paper concerns Carleman estimate and its applications for two kinds of bidomain model in electrocardiology,which describes the electrical activity in the cardiac tissue.The thesis is organized as follows:In chapter 1,we first summarize the research fields on the inverse problems in mathematical physics.Then we review the background and some development of the related problems and summarize the main work of the present dissertation.In chapter 2,we consider Carleman estimate and its applications for a linearized bidomain model in electrocardiology.First,we concern Carleman estimate for a lin-earized bidomain model in electrocardiology.Second,by means of this Carleman es-timate,we study two problems for the linearized bidomian model,a Cauchy problem and an inverse conductivities problem.We prove a conditional stability result for the Cauchy problem and a Holder stability result for the inverse conductivities problem.In chapter 3,we concern the null controllability for a degenerate reaction-diffusion system in electrocardiology.To deal with nonlinearity and degeneracy of this system,we first consider an approximate problem of the linearized system.We prove a uniform Carleman estimate and then an observability inequality for the adjoint system of this approximate problem.Based on this Carleman estimate and a fixed point argument,we finally obtain the null controllability for the simplified cardiac model.