The Null Controllability Of A Coupled Degenerate Parabolic System With The Lower Order Terms

This paper studies the null controllability of the following coupled degenerate parabolic systems with lower order terms where T>0,a satisfies bi?L?(0,T;W1,?(0,1))(i=1,2),cj? L?((0,1)×(0,T))(j=1,2,3,4),h is a control function,and ?? is the characteristic function on the non empty subset ? ?(0,1).This parabolic system has a more general diffusion coefficient a which is degenerate only when x=0.There are two types of degeneracy for a,namely weak degenerate case(W1,W2)and strong degenerate case(S1,S2,S3).Here,the convection term can be controlled by the diffusion term,so it will not affect the establishment of Carleman estimate,but it is still necessary to overcome the difficulties caused by the general diffusion coefficient.We first consider the system with two control functions.According to the classical research methods and the study of the single degenerate equation,the Carleman estimate of its conjugate problem is established,and the observable inequality is derived to prove its null controllability.Then,using the solution of the coupled system with two control functions and its control functions,the null controllability of the system with one control function is proved by constructing the control function under appropriate conditions.