Approximate Controllability Of A Class Of Linea Svstems With Boundary Degeneracy

In this paper,we consider the approximate controllability of the degenerate linear parabol-ic system(?)u/(?)t-(?)/(?)x(a(x)Vu/(?)x)-(?)/(?)y(b(y)(?)u/(?)u)+c(x,y,t)t=h(x,y,t)?D,(x,y,t)?QT,u(x,y,t)= 0,(x,y,t)??,(0.1)u(x,y,0)= u0(x,y),(x,y)? Q,where QT ?:(0,1)×(0,1)×(0,T).The coefficients a(x)and b(y)are nonnegative continuous functions on[0,1],which may be weakly degenerate or non-degerate on the lateral bound-ary.This brings much difficulty in investigating the wellposedness and controllability of the problem.This thesis is divided into three chapters.In the first chapter we confine ourselves to overviewing some physical backgrounds of the problem under consideration and some related works in recent years.Some necessary preliminaries are also given in this chapter.In the second chapter,we first state the definition of weak solutions of our problem.Then,by applying the parabolic regularization method as welll as the weak convergence trickes,we obtain the existence of weak solutions.Finally the uniqueness of weak solutions is proved by using Holmgren's method.In chapter 3,we first obtain the existence of weak solution to the conjugate problem.Then we constructing a control prove the approximate controllabil-ity of our problem by function using the solutions of the conjugate problem.There are the following conclusions in the paper:Theorem 0.1.Assume a,b ? C([0,1])and is positive in(0,1)and c ?L??QT)?h?L2(QT)and u0 ?L2(Q??there exists uniquely aweak solutionn uof the problem(0.4),and the solution u satisfies(?)?u?L?2((0,T);L2(Q))+?a(x)|(?)u/(?)x|2?L1(QT+?b(y)|(?)u/(?)y|?L1(QT)?C(?h?L22(DT)+?u0?L22(Q)),(0.5)where C>0 is related to T and ?c?L?(QT).(ii)Assume a|(?)u0/(?)x|2 ?L1(Q),then?)u/(?)t?L2(QT),?(?)u/(?)t?L22(QT)+?a|(?)u/(?)t|2?L?((0,T);L1(Q))+?B|(?)u/(?)t|2?L?((0,T);L1(Q))?C(?h?L22(DT)+?u0?L22(q)+?a|(?)u0/(?)x|2?L1(Q)+?b|(?)u0/(?)y|2?L1(Q)),(0.6)wherw C>0 is constant and related to T and ?c?L?(QT).Theorem 0.2.Assume a,b ?C([0,1])and is positive in(0,1),c ? L?(QT).The control system(0.4)is approximately controllable.namely,for and given initial datum u0,ud ? L2(Q)and ?>0,the exists h ? L2(QT)such that the weak solution u satisfies?u(·,T)-ud(·)?L2(Q)? ?.