Local Null Controllability For A Coupled Semilinear Parabolic System

This paper concerns the local null controllability for the following systemwhere is a bounded nonempty open set with a smooth boundary , , m(x) is the characteristic function of the subset , u, v are the control functions acting in [0,T]. Here T > 0, a, b, c, a, b, c are constants.The local null controllability problem for (I) can be formulated as follows: Given , m a neighbourhood of (0,0), to find controls u and such that the solution (y,z) to (I) satisfiesThis paper is made up of three parts.Part one introduces the background of the system and the revelant research progress. Furthermore, we state our main results following from some retropection of the results obtained by previous mathematicians, i.e., the local null controllability for (I) holds under some assumptions on the initial data y0 and Z0.Part two firstly linearizes the system (I), then introduces some known results, mainly the Carleman inequality for linear parabolic system(Lemma 2.2), which will be used in the following text. By applying this lemma, we obtain the the Carleman inequality for the linearized system(Lemma 2.4), from which we prove the null controllability for the linearized system (Theorem 2.3).Part three proves the local null controllability for (I), i.e., Theorem 3.3, by applying Kakutani fixed point theorem.