Approximate Controllability For Semilinear Parabolic Systems By One Control Force

In this paper, we consider the approximate controllability of the following semilinear parabolic systemwhere Rn be an open and bounded domain with a C2 smooth boundary , and be a nonempty open subset. m(x) denote the characteristic function of the nonempty open subset of . The functions f, g : are assumed to be globally Lipschitz all along the paper, i.e. L > 0 such thatfurther more, we assume that g is strictly decreasing of y.The approximate controllability of (1.1) can be described as follows: Given T > 0, (y1,z1) L2 ( ) x L2( ), and , to find a control such that the solution of (1.1) satisfiesThis paper is organized as follows:In section 1, we introduce the background of the approximate controllability and the relavant research progress, what'more, we state our main result.In section 2, we prove the approximate controllability of the linearized system by means of a cost function.In section 3, we adopt the fixed method to prove the approximate controllability of system (1.1).In section 4, we will prove the unique continuation property for the linearized system to (1.1), which has been quoted as Lemma 2.1 in section 2.