The paper studies the -approximate controllability problem in time for a class of semilinear parabolic systems in a bounded domain in . Considered are two types of problems: and . In each problem, the control functions are fairly general, given by the form . In , the control and the nonlinear term , being the state of the system, enter the equation under a homogeneous boundary condition of the Dirichlet type, and , supposed to be nonnegative on , belongs to a dense subset of . In , and appear on the boundary condition of the Neumann type, and belong to a dense subset of . For example, is written as
In both cases, the problem in the linear case where is first solved. Based on this result, the semilinear problem is solved as a perturbation of a linear problem cancelling the nonlinear term. The problem is also studied via the Kakutani fixed-point theorem for other semilinear parabolic systems including multivalued coefficients (a case of flux boundary controls).