Approximate Controllability Properties of the Semilinear Heat Equation With Lumped Controls

In this article, we study the global controllability properties of aone-dimensional semilinear heat equation with sublinear reaction term, governed in a bounded domain by internal lumped controls. We prove thatit is possible to exactly control any finite dimensional portion of its solution (when expanded along the sequence of the eigenfunctions of the associated Laplacian), provided that the truncated linear equationis approximately controllable in L^2 (0,1). We also describe a certain topology (weaker than L^2 (0,1)) in which this system is, in fact, globally approximately controllable at any positive time. Some extensions to the case of several dimensions are also given.