On the approximate controllability for second order nonlinear parabolic boundary value problems.

In this communication we develop and improve some of the results of [4] on the approximate controllability of several semilinear parabolic boundary value problems where the nonlinear term appears either at the second order parabolic equation or at the put boundary condition. We also distinguish the cases where the control function acts on the interior of the parabolic set Q := R x (0, T) from the one in which the control acts on the boundary Sigma := partial derivative Omega x (0, T). Most of our results will concern to control problems with final observation i.e. our goal is to prove that the set {y(T, : v)} generated by the value of solutions at time T is dense in L(2)(Omega) when v runs through the set of controls. Nevertheless we also consider a control problem with a boundary observation. In that case we shall prove that if Sigma(1) subset of Sigma then the set {y(.,.:v}/(Sigma 1)} generated by the trace of solutions on Sigma(1) is a dense subset of L(2)(Sigma(1)) when v runs through the set of controls.