ON THE APPROXIMATE CONTROLLABILITY FOR 2ND-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 b oundary value problems where the nonlinear term appears either at the second order parabolic equation or at the put boundary condition. We a lso distinguish the cases where the control function acts on the inter ior of the parabolic set Q := R x (0, T) from the one in which the con trol acts on the boundary Sigma := partial derivative Omega x (0, T). Most of our results will concern to control problems with final observ ation 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 t hrough the set of controls. Nevertheless we also consider a control pr oblem 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.