The boundary approximate controllability of the Laplace equation observed on an interior curve is studied in this paper. First we consider the Laplace equation with a bounded potential. The Lp (1 < p < ∞) approximate controllability is established and controls of Lp-minimal norm are built by duality. At this point, a general result which clarifies the relationship between this duality approach and the classical optimal control theory is given. The results are extended to the Lp (1≤ p < ∞) approximate controllability with quasi bang-bang controls and finally to the semilinear case with a globally Lipschitz non linearity by a fixed point method. A counterexample shows that the globally Lipschitz hypothesis is essential. To compute the control, a numerical method based in the duality technique is proposed. It is tested in several cases obtaining a fast behavior in the case of fixed geometry.
@article{urn:eudml:doc:44458,
title = {On the controllability of the Laplace equation observed on an interior curve.},
journal = {Revista Matem\'atica de la Universidad Complutense de Madrid},
volume = {11},
year = {1998},
pages = {403-441},
zbl = {0919.35019},
mrnumber = {MR1666505},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:44458}
}
Osses, A.; Puel, J.-P. On the controllability of the Laplace equation observed on an interior curve.. Revista Matemática de la Universidad Complutense de Madrid, Tome 11 (1998) pp. 403-441. http://gdmltest.u-ga.fr/item/urn:eudml:doc:44458/