Full paper in PDF:
$%A. Osses and J.-P. Puel, On the controllability of the Laplace equation observed on an interior
curve, Rev. Mat. Complut. 11 (1998), no. 2, 403–441.%$
On the Controllability of the Laplace
Equation Observed on an Interior Curve
Axel OSSES and Jean-Pierre PUEL
Centre de Mathématiques Appliquées
Ecole Polytechnique
91128 Palaiseau Cedex
Paris — France | Université de Versailles Saint-Quentin
and Centre de Mathématiques Appliquées
Ecole Polytechnique
91128 Palaiseau Cedex
Paris — France |
| |
Received: February 11, 1997
Revised: April 15, 1997
ABSTRACT
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
(
) approximate controllability is
established and controls of
-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
(
) 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.
1991 Mathematics Subject Classification: 35B37, 93B40, 93B06, 93C10.