Resumen de On the controllability of the Laplace equation observed on an interior curve
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 < µ) 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 < µ) 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 behaviour in the case of fixed geometry.
