Positivity 8: 229–242, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Optimal control of the obstacle in semilinear
and SUZANNE LENHART
Département de Mathématiques, Université d’Orléans, Laboratoire MAPMO, UFR Sciences, BP
6759, 45067 Orléans, France. E-mail: Maitine.Bergounioux@labomath.univ-orleans.fr
University of Tennessee, Mathematics Department, Knoxville, TN 37996-1300, USA.
(Received 24 June 2002; accepted 23 March 2003)
Abstract. We consider an optimal control problem where the state satisﬁes an obstacle type semi-
linear variational inequality and the control function is the obstacle. The state is chosen to be close
to a desired proﬁle while the obstacle is not too large in H
, and H
-bounded. We prove that an
optimal control exists and give necessary optimality conditions, using approximation techniques.
Key words: Optimal control, obstacle problem, variational inequalities, semilinear elliptic equations
We consider an optimal control problem where the state satisﬁes an obstacle type
variational inequality and the control function is the obstacle. The state is chosen
to be close to a desired proﬁle while the obstacle is not too large in H
problem has been studied in  in the case where the variational inequality is linear
and associated to the Laplace operator with the surprising result that the “optimal
obstacle = optimal state”. In this case, speciﬁc techniques give convergence results
without any additional compactness assumption. When the variational inequality
is governed by a semilinear operator, we cannot obtain existence and necessary
conditions for the optimal solution without a priori estimates (and bounds ) which
are strong enough to ensure good convergence properties of approximating pro-
cesses. Therefore, in this paper, we consider admissible controls (obstacles) which
are bounded in the Sobolev space H
. The recent paper by Adams and Lenhart
 treats control of H
-obstacles in Laplacian case with source terms, while the
parabolic case is treated in . The paper by Lou  generalized the regularity
results of  when the target proﬁle is in L
. See also Lou’s work  on
obstacle control in quasilinear variational inequalities for p-laplacian operator in
which the “optimal obstacle = optimal state”. See works by Bergounioux  and
Chen  and the references therein for semilinear elliptic variational inequalities
with control entering in lower order terms.
Let us specify the problem under consideration: