Positivity 13 (2009), 321–338
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/020321-18, published online October 4, 2008
Optimal control of unilateral obstacle problem
with a source term
Abstract. We consider an optimal control problem for the obstacle problem
with an elliptic variational inequality. The obstacle function which is the
control function is assumed in H
. We use an approximate technique to intro-
duce a family of problems governed by variational equations. We prove optimal
solutions existence and give necessary optimality conditions.
Mathematics Subject Classiﬁcation (2000). Primary 35R35, Secondary 49J40.
Keywords. Optimal control, variational inequality.
The study of variational inequalities and free boundary problems ﬁnds application
in a variety of disciplines including physics, engineering, and economies as well as
potential theory and geometry. In the past years, the optimal control of variational
inequalities has been studied by many authors with diﬀerent formulations. For
example optimal control problems for obstacle problems (where the obstacle is a
given (ﬁxed) function) were considered with the control variables in the variational
inequality. Roughly speaking, the control is diﬀerent from the obstacle, see for
example works by [4,8,15,16] and the references therein.
Here we deal with the obstacle as the control function. This kind of problem
appears in shape optimization for example. It may concern a dam optimal shape.
The obstacle gives the form to be designed such that the pressure of the ﬂuid inside
the dam is close to a desired value. This is equivalent in some sense to controlling
the free boundary .
The main diﬃculty of this type of problem comes from the fact that the
mapping between the control and the state (control-to-state operator) is not
diﬀerentiable but only Lipschitz-continuous and so it is not easy to get ﬁrst order
The author is grateful to Prof. M. Bergounioux for her instructive suggestions.