Appl Math Optim 56:1–17 (2007)
2007 Springer Science+Business Media, Inc.
Optimal Control of Obstacle Problems by H
and Karl Kunisch
Department of Mathematics, North Carolina State University,
Raleigh, NC 27695-8205, USA
Institut f¨ur Mathematik und wissenschaftliches Rechnen,
Karl-Franzens-Universit¨at Graz, A-8010 Graz, Austria
Abstract. Optimal control of variational inequalities with the controls given by
the obstacles is considered. Existence optimal solutions are proved for obstacles
regularity and ﬁrst-order optimality conditions are derived which, under
additional assumptions, are also sufﬁcient. A numerical algorithm is proposed and
its practical feasibility is investigated.
Key Words. Control of variational inequalities, Optimality system, Moreau–Yosida
AMS Classiﬁcation. 49J24, 49K24, 90C99.
In this paper we consider a control problem where the state satisﬁes a variational inequal-
ity of obstacle-type and the control variable is the obstacle itself. While the techniques
that we employ can be applied to a wider class of variational inequalities, we describe
them in detail for second-order elliptic obstacle problems: ﬁnd y ∈ K such that
Ay − f,ϕ− y≥0 for all ϕ ∈ K, (1.1)
Kazufumi Ito’s research was partially supported by the Army Research Ofﬁce under DAAD19-02-1-
0394. Karl Kunisch was supported in part by the Fonds zur F ¨orderung der wissenschaftlichen Forschung under
SFB 03, “Optimierung und Kontrolle”.