Appl Math Optim 38:121–140 (1998)
1998 Springer-Verlag New York Inc.
D. R. Adams,
S. M. Lenhart,
and J. Yong
Department of Mathematics, University of Kentucky,
Lexington, KY 40506, USA
Department of Mathematics, University of Tennessee,
Knoxville, TN 37996, USA
Department of Mathematics, Fudan University,
Shanghai 200433, China
Communicated by D. Kinderlehrer
Abstract. An optimal control problem for an elliptic obstacle variational inequal-
ity is considered. The obstacle is taken to be the control and the solution to the
obstacle problem is taken to be the state. The goal is to ﬁnd the optimal obstacle
() so that the state is close to the desired proﬁle while the H
() norm of
the obstacle is not too large. Existence, uniqueness, and regularity as well as some
characterizations of the optimal pairs are established.
Key Words. Obstacle problem, Green potentials, Approximation.
AMS Classiﬁcation. 93C20, 31B05.
Suppose ⊂ R
is a bounded domain with a C
boundary ∂. Let z ∈ L
() be a
given target proﬁle. For any ψ ∈ H
(), we deﬁne
() | v ≥ ψ,a.e. x ∈ }. (1.1)
The second author is supported in part by the National Science Foundation, and the third author is
supported in part by the NNSF of China, the Chinese State Education Commission Science Foundation, and
the Trans-Century Training Programme Foundation for the Talents by the State Education Commission of