Appl Math Optim 39:211–227 (1999)
1999 Springer-Verlag New York Inc.
Second-Order Necessary Optimality Conditions for
Some State-Constrained Control Problems of
Semilinear Elliptic Equations
and F. Tr¨oltzsch
Departamento de Matem´atica Aplicada y Ciencias de la Computaci´on,
E.T.S.I. Industriales y de Telecomunicaci´on, Universidad de Cantabria,
Av. Los Castros s/n, 39071 Santander, Spain
Fakult¨at f¨ur Mathematik, Technische Universit¨at Chemnitz,
D-09107 Chemnitz, Germany
Communicated by I. Lasiecka
Abstract. In this paper we are concerned with some optimal control problems
governed by semilinear elliptic equations. The case of a boundary control is studied.
We consider pointwise constraints on the control and a ﬁnite number of equality
and inequality constraints on the state. The goal is to derive ﬁrst- and second-order
optimality conditions satisﬁed by locally optimal solutions of the problem.
Key Words. Boundary control, Semilinear elliptic equations, Optimality condi-
tions, State constraints.
AMS Classiﬁcation. 49K20, 35J25.
In this paper we discuss ﬁrst- and second-order necessary optimality conditions for aclass
of optimal boundary control problems governed by a linear elliptic partial differential
This research was partially supported by the European Union, under HCM Project Number
ERBCHRXCT940471, and by the Deutsche Forschungsgemeinschaft, under Project Number Tr 302/1-2.
The ﬁrst author was also supported by Direcci´on General de Investigaci´on Cient´ıﬁca y T´ecnica (Spain).