Appl Math Optim 45:125–143 (2002)
2002 Springer-Verlag New York Inc.
Dirichlet Boundary Control of Semilinear Parabolic Equations
Part 1: Problems with No State Constraints
N. Arada and J.-P. Raymond
UMR CNRS MIP, Universit´e Paul Sabatier,
31062 Toulouse cedex 4, France
Abstract. This paper is concerned with distributed and Dirichlet boundary con-
trols of semilinear parabolic equations, in the presence of pointwise state constraints.
The paper is divided into two parts. In the ﬁrst part we deﬁne solutions of the state
equation as the limit of a sequence of solutions for equations with Robin boundary
conditions. We establish Taylor expansions for solutions of the state equation with
respect to perturbations of boundary control (Theorem 5.2). For problems with no
state constraints, we prove three decoupled Pontryagin’s principles, one for the dis-
tributed control, one for the boundary control, and the last one for the control in the
initial condition (Theorem 2.1). Tools and results of Part 1 are used in the second
part to derive Pontryagin’s principles for problems with pointwise state constraints.
KeyWords. Dirichlet boundary controls, Semilinear parabolic equation, Pontrya-
AMS Classiﬁcation. 49K20, 93C20, 35K20.
This paper is concerned with an optimal control problem for the following parabolic
+ Ay + (x, t, y, u)=0inQ, y = v on , y(0) = w in , (1)
where Q = × ]0, T[, is a bounded domain in R
, = × ]0, T [, and is
the boundary of . The distributed control u belongs to a subset U
q > N /2 + 1, the boundary control v belongs to V
(), the control w in the