boundary value control and where the observation is the final state. The methods ... (1.2) are generated by solving Unear initial value problems of the form (1.1).
MATHEMATICS OF COMPUTATION, VOLUME 34, NUMBER 149 JANUARY 1980, PAGES 115-125
Initial Value Methods for Parabolic Control Problems By Ragnar Winther Abstract.
We study iterative
boundary
value control
based on transforming control) straints).
tial value problems
for parabolic
control
problems
is the final state.
with a Neumann The methods
the original control problem (which may have constraints
into an equivalent The methods
methods
and where the observation
problem of minimizing a strictly convex functional
are semidiscrete
are
on the (no con-
in the sense that we assume that parabolic
ini-
can be solved exactly.
1. Introduction. The purpose of this paper is to study approximations of certain parabolic control problems. In order to describe these problems, let £2 be a bounded domain in Rd with sufficiently smooth boundary, 8Í2, and for a fixed T > 0
let Ô = (0, T) x n and 2 = (0, T) x 3Í2. On the domain £2 let L denote the second order differential operator
Lu = -
i,7=i dxi \
bxjJ
We will assume that ai • and c are sufficiently smooth real-valued functions on
Í2 and that the operator L is strongly elUptic; i.e., there is a constant cx > 0 such
that í,7= i
i'= l
for all x G Í2 and %G Rd. Now let i>! G L2(ü) and /G L2(Q) be given and for any g G ¿2(E), let ug = u (t, x) denote the unique weak solution of the problem
(1.1)
du/dt + Lu = f
on Q,
du/dv=g
on 2,
«(0, •) = vx
on il.
Here d/dv = Ef •_, a¡ n¡(blbxA, where n¡ is the ith component of the outward unit
normal on 8Í2. We shall consider optimal control problems of the form
(1.2)
min{\\ugiT,-)-v2\\l2(n)+a\\g\\2L2
},
g£K
Received November 27, 1978. AMS (MOS) subject classifications
(1970).
Primary 49D05. © 1980 American Mathematical 0025-571 8/80/0000-0006/$03.75
115 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
Society
116
RAGNAR WINTHER
where v2 G Z,2(S2) and a > 0 are given and where A' is a closed convex set of L2(Z).
In Section 2 we shaU consider the case when K = L2ÇZ), while in Section 4 we con-
sider K = {g&L2ÇZ)\g >0 a.e. on Z}. In both cases we shall derive algorithms where approximations of the solution of (1.2) are generated by solving Unear initial value problems of the form (1.1). This will
be done by showing that (1.2) (in both cases mentioned above) is equivalent to minimizing a strictly convex functional over the entire space L2(£l).
The minimum of
this functional can then be found by standard iterative techniques.
In the case K =
Z,2(2) the associated functional is quadratic, and in Section 3 we study how this functional can be minimized by the conjugate gradient method. We shall particularly show certain superlinear convergence estimates for this method by applying some results recently given by the author in [13]. In order to evaluate the functionals mentioned above we have to solve two initial value problems of the form (1.1). Hence, the approximation of (1.2) is reduced
to the well-studied problem of solving problems of the form (1.1) numerically. Such numerical methods will not be considered here. We shall instead assume that problems of the form (1.1) can be solved exactly.
However, we mention that a detailed study of fuUy discrete analogs of the methods considered here was done by the author in [11] and [12] in the case when K = L2(Z), and some numerical examples can be
found in [11]. 2. Preliminaries. We start with some notation. If Hx and H2 axe two Hubert spaces then l(//j, H2) will denote the set of bounded linear maps from Hx to H2. On the spaces L (Í2) and ¿2(9Í2), respectively, we shall use the notation
(, i/O= Jn dx
and
(= jan y\pdo
for the inner products, and the associated norms wiU be denoted by II• II and I-I. If p > 0, let /^(íí) denote the Sobólev space of order p of real-valued functions on £2 (see for example Lions and Magenes [5]). If p < 0, //p(f2) is defined to be the dual of/T~P(Í2) with respect to the inner product of L2(Í2).
Furthermore,
W(0,T) = J/l/GL2(0, T-H'im. JteL2(*°>^/T1^))!
we let
.
where d/dt is taken in the sense of distributions on (0, T) with values in //'(Í2).
We
recaUfrom [5] that W(0, T) C C(0, T; ¿2(S2)). Let also B: HliQ,) x H1^)
—►R denote the biUnear form associated with the
operator L; i.e.,
r
[
"
dip d\b
dx.
The problem (1.1) can now be written in the variational form
(2.1)
(idu/dt, 0 such that \\F(zx) - F(z2)\\ < (5.5)
sup of1 i7" \E(t)(zx - z2)\ \E(t^p\dt il^ll=i "
