Next: Elliptic control problems with
Elliptic control problems with Neumann boundary conditions
The following elliptic control problem with control and state constraints
constitutes a generalization of elliptic problems considered in
Casas , Casas et al. [10,11],
Ito, Kunisch , Kunisch, Volkwein .
The problem is to determine a control
that minimizes the functional
subject to the state equation
and the inequality constraints on control and state
In this setting,
is a bounded domain
with piecewise smooth boundary
The derivative in the direction of the outward unit normal of
is denoted by
Note that the state inequality constraints (2.4) are supposed
to hold on the closure of .
The Laplacian in (2.2)
can be replaced by an elliptic operator
where the coefficients
following coercivity condition with some :
However, in the sequel we restrict the discussion to the operator
which simplifies the form of the necessary conditions
and the presentation of the numerical approach in section 4.
are assumed to be -functions. It is straightforward to include more than one inequality constraint into (2.3) or (2.4).
However, since both the state and control variable are scalar variables,
the active sets for different inequality constraints are disjoint and
hence can be treated separately.
Then under appropriate assumptions on the function
it can be shown that the state equation (2.2) admits for each
a weak solution
(cf. Casas et al. ), i.e., it holds
An optimal solution of problem (2.1)-(2.4) will be
denoted by and .
From  we infer the further assumption that the function
in the Neumann condition (2.2) is sufficiently smooth and
satisfies the following inequality with suitable
Questions of existence of optimal
solutions will not be discussed here.
The active sets for the inequality constraints (2.3) and (2.4) are defined by
It is required that the following regularity conditions hold:
Here and in the following, partial derivatives are denoted by subscripts.
First order optimality conditions for a local optimal solution
and can be derived by generalizing the line of
proof in Casas , Casas et al. [10,11].
Problem (2.1)-(2.4) is considered
as a mathematical programming problem in Banach spaces to which the first
order Kuhn-Tucker conditions are applicable.
In particular, this approach requires that the regularity condition given
in Zowe, Kurcyusz  is satisfied;
cf. Casas et al. .
We do not discuss this regularity condition in detail although
condition (2.6) forms part of it.
The first order necessary conditions imply that there exist an
and a bounded Borel measure on
such that the following three conditions hold,
1. adjoint equation:
2. minimum condition on :
3. complementarity condition:
The adjoint equations (2.7), (2.8) are understood
in the weak sense, cf. Casas et al. .
According to Bourbaki , Chapter 9, the bounded Borel measure
appearing in the adjoint equation (2.7), (2.8)
has the decomposition
where represents the Lebesgue measure and
with respect to ; the functions
The problem of obtaining the decomposition (2.11) explicitly
is related to the difficulty of determining the structure of the active set
To our knowledge, the literature does not contain any numerical examples
where the decomposition (2.11) has actually been computed.
In section 4, we shall make an attempt to approximate the measure by the
multipliers of the discretized problem.
In later applications, we shall mostly deal with cost functionals of
tracking type (cf. Ito, Kunisch ),
with given functions
and a nonnegative weight
The control and state constraints (2.3) and (2.4)
are supposed to be box constraints of the simple type
For these data the adjoint equation (2.7), (2.8) become
If the function in (2.2) has the form
, i.e., if
then the minimum condition (2.9) reduces to
: condition (2.15) shows that the control
is given by the projection of
on the interval
which can be stated more explicitly as
Case : we obtain an optimal control of bang-bang
or singular type:
Thus, for the adjoint function
on the boundary
plays the role of a switching function. The isolated zeros of
are the switching points of a bang-bang control.
Next: Elliptic control problems with
Hans D. Mittelmann