subject to the elliptic state equation,

boundary conditions of Neumann or Dirichlet type,

and control and state inequality constraints,

The functions and

are assumed to be -functions. It is straightforward to include more than one inequality constraint in (5) or (6). 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.

The Laplacian in (2)
can be replaced by any elliptic operator

where the coefficients satisfy the 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 numerical analysis.

An optimal solution of problem (1)-(6) will be
denoted by and .
The *active sets* for the inequality constraints
(5), (6) are defined by

Here and in the following, partial derivatives are denoted by subscripts.

First order necessary conditions for the rather general problem
(1)-(6) are not yet available in the literature.
The main difficulty results from the Dirichlet condition (4)
which prevents solution from being sufficiently regular.
First order necessary conditions for problems with *linear* elliptic
equations
and pure Neumann conditions
may be found in Casas [14], Casas et al. [15,16].
A weak formulation for *linear* elliptic equations and Dirichlet conditions
is due to Bergounioux, Kunisch [4].

We shall present first order conditions in a form that can be derived at least in a purely formal way. This form will turn out to be consistent with the first order conditions of Kuhn-Tucker for the discretized elliptic control problem in section 3.1. We assume that there exists an adjoint state , a multiplier , and a regular Borel measure on such that the following conditions hold:

*adjoint equation and boundary conditions*:

The adjoint equations (9)-(11) are understood in the weak sense, cf. Casas et al. [16]. According to Bourbaki [10], Chapter 9, the regular Borel measure appearing in the adjoint equation (9) has the decomposition

where represents the Lebesgue measure and is singular with respect to ; the functions are measurable on . The problem of obtaining the decomposition (15) explicitly is related to the difficulty of determining the structure of the active set . In section 3, we shall make an attempt to approximate the measure by the multipliers of the discretized control problem.

In many applications, the cost functional (1) is of
*tracking type*, cf. [2,4,22,24],

with functions and . Here, in particular we assume that the functions and in (1) coincide. For these data the adjoint equations (9)-(11) become

If the function in the Neumann condition (3) has the special form , then the minimum condition (12) reduces to

Likewise, if the function in (4) is given by , the minimum condition (13) yields

**Case
**: The previous conditions determine the following
control laws:

for
,

**Case **: We obtain an optimal control of
*bang-bang* or *singular* type:

for
,

Hence in case , the so-called

2000-12-09