Necessary conditions for elliptic control problems with control and state constraints

We consider the optimal control problem of determining a
*distributed control* function
that minimizes the functional

Neumann and Dirichlet boundary conditions,

and mixed control-state inequality constraints, resp.

The split boundary formulation permits simultaneous treatment of various boundary conditions while in [23] this was done in separate sections. The functions and are supposed to be -functions and is assumed in the Dirichlet condition (2.4). We have to admit that no numerical example with a spliting of the boundary into will be considered in this paper. However, the spltting has been introduced to allow for a general discussion of necessary conditions. A practical example with splitted boundary may be found in [25].

The Laplacian
in the elliptic equation (2.2)
can be replaced by an elliptic operator in divergence form.
We refer to section 2 of [23] for a precise definition.
The above distributed control problem is slightly more general than the one
considered in Bonnans and Casas [6] where first order
conditions have been given in terms of a weak and strong Pontryagin principle.
For *linear* elliptic equations, first order conditions may also
be found in Bergounioux et al. [3], Bonnans and Casas
[5].
Nonlinear elliptic equations of Lotka-Volterra type have been considered in
Cañada et al. [8] and Leung, Stojanovic [21,28].

Throughout this paper, it wil be assumed that an optimal solution
and
of problem (2.1)-(2.6) exists.
To ensure well-posedness of the elliptic problem
(2.1)-(2.3) we require as in Bonnans, Casas [6], condition (2.3), that

The *active sets* for the inequality constraints
(2.5), (2.6) are given by

Extending in a purely formal way the first order necessary conditions in Bonnans and Casas [6], we arrive at the following. There exists an adjoint state , a multiplier , and a regular Borel measure in such that the following conditions hold:

The adjoint equations (2.10)-(2.12) are understood in the weak sense. According to Bourbaki [7], Chapter 9, the regular Borel measure in the adjoint equations (2.10) and (2.11) possesses a decomposition

where represents the Lebesgue measure, the measure is singular with respect to and are measurable.

In many applications, the control and state constraints (2.5) and (2.6) are simple box constraints of the type

while the the minimum condition (2.13) yields the control law

A further specialization refers to a cost functional (2.1) of
*tracking type* which has been considered frequently, cf. [1,2,19,20],

2000-10-06