subject to the elliptic state equation,

Neumann and Dirichlet boundary conditions,

and mixed control-state inequality constraints resp.

The functions and are assumed to be -functions, while the Dirichlet condition (28) holds with .

The above distributed control problem is slightly more general than the one
considered in Bonnans and Casas [9] 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
[8].
Nonlinear elliptic equations of Lotka-Volterra type have been treated in
Canada et al. [12] and Leung, Stojanovic [25,34].

Denote an optimal solution of problem (25)-(30)
by and .
The *active sets* corresponding to the inequality constraints
(29), (30) are given by

Then first order necessary conditions can be stated in the following form. There exist an adjoint state , a multiplier and a regular Borel measure in such that the following conditions hold:

The adjoint equations (33)-(35) are understood in the weak sense. The regular Borel measure in the adjoint equation (33) has a decomposition similar to that in (15),

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

With regard to Example 4.2 in section 4 we shall discuss the minimum condition
(36) in case that the control and state constraints (29) and (30) are box constraints

It is straightforward to obtain analogous control laws for tracking functionals similar to (16).

2000-12-09