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## Distributed Control Problem

Here the problem is to determine a distributed control function that minimizes the functional
 (25)

subject to the elliptic state equation,
 (26)

Neumann and Dirichlet boundary conditions,
 (27) (28)

and mixed control-state inequality constraints resp. pure state inequality constraints,
 (29) (30)

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

 (31)

It is required that the following regularity conditions analogous to (8) hold:
 (32)

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:
 (33) (34) (35)

minimum condition for :
 (36)

complementarity conditions:
 (37)

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),
 (38)

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

 (39)

with functions and . We immediately derive from (36) the control law
 (40)

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

Next: Discretization and optimization techniques Up: Necessary conditions for elliptic Previous: Boundary control problem
Hans D. Mittelmann
2000-12-09