Elliptic control problems with Dirichlet boundary conditions

The following elliptic control problem with control and state constraints
generalizes the elliptic problems considered in
Bergounioux, Kunisch [3], Ito, Kunisch [16].
When admitting Dirichlet boundary conditions,
the functions in the cost functional (2.1),
in the boundary conditions (2.2),
and in the constraints (2.3) do not depend on the
state variable . This leads to the problem of
finding a control
that minimizes the functional

and the inequality constraints on control and state

When treating Dirichlet boundary conditions, an intrinsic difficulty
arises from the fact that the solution operator
for (3.2) is not sufficiently smooth
as to give an appropriate form of first order necessary conditions:
cf., Lions [19], Lions, Magenes [20], Chapter 2.
A weak formulation of first
order necessary conditions for *linear* elliptic equations
may be found in Bergounioux, Kunisch [3].
Instead of trying to prove first order necessary conditions
under strong assumptions
we content ourselves with deriving first order
necessary conditions in a purely formal way.
This form of the necessary conditions will be
justified by its analogy in
the first order necessary conditions for the *discretized version*
of the elliptic problem; cf. Section 4.2.

Denote an optimal solution of problem (3.1)-(3.4)
by and . The *active sets*
of inequality constraints (3.3) and (3.4) are

We require the following regularity conditions:

Let and be multipliers associated with the elliptic
equation and the Dirichlet boundary
condition in (3.2), and let resp. be the
multiplier resp. the Borel measure associated with the control and state inequality constraints
(3.3) and (3.4). Then the Lagrangian for
problem (3.1)-(3.4) becomes

The first order necessary condition with respect to the state variable gives

Function evaluation along a stationary solution will be denoted henceforth by a bar, e.g., etc. Using partial integration (Green's theorem),

we rewrite as

which holds for all functions . From this we obtain the

Moreover, we find the multiplier on . Then the optimality condition for the control variable is evaluated as

which yields the

The

The Borel measure has a decomposition similar to the one in (2.11).

Note that the minimum condition (3.8) agrees with the condition (2.9) if we replace formally by . Hence, assuming functionals of tracking type (2.12), box constraints (2.13) for control and state and the property , we obtain the counterparts of the control laws (2.16), (2.17).

**Case
**: the control is determined by

**Case **: the optimal control is of *bang-bang*
or *singular* type

2002-11-25