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# 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

 (3.1)

subject to the state equation
 (3.2)

and the inequality constraints on control and state
 (3.3) (3.4)

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:
 (3.5)

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 adjoint equations:
 (3.6) (3.7)

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

which yields the minimum condition:
 (3.8)

The complementarity condition is similar to (2.10):
 (3.9)

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

 (3.10)

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

 (3.11)

Thus, for the outward normal derivative on the boundary plays the role of a switching function. The isolated zeros of are the switching points of a bang-bang control.

Next: Discretization and optimization techniques Up: paper84 Previous: Elliptic control problems with
Hans D. Mittelmann
2002-11-25