Next: Elliptic control problems with Up: paper84 Previous: Introduction

Elliptic control problems with Neumann boundary conditions

The following elliptic control problem with control and state constraints constitutes a generalization of elliptic problems considered in Casas [9], Casas et al. [10,11], Ito, Kunisch [16], Kunisch, Volkwein [18]. The problem is to determine a control that minimizes the functional

 (2.1)

subject to the state equation
 (2.2)

and the inequality constraints on control and state
 (2.3) (2.4)

In this setting, is a bounded domain with piecewise smooth boundary . The derivative in the direction of the outward unit normal of is denoted by in (2.2). Note that the state inequality constraints (2.4) are supposed to hold on the closure of .

The Laplacian in (2.2) can be replaced by an elliptic operator

where the coefficients satisfy the following coercivity condition with some :

However, in the sequel we restrict the discussion to the operator which simplifies the form of the necessary conditions and the presentation of the numerical approach in section 4. The functions and are assumed to be -functions. It is straightforward to include more than one inequality constraint into (2.3) or (2.4). However, since both the state and control variable are scalar variables, the active sets for different inequality constraints are disjoint and hence can be treated separately.

Then under appropriate assumptions on the function it can be shown that the state equation (2.2) admits for each a weak solution (cf. Casas et al. [10]), i.e., it holds

for all An optimal solution of problem (2.1)-(2.4) will be denoted by and . From [10] we infer the further assumption that the function in the Neumann condition (2.2) is sufficiently smooth and satisfies the following inequality with suitable ,
 (2.5)

Questions of existence of optimal solutions will not be discussed here. The active sets for the inequality constraints (2.3) and (2.4) are defined by

It is required that the following regularity conditions hold:
 (2.6)

Here and in the following, partial derivatives are denoted by subscripts.

First order optimality conditions for a local optimal solution and can be derived by generalizing the line of proof in Casas [9], Casas et al. [10,11]. Problem (2.1)-(2.4) is considered as a mathematical programming problem in Banach spaces to which the first order Kuhn-Tucker conditions are applicable. In particular, this approach requires that the regularity condition given in Zowe, Kurcyusz [24] is satisfied; cf. Casas et al. [11]. We do not discuss this regularity condition in detail although condition (2.6) forms part of it. The first order necessary conditions imply that there exist an adjoint state , a multiplier , and a bounded Borel measure on such that the following three conditions hold,

 (2.7) (2.8)

2. minimum condition on :
 (2.9)

3. complementarity condition:
 (2.10)

The adjoint equations (2.7), (2.8) are understood in the weak sense, cf. Casas et al. [11]. According to Bourbaki [6], Chapter 9, the bounded Borel measure appearing in the adjoint equation (2.7), (2.8) has the decomposition
 (2.11)

where represents the Lebesgue measure and is singular with respect to ; the functions are measurable on . The problem of obtaining the decomposition (2.11) explicitly is related to the difficulty of determining the structure of the active set . To our knowledge, the literature does not contain any numerical examples where the decomposition (2.11) has actually been computed. In section 4, we shall make an attempt to approximate the measure by the multipliers of the discretized problem.

In later applications, we shall mostly deal with cost functionals of tracking type (cf. Ito, Kunisch [16]),

 (2.12)

with given functions , and a nonnegative weight The control and state constraints (2.3) and (2.4) are supposed to be box constraints of the simple type
 (2.13)

with functions and . For these data the adjoint equation (2.7), (2.8) become
 (2.14)

If the function in (2.2) has the form , i.e., if holds, then the minimum condition (2.9) reduces to
 (2.15)

Case : condition (2.15) shows that the control is given by the projection of on the interval which can be stated more explicitly as
 (2.16)

Case : we obtain an optimal control of bang-bang or singular type:
 (2.17)

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

Next: Elliptic control problems with Up: paper84 Previous: Introduction
Hans D. Mittelmann
2002-11-25