### Abstract

In this paper we study an optimal control problem for a nonlinear monotone Dirichlet
problem where the control is taken as

**MSC: **
35J70, 49J20, 49J45, 93C73.

##### Keywords:

nonlinear Dirichlet problem; control in coefficients; optimality conditions### 1 Introduction

The aim of this paper is to derive a first-order optimality system for a nonlinear
Dirichlet optimal control problem where the control is taken as an
*y* is the solution of a nonlinear Dirichlet problem by choosing an appropriate coefficient

subject to the constraints

where

Optimal control in coefficients for partial differential equations is a classical
subject initiated by Lurie [1,2], Lions [3], Zolezzi [4]. Tartar and Murat [5-7] showed examples of the non-existence for such problems (see *e.g.*[8] also for the historical development). Since the range of OCPs in coefficients is
very wide, including as well optimal shape design problems, optimization of certain
evolution systems, some problems originating in mechanics and others, this topic has
been widely studied by many authors. In particular, it leads to the possibility to
optimize material properties what is extremely important for material sciences. The
crucial point is to give the right interpretation of the optimal coefficients in the
context of applications (see, for instance, [9,10]). Usually this aspect is closely related with the structural assumptions that have
to be considered during the optimization process in terms of constraints. One way
of doing so is via proper parametrization of the material, respectively, the coefficients,
using mixtures, represented by characteristic functions. This has been pursued by
Allaire [11] and many other authors in recent years. Another restriction can be realized via regularity
of the coefficients and hard constraints. This procedure has been followed first by
Casas [12] for a scalar problem, as one of the first papers in that direction, and later by
Haslinger *et al.*[13] in the context of what has come to be known as *Free Material Optimization* (FMO). However, most of the results and methods rely on linear PDEs, while only very
few articles deal with nonlinear problems, see Kogut [14] and Kogut and Leugering [15]. Another point of interest is degeneration in the coefficients which is typically
avoided by assuming lower bounds on the coefficients. However, degeneration occurs
genuinely in topology optimization, damage and crack problems. In Kogut and Leugering
[16-18] and in Kupenko and Manzo [19] this problem has been considered in the context of linear problems (see also [20]). The nonlinear case was considered in [21-24]. In this article, we extend our results to scalar nonlinear problems, where degeneration
occurs already with respect to the states.

Another important point, arising after the solvability of the optimization problem
had been proved, is the question as regards optimality conditions. The classical approach
to deriving such conditions is based on the Lagrange principle. However, in the case
when the control is considered in the coefficients of the main part of the state equation,
the classical adjoint system often cannot be directly constructed due to the lack
of differential properties of the solution to the boundary value problem with respect
to control variables. It was the main reason why Serovajskiy has proposed the concept
of the so-called quasi-adjoint system [25] and showed that optimality conditions for the linear elliptic control problem in
coefficients can be derived, provided the mapping

The paper is organized as follows. In Section 2 we give some preliminaries and prescribe
the class of admissible controls to problem (1.1)-(1.4). In Section 3 we analyze the
solvability properties of optimal control problem (1.1)-(1.4), using the monotonicity
of generalized

where

and show that it admits the Gâteaux derivative with respect to the so-called non-degenerate
directions
*y*.

In Section 5 we discuss the formal approach in deriving first-order optimality conditions for optimal control problem (1.1)-(1.4). In order to derive an optimality system, we apply the Lagrange principle. It is well known that the proof of this principle is different for different classes of optimal control problem (see, for instance, [10,12,23,28-30]). The complexity of this procedure significantly depends on the form of the extremal problem under consideration. The procedure is rather simple if the controllable system is described by a linear well-posed controllable boundary value problem, but it becomes much more complicated if the controllable system is either ill-posed or nonlinear and singular.

With that in mind, we introduce the notion of a quasi-adjoint state

where the degeneration occurs in a natural way with respect to the states.

This concept was proposed for linear problems by Serovajskiy [25], where it was shown that an optimality system for the optimal control problems in
coefficients can be recovered in an explicit form if the mapping
*θ* tends to zero. However, in contrast to Casas and Fernandez [29], we do not apply a perturbation of the differential operator that removes the singularity
at the origin.

In particular, following the well-known Hardy-Poincaré Inequality, we show that the
sequence of quasi-adjoint states

where

for some positive constants

Note that the fulfilment of this assumption is feasible if the matrix

### 2 Notation and preliminaries

Throughout the paper Ω is a bounded open subset of
*i.e.*

Let
*N*-dimensional Lebesgue measure.

For any vector field

where

*Functions with bounded variations*. Let

where

According to the Radon-Nikodym Theorem, if
*Df* is a measure and there exist a vector-valued function
*N*-dimensional Lebesgue measure

**Definition 2.1** A function
*i.e.*

Under the norm
*BV*-functions (see [31]).

**Proposition 2.2**

(i) *Let*
*be a sequence in*
*strongly converging to some**f**in*
*and satisfying condition*
*Then*

(ii) *for every*
*there exists a sequence*
*such that*

(iii) *for every bounded sequence*
*there exist a subsequence*, *still denoted by*
*and a function*
*such that*
*in*

*Admissible Controls and Generalized**p-Laplacian.* Let *α*, *β*, *γ*, and *m* be given positive constants such that

It is clear that

We say that a nonlinear operator
*p*-Laplacian if it has a representation

or via the pairing

It is easy to see that for every admissible control
*i.e.*

and demi-continuous, where by the demi-continuity property we mean the fulfilment
of the implication:
*p*-Laplacian

it is easy to check the validity of the following estimate:

As a result, since

Then by well-known existence results for nonlinear elliptic equations with strictly
monotone demi-continuous coercive operators (see [32,33]), one can conclude: for every

admits a unique weak solution in
*y* is the weak solution of (2.3) if

### 3 Setting of the optimal control problem

We consider the following optimal control problem:

subject to the constraints

where

Hereinafter,

**Remark 3.1** As was mentioned in the previous section, the characteristic feature of optimal control
problem (3.1)-(3.3) is the fact that the set of admissible controls

Let *τ* be the topology on the set

**Proposition 3.1***For any*
*and*
*a weak solution*
*to the variational problem* (3.2)-(3.3) *satisfies the estimate*

*Proof* To prove the proposition it is enough to put in equality (3.2) as a test function
the element *y* and then use the properties of the class

Now to get the desired estimate we divide each part of the obtain relation into

**Proposition 3.2***If*
*and*
*in*
*then*
*in*
*for any*
*and*
*in*

*Proof* Since

by Proposition 2.2(i) it follows that

Hence,

implies that

To end the proof, it is enough to note that strong convergence

that is
*u* is the weak-∗ limit for the whole sequence

**Proposition 3.3**
*is a sequentially compact subset of*
*for any*
*and it is a sequentially weakly*-∗ *compact subset of*

*Proof* Let

**Proposition 3.4***For every*
*the set* Ξ *is sequentially compact*, *i*.*e*. *for each sequence*
*it can be found a subsequence*
*such that*
*in*
*in*
*where*
*that is*,
*is a weak solution to the Dirichlet boundary value problem* (3.2) *with*

*Proof* Let

Taking into account the inequality (see [34])

and definition of the class of admissible controls

Since

Hence, passing to the limit in (3.7) as

As a result, we finally have

that is, the limit pair

Taking into account Propositions 3.1-3.3, in a similar manner to [15,21], it is easy to conclude the following existence result.

**Theorem 3.5***The optimal control problem* (3.1)-(3.3) *admits at least one solution*

### 4 Auxiliary results

The main goal of this paper is to derive the optimality conditions for optimal control
problem (3.1)-(3.3). However, we deal with the case when we cannot apply the well-known
classical approach (see, for instance, [35,36]), since for a given distribution

To begin with, we discuss the differentiable properties of the Lagrange functional associated with problem (3.1)-(3.3). Since (3.3) can be seen as a constraint, we define the Lagrangian as follows:

where

For given

Clearly, we cannot claim that
*y* does not imply, in general, the *χ*-convergence of subsets
*θ* small enough. Hence, in this case the convergence

**Definition 4.1** We say that an element

We have the following result.

**Proposition 4.2***Assume that**y**is an element of*
*such that the set*

*has zero Lebesgue measure*. *Then*
*is a regular point of the Lagrangian* (4.1) *in the sense of Definition *4.1.

*Proof* Let *h* be a given element of
*y* and ∇*h*. However, the Lebesgue Differentiation Theorem states that, given any
*y* are the Lebesgue points of
*θ* small enough, and

**Remark 4.1** It is worth noting that due to the results of Manfredi (see [37]), the assumptions of Proposition 4.2 appears natural and it is not a restrictive
supposition in practice. Indeed, following [37], we can ensure that the set
*p*-Laplace equation (a *p*-harmonic function) has zero Lebesgue measure. Moreover, it is also easy to observe
that if *y* and *v* in

We are now ready to study the differentiability properties of the Lagrangian

**Lemma 4.3***Let*
*be a given element*, *and let*
*be a regular point of the Lagrangian* (4.1). *Then the mapping*

*is Gâteaux differentiable at**y**and its Gâteaux derivative*

*exists and takes the form*

*Proof* Let

With that in mind, let us consider the vector-valued function

for which the Taylor expansion with the remainder term in the Lagrange form leads to the relation

where

Let

where the set

Following [[34], p.598] (see also [38]), we have: for any

for all

Since Gâteaux differentiability of the operator

such that

we arrive at the following obvious consequence of Lemma 4.3.

**Corollary 4.4***Let*
*and*
*be given elements*, *and let*
*be a regular point of the Lagrangian* (4.1). *Then the mapping*

*is Gâteaux differentiable at**y**and its Gâteaux derivative*,

*exists and takes the form*

**Remark 4.2** In view of the equality

the last term in (4.7) can be rewritten as follows:

Before deriving the optimality conditions, we need the following auxiliary result.

**Lemma 4.5***Let*
*and*
*be given distributions*. *Assume that each point of the segment*
*is regular for the mapping*
*Then there exists a positive value*
*such that*

*with*

*Proof* For given *u*, *μ*,
*y*, and *v*, let us consider the scalar function

To conclude the proof, it remains to take into account (4.7) and apply the Mean Value Theorem:

□

### 5 Formalism of the quasi-adjoint technique

We begin with the following assumption:

(H1) The distribution

Let

Hence, splitting it in two terms, we obtain

for all

Since the set of admissible controls

where

Using (4.7), we obtain

where

Now we introduce the concept of quasi-adjoint states that was first considered for linear problems by Serovajskiy [25].

**Definition 5.1** We say that, for given

or in the operator form

Here,

Let us assume, for a moment, that quasi-adjoint state
*λ* in (5.4) as the quasi-adjoint state, that is, we can set

Thus, in order to derive the necessary optimality conditions and provide their substantial
analysis, it remains to pass to the limit in (5.5)-(5.7) as

### 6 The Hardy Inequality and asymptotic behavior of quasi-adjoint states

The main goal of this section is to study the well-posedness of variational problem
(5.5) and describe the asymptotic behavior of its solutions as parameter *θ* tends to zero.

We begin with the following evident consequence of Proposition 3.4.

**Lemma 6.1***Assume that*
*and*
*strongly in*
*Then*, *for the corresponding solutions of boundary value problem* (3.2)-(3.3), *we have strong convergence*
*in*

Since, by the initial suppositions,

**Corollary 6.2**

(i)
*in*
*as*

(ii)
*in*
*as*

Further, we note that

(A_{1}) if

and, hence, by the Hölder Inequality,

(A_{2}) If

and, therefore,

Then, in view of Corollary 6.2 and estimates (6.2) and (6.4), we can give the following obvious conclusion.

**Corollary 6.3***For any*
*with*
*we have*
*in*
*as*

Our next intention is to study the variational problem (5.5). With that in mind, we rewrite it in the form

where

To begin with, we make use of the following observation.

**Proposition 6.4***Let*
*be an optimal pair for problem* (3.1)-(3.3). *Then*, *for any*
*and*
*we have*:

(1)
*where*
*denotes the set of all*
*symmetric matrices*, *which are obviously determined by*
*scalars*.

(2)
*for all*

*Proof* The first property is obvious. To prove the second one, it enough to take into account
the definition of the class of admissible controls

□

**Proposition 6.5**

(a)
*in*
*as*

(b)
*in*
*for any*
*and*
*in*
*as*

(c)
*in*
*as*

*Proof* Validity of assertions (a)-(b) immediately follows from Corollary 6.3 and Propositions 3.2
and 6.4. The strong convergence property in (c) is a direct consequence of the Lebesgue
Convergence Theorem. □

The following results are crucial for our further analysis.

**Lemma 6.6***Assume that*, *for given*
*and*
*Then each element**ψ**of*
*can be represented in a unique way as follows*:

*Proof* Let us fix an element

where the constant *C* comes from the Poincaré Inequality. Using the evident equality

and applying the Hölder Inequality with exponents

Thus,