Research

# On optimality conditions for optimal control problem in coefficients for Δp-Laplacian

Olha P Kupenko12 and Rosanna Manzo3*

Author Affiliations

1 Department of System Analysis and Control, National Mining University, Karl Marx av., 19, Dnipropetrovsk, 49005, Ukraine

2 Institute of Applied and System Analysis, National Technical University of Ukraine ‘Kiev Polytechnical Institute’, Peremogy av., 37, build. 35, Kiev, 03056, Ukraine

3 Dipartimento di Ingegneria dell’Informazione, Ingegneria Elettrica e Matematica Applicata, Università degli Studi di Salerno, Via Giovanni Paolo II, 132, Fisciano, 84084, Italy

For all author emails, please log on.

Boundary Value Problems 2014, 2014:72  doi:10.1186/1687-2770-2014-72

 Received: 9 January 2014 Accepted: 11 March 2014 Published: 28 March 2014

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

### Abstract

In this paper we study an optimal control problem for a nonlinear monotone Dirichlet problem where the control is taken as -coefficient of -Laplacian. Given a cost function, the objective is to derive first-order optimality conditions and provide their substantiation. We propose some ideas and new results concerning the differentiability properties of the Lagrange functional associated with the considered control problem. The obtained adjoint boundary value problem is not coercive and, hence, it may admit infinitely many solutions. That is why we concentrate not only on deriving the adjoint system, but also, following the well-known Hardy-Poincaré Inequality, on a formulation of sufficient conditions which would guarantee the uniqueness of the adjoint state to the optimal pair.

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 -coefficient in a nonlinear state equation. The optimal control problem we consider in this paper is to minimize the discrepancy , where , Ω is an open bounded Lipschitz domain in with , is a given distribution, and y is the solution of a nonlinear Dirichlet problem by choosing an appropriate coefficient of -Laplacian. Namely, we consider the following minimization problem:

(1.1)

subject to the constraints

(1.2)

(1.3)

(1.4)

where is a class of admissible controls and , .

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 possesses the weakened continuity property. However, the verification of this property is not easy matter even for linear systems. In the case of quasi-linear or nonlinear state equations, we are faced with another problem - the Lagrange functional to the indicated problem is not Gâteaux differentiable at the origin. To overcome this difficulty, Casas and Fernández introduced the special family of perturbed optimal control problems and derived the optimality conditions passing to the limit in optimality conditions for approximating control problems. In order to apply this approach to optimal control problem (1.1)-(1.4) it would suffice to assume the following extra conditions: and that look rather restrictive from physical point of view. The second option coming from the approach of Casas and Fernández is the fact that the linear elliptic equation for the adjoint state is not coercive in general and, hence, the adjoint boundary value problem may admit infinitely many solutions. As a result, the attainability of some solutions is rather a questionable matter. That is why in this paper we concentrate not only on deriving of the adjoint system, but also on formulation of sufficient conditions which would guarantee the uniqueness of the adjoint state to the optimal pair.

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 -Laplacian (see for comparison [26,27]). The aim of Section 4 is to give a collection of preliminary results concerning the differentiability properties of the Lagrange functional associated with problem (1.1)-(1.4)

where

and show that it admits the Gâteaux derivative with respect to the so-called non-degenerate directions at the point 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 to an optimal solution as a solution of the following Dirichlet boundary value problem for degenerate linear elliptic equation:

(1.5)

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 possesses the so-called weakened continuity property. However, it should be stressed that the fulfilment of this property is not proved for the case of -Laplacian with and, thus, should be considered as some extra hypothesis. Moreover, from a practical point of view, the verification of the weakened continuity property for quasi-adjoint states is not an easy matter, in general. That is why, in order to derive optimality conditions in the framework of more appropriate assumptions, we provide in Section 6 the analysis of the well-posedness of variational problem (1.5) and describe the asymptotic behavior of its solutions as parameter 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 to can be defined in a unique way (as unique solutions for (1.5)) and is bounded in provided, for given distributions and with and , an optimal solution to the nonlinear Dirichlet boundary value problem (1.3)-(1.4) satisfies the properties

where

for some positive constants , , and some collection of points .

Note that the fulfilment of this assumption is feasible if the matrix has a non-degenerate spectrum for each (for the details, we refer to [17]). The main argument given in Sections 6-7 is to look for the solutions of the quasi-adjoint problem (1.5) in the form , where . As a result, we show that each of the variational problems for the corresponding quasi-adjoint states has a unique solution, and these solutions form a weakly convergent sequence in . This property suffices in order to establish that the optimality system for problem (1.1)-(1.4) remains valid even if the matrix has a degenerate spectrum.

### 2 Notation and preliminaries

Throughout the paper Ω is a bounded open subset of , . The space of distributions in Ω is the dual of the space . For real numbers , and such that , the space is the closure of in the Sobolev space , while is the space of distributions of the form , with (i.e. is the dual space of ). As a norm in the space we can take the following one:

Let be the characteristic function of a set and let be its N-dimensional Lebesgue measure.

For any vector field , the divergence is an element of the space defined by the formula

(2.1)

where denotes the duality pairing between and , and denotes the scalar product of two vectors in .

Functions with bounded variations. Let be a function of . Define

where .

According to the Radon-Nikodym Theorem, if then the distribution Df is a measure and there exist a vector-valued function and a measure , singular with respect to the N-dimensional Lebesgue measure restricted to Ω, such that .

Definition 2.1 A function is said to have a bounded variation in Ω if . By we denote the space of all functions in with bounded variation, i.e..

Under the norm , is a Banach space. For our further analysis, we need the following properties of BV-functions (see [31]).

Proposition 2.2

(i) Letbe a sequence instrongly converging to somefinand satisfying condition. Then

(ii) for every, , there exists a sequencesuch that

(iii) for every bounded sequencethere exist a subsequence, still denoted by, and a functionsuch thatin.

Admissible Controls and Generalizedp-Laplacian. Let α, β, γ, and m be given positive constants such that and . We define the class of admissible controls as follows:

(2.2)

It is clear that is a nonempty convex subset of with empty topological interior.

We say that a nonlinear operator is the generalized p-Laplacian if it has a representation

or via the pairing

It is easy to see that for every admissible control , the operator turns out to be coercive, i.e.

and demi-continuous, where by the demi-continuity property we mean the fulfilment of the implication: strongly in implies that weakly in (see [26,32]). Moreover, p-Laplacian is a strictly monotone operator for each . Indeed, having applied the trick

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

As a result, since , it follows that

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 and , the nonlinear Dirichlet boundary value problem

(2.3)

admits a unique weak solution in . Let us recall that a function y is the weak solution of (2.3) if

(2.4)

(2.5)

### 3 Setting of the optimal control problem

We consider the following optimal control problem:

(3.1)

subject to the constraints

(3.2)

(3.3)

where and are given distributions.

Hereinafter, denotes the set of all admissible pairs to optimal control problem (3.1)-(3.3).

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 is a convex set with an empty topological interior. As we will see later on, this circumstance entails some technical difficulties in the substantiation of optimality conditions for the given problem.

Let τ be the topology on the set which we define as a product of the strong topology of and the weak topology of . Further we make use of the following results, which play a key role for the solvability of optimal control problem (3.1)-(3.3) (see [26,32] and [15,21] for comparison).

Proposition 3.1For anyand, a weak solutionto the variational problem (3.2)-(3.3) satisfies the estimate

(3.4)

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 on the left-hand side of the relation and the Cauchy-Bunjakowsky Inequality on the right-hand side. Indeed, we get

Now to get the desired estimate we divide each part of the obtain relation into and raise each side to the power . □

Proposition 3.2Ifandin, theninfor anyandin.

Proof Since in and

by Proposition 2.2(i) it follows that

Hence, . Moreover, for any , the estimate

implies that in .

To end the proof, it is enough to note that strong convergence in implies, up to a subsequence, convergence almost everywhere in Ω. Hence, by the Lebesgue Theorem, we have

that is in . Since this conclusion is true for any weakly-∗ convergent subsequence of , it follows that u is the weak-∗ limit for the whole sequence . □

Proposition 3.3is a sequentially compact subset offor any, and it is a sequentially weakly-∗ compact subset of.

Proof Let be any sequence of . Then is bounded in . As a result, the statement immediately follows from Propositions 3.2 and 2.2(iii). □

Proposition 3.4For everythe set Ξ is sequentially compact, i.e. for each sequenceit can be found a subsequencesuch thatin, in, where, that is, is a weak solution to the Dirichlet boundary value problem (3.2) with.

Proof Let be an arbitrary sequence of admissible pairs. Then Proposition 3.3 and a priori estimate (3.4) lead to the existence of a subsequence, still denoted by and a pair such that

(3.5)

(3.6)

Taking into account the inequality (see [34])

and definition of the class of admissible controls , we conclude: there exists a constant independent of such that

(3.7)

Since , , and in , it follows that

Hence, passing to the limit in (3.7) as , we arrive at a conclusion (see [26]):

As a result, we finally have

that is, the limit pair is an admissible to optimal control problem (3.1)-(3.3). The proof is complete. □

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 3.5The 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 the mapping is not Fréchet differentiable on the class of admissible controls, in general, and the class has an empty topological interior. With that in mind, we apply the so-called differentiation concept on convex sets and introduce the notion of a quasi-adjoint state to an optimal solution that was proposed for linear problems by Serovajskiy [25].

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:

(4.1)

where is a Lagrange multiplier and

For given and , let us consider the following sets:

Clearly, we cannot claim that in for some , because convergence of the sequence to ∇y does not imply, in general, the χ-convergence of subsets to as [[15], p.218]. Indeed, let be such that almost everywhere in Ω and . Then whereas for all positive θ small enough. Hence, in this case the convergence fails. In view of this, we make use of the following notion (see [23]).

Definition 4.1 We say that an element is a regular point for the Lagrangian (4.1) if for each the direction is non-degenerate in the following sense:

(4.2)

We have the following result.

Proposition 4.2Assume thatyis an element ofsuch that the set

(4.3)

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

Proof Let h be a given element of . If , then, by definition, . Thus, there is a value such that for all . It is worth to note that this pointwise inequality makes a sense if only is a Lebesgue point of both ∇y and ∇h. However, the Lebesgue Differentiation Theorem states that, given any , almost every is a Lebesgue point. Hence, almost all Lebesgue points of ∇y are the Lebesgue points of for θ small enough, and for all . Since the set has zero Lebesgue measure, it follows that and for small enough. Therefore, almost everywhere in Ω and hence, strongly in . □

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 for non-constant solutions of the p-Laplace equation (a p-harmonic function) has zero Lebesgue measure. Moreover, it is also easy to observe that if y and v in are two regular points of the functional , then there exists a positive number () such that each point of the segment is also regular for .

We are now ready to study the differentiability properties of the Lagrangian . We begin with the following result.

Lemma 4.3Letbe a given element, and letbe a regular point of the Lagrangian (4.1). Then the mapping

is Gâteaux differentiable atyand its Gâteaux derivative

exists and takes the form

(4.4)

Proof Let be a regular point for the Lagrangian (4.1) and let be an arbitrary distribution. Following the definition of the Gâteaux derivative, we have to deduce the following equality:

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 and

Let be an arbitrary value. Let us consider the following decomposition:

where the set is defined by (4.3), and and are measurable subsets of Ω such that

Following [[34], p.598] (see also [38]), we have: for any there exists a positive value such that

(4.5)

(4.6)

for all , , and small enough. Since by the initial assumptions, it follows from (4.5)-(4.6) that the vector-valued function is Gâteaux differentiable. Hence, the operator is Gâteaux differentiable for any regular point and for any admissible control , and its Gâteaux derivative takes the form (4.4). □

Since Gâteaux differentiability of the operator implies the existence of Gâteaux derivative for the functional

such that

we arrive at the following obvious consequence of Lemma 4.3.

Corollary 4.4Letandbe given elements, and letbe a regular point of the Lagrangian (4.1). Then the mapping

is Gâteaux differentiable atyand its Gâteaux derivative,

exists and takes the form

(4.7)

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 4.5Let, , andbe given distributions. Assume that each point of the segmentis regular for the mapping. Then there exists a positive valuesuch that

(4.8)

with.

Proof For given u, μ, , y, and v, let us consider the scalar function . Since by Corollary 4.4, the functional is Gâteaux differentiable at each point of the segment , it follows that the function is differentiable on and

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 is such that, for each admissible control , the corresponding weak solution of the nonlinear Dirichlet boundary value problem (2.3) is a regular point of the mapping .

Let be an optimal pair for problem (3.1)-(3.3). Then

(5.1)

Hence, splitting it in two terms, we obtain

(5.2)

for all and such that .

Since the set of admissible controls has an empty topological interior, we justify the choice of perturbation for an optimal control as follows:

where is an arbitrary admissible pair, and . Then due to Hypothesis (H1) and Remark 4.1, we can suppose that each point of the segment is regular for the mapping , where is the corresponding solution of boundary value problem (3.2)-(3.3). Hence, by Lemma 4.5, there exists a value such that condition (5.1) can be represented as follows:

(5.3)

Using (4.7), we obtain

(5.4)

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 and , a distribution is the quasi-adjoint state to if satisfies the following integral identity:

(5.5)

or in the operator form

(5.6)

Here, is the identity matrix, is the solution of problem (3.2)-(3.3), , and is a constant coming from equality (5.3).

Let us assume, for a moment, that quasi-adjoint state to is a distribution of . Then we can define the element λ in (5.4) as the quasi-adjoint state, that is, we can set . As a result, the increment of Lagrangian (5.4) can be simplified to the form

(5.7)

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 , and to show that the sequence of quasi-adjoint states is defined in a unique way through relation (5.5) and it is compact with respect to the weak topology of .

### 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 6.1Assume thatandstrongly in. Then, for the corresponding solutions of boundary value problem (3.2)-(3.3), we have strong convergencein.

Since, by the initial suppositions, in as , we immediately arrive at the following consequence of Lemma 6.1.

Corollary 6.2

(i) inas;

(ii) inas.

Further, we note that

(A1) if , where , and , then

(6.1)

and, hence, by the Hölder Inequality,

(6.2)

(A2) If and then

(6.3)

and, therefore,

(6.4)

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

Corollary 6.3For anywith, we haveinas.

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

(6.5)

where

(6.6)

(6.7)

(6.8)

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

Proposition 6.4Letbe an optimal pair for problem (3.1)-(3.3). Then, for anyand, we have:

(1) , wheredenotes the set of allsymmetric matrices, which are obviously determined byscalars.

(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 and the following chain of estimates:

□

Proposition 6.5

(a) inas;

(b) infor anyandinas;

(c) inas.

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 6.6Assume that, for givenand, . Then each elementψofcan be represented in a unique way as follows:

(6.9)

Proof Let us fix an element . Then for , by the Hölder Inequality with and , we have

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

and applying the Hölder Inequality with exponents and for the first term and with and for the second one, we get

Thus, . Since the element inherits the trace properties along Ω from its parent element ψ, we finally obtain . The proof is complete. □

As an obvious consequence of this result and continuity of the embedding of Sobolev spaces , we can give the following conclusion.

Corollary 6.7If, for givenand, , then there exists a dense subsetofsuch that

(6.10)

Taking this fact into account, it is plausible to introduce the following linear mapping:

(6.11)

Since domain of is dense in Banach space , it follows that for , as for a densely defined operator, there exists an adjoint operator

such that

where

Notice that, in general, the adjoint operator is not densely defined.

Let us consider the following linear operator:

where and are defined by (6.6)-(6.7). By Proposition 6.4, we have

Hence, and this operator is obviously monotone,

and demi-continuous. However, because of the multiplier , this operator can lose the coercivity property.

Let λ be a positive constant such that . Let be a given collection of points. Let be a given control. We define a nonempty subset as follows: if and only if

(6.12)

for some positive constant , where

(6.13)

(6.14)

We are now in a position to give an important property of the operator .

Lemma 6.8Assume that, for given, , and, the distributionis such that

Then

(6.15)

where

(6.16)

(6.17)

and the linear operatordefines an isomorphism frominto its dual.

Proof Let v and z be arbitrary elements of . Then by Corollary 6.7, we have . Further, following the definition of the operator , we can provide the following chain of transformations:

Hence,

To conclude the proof it remains to show that operator defines an isomorphism from into its dual . With that in mind, we make use of the following version of the Hardy-Poincaré Inequality: for a given internal point there exists a constant such that for every

where and .

According to this result and the fact that , we have

(6.18)

and, therefore,

(6.19)

Thus, in view of (6.18)-(6.19),

is equivalent to the standard norm of , and therefore, the operator given by (6.16) defines an isomorphism from into its dual . □

The next step of our analysis is to show that, for every , , and , the quasi-adjoint state to can be defined as a unique solution to the Dirichlet boundary value problem (6.5). With that in mind, we make use of the following hypothesis.

(H2) For a given distribution with and , the weak solutions of the nonlinear Dirichlet boundary value problem (2.3) satisfy the property: and for every pair of admissible controls .

Lemma 6.9Assume that the Hypothesis (H2) is valid. Then the Dirichlet boundary value problem (6.5) has a unique solutionfor every, , and.

Proof Let be an optimal pair to problem (3.1)-(3.3). Let be the solution of problem (3.2)-(3.3) for given , , and let , where takes an arbitrary value in . Since, by the initial assumptions, , it follows that

for all . Then Lemma 6.8 implies that

provided . By the Hardy-Poincaré Inequality (see (6.19)), the expression

can be considered as a scalar product in . Then, in view of the Riesz Representation Theorem, we conclude to the existence of a unique element such that

As a result, we have: is a unique solution to the Dirichlet boundary value problem (2.3). Moreover, by Corollary 6.7, we finally get . □

In view of Lemma 6.9, it makes a sense to accept the following hypothesis.

(H3) For given distributions and with and , the weak solution to the nonlinear Dirichlet boundary value problem (2.3) satisfies the properties

Lemma 6.10Assume that Hypotheses (H2)-(H3) are valid. Then there exists a constantindependent ofθsuch that, i.e. the sequence of quasi-adjoint statestois relatively compact with respect to the weak convergence of.

Proof Due to Hypothesis (H2) and Lemma 6.9, the sequence of quasi-adjoint states to can be defined in a unique way for all , where is the solution of the following Dirichlet problem:

(6.20)

Therefore, in order to prove this lemma, it is enough to show that the sequence is uniformly bounded in . To this end, we note that the integral identity (6.20) leads to the corresponding energy equality

(6.21)

Since

and

it follows that there exists a constant independent of θ such that

As a result, in view of (6.19), the energy equality (6.21) immediately leads to the estimate.

This concludes the proof. □

### 7 Optimality conditions

We are now in a position to derive the first-order optimality conditions for optimal control problem (3.1)-(3.3).

Theorem 7.1Let us suppose that, , andare given with. Letbe an optimal pair to problem (3.1)-(3.3). Letbe a quasi-adjoint state todefined for eachandby (5.5). Then Hypotheses (H1)-(H3) imply the existence of an elementsuch that (within a subsequence) inas, and

(7.1)

(7.2)

(7.3)

Proof Let be an admissible pair. Let be the solution of problem (3.2)-(3.3) for given and . Then, as was shown at the end of Section 5, there exists a value such that the increment of Lagrangian can be simplified to the form (5.7), provided the element λ has been defined as the quasi-adjoint state, that is, . By Lemmas 6.9-6.10, the sequence of quasi-adjoint states to can be defined in a unique way and is bounded in . Hence, there exists an element such that, up to a subsequence of , we have in as . It remains to pass to the limit in (5.5), (5.7) as and to show that in the limit we will arrive at the relations (7.3) and (7.1), respectively. To this end, we note that

(A1) by the initial suppositions, in as ;

(A2) by Corollary 6.2, in as , and, hence, in as ;

(A3) up to a subsequence

by the strong convergence of in and convergence almost everywhere in Ω;

(A4) if , then

(7.4)

(A5) if , then

(7.5)

Then the limit passage in (5.7) immediately leads to (7.1). Therefore, in order to end the proof, it remains to establish the validity of relation (7.3). With that in mind, we rewrite (5.5) as follows:

where

and φ is an arbitrary element of .

Since

let us show that (), and, hence, as .

Using the Hölder Inequality and estimate (2) in Proposition 6.4, we have

(7.6)

Therefore, if , then, by (A4) and Hölder’s Inequality with Hölder conjugates and , we can estimate (7.6) as follows:

(7.7)

Since by Lemma 6.10, and in (see the condition (A2)), it follows that

Thus, in view of estimate (7.7), we conclude: .

As for the case , the inequality (7.6) and condition (A5) lead to the estimate

Further it remains to repeat the trick like in (7.7). As a result, we obtain

Therefore, having applied the arguments given before, we can conclude: if , then .

As for the term , we have

(7.8)

To clarify the asymptotic behavior of the term as θ tends to zero, we note that

Since , , and , it follows that the inclusion holds true with . Hence, the condition is ensured by the weak convergence in .

Thus, summing up the results given above, we finally obtain

(7.9)

As for the term

we see that

Hence, strong convergence in implies strong convergence

(7.10)

As a result, we finally get

(7.11)

Thus, combining relations (7.9)-(7.11), it is easy to see that the limit passage in (5.5) leads to the variational statement of the Dirichlet boundary value problem (7.3). The proof is complete. □

### 8 Conclusions

In this paper the optimal control problem (3.1)-(3.3) for a nonlinear monotone elliptic equation with homogeneous Dirichlet conditions and -control in coefficients of -Laplacian has been studied. Having defined the class of admissible control in form (2.2), we have proved solvability of the considered problem. After that, using Lagrange principle, the concept of quasi-adjoint system and the well-known Hardy-Poincaré Inequality, we have derived the corresponding optimality system and formulated sufficient conditions under which the degenerate adjoint boundary value problem admits a unique solution.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

### References

1. Lurie, KA: Optimum control of conductivity of a fluid moving in a channel in a magnetic field. J. Appl. Math. Mech.. 28, 316–327 (1964). Publisher Full Text

2. Lurie, KA: Applied Optimal Control Theory of Distributed Systems, Plenum, New York (1993)

3. Lions, JL: Optimal Control of Systems Governed by Partial Differential Equations, Springer, New York (1971)

4. Zolezzi, T: Teoremi d’esistenza per problemi di controllo ottimo retti da equazioni ellittiche o paraboliche. Rend. Semin. Mat. Univ. Padova. 44, 155–173 (1970)

5. Murat, F: Un contre-exemple pour le problème du contrôle dans les coefficients. C. R. Acad. Sci. Paris Sér. A-B. 273, A708–A711 (1971) (in French)

6. Murat, F: Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients. Ann. Math. Pures Appl.. 112, 49–68 (1977). Publisher Full Text

7. Tartar, L: Problèmes de contrôle des coefficients dans des équations aux dérivées partielles. In: Bensoussan A, Lions JL (eds.) Control Theory, Numerical Methods and Computer Systems Modelling, Springer, Berlin (1975)

8. Tartar, L: The General Theory of Homogenization a Personalized Introduction, Springer, Berlin (2009)

9. Litvinov, VG: Optimization in Elliptic Boundary Value Problems with Applications to Mechanics, Mir, Moskow (1987)

10. Pironneau, O: Optimal Shape Design for Elliptic Systems, Springer, Berlin (1984)

11. Allaire, G: Shape Optimization by the Homogenization Method, Springer, Berlin (2002)

12. Casas, E: Optimal control in the coefficients of elliptic equations with state constraints. Appl. Math. Optim.. 26, 21–37 (1992). Publisher Full Text

13. Haslinger, J, Kocvara, M, Leugering, G, Stingl, M: Multidisciplinary free material optimization. SIAM J. Appl. Math.. 70(7), 2709–2728 (2010). Publisher Full Text

14. Kogut, OP: Qualitative analysis of one class of optimization problems for nonlinear elliptic operators. PhD thesis at Gluskov Institute of Cybernetics NAS, Kiev (2010)

15. Kogut, PI, Leugering, G: Optimal Control Problems for Partial Differential Equations on Reticulated Domains. Approximation and Asymptotic Analysis, Birkhäuser, Boston (2011)

16. Kogut, PI, Leugering, G: Optimal -control in coefficients for Dirichlet elliptic problems: W-optimal solutions. J. Optim. Theory Appl.. 150(2), 205–232 (2011). Publisher Full Text

17. Kogut, PI, Leugering, G: Optimal -control in coefficients for Dirichlet elliptic problems: H-optimal solutions. Z. Anal. Anwend.. 31(1), 31–53 (2012)

18. Kogut, PI, Leugering, G: Matrix-valued -optimal controls in the coefficients of linear elliptic problems. Z. Anal. Anwend.. 32(4), 433–456 (2013). Publisher Full Text

19. Kupenko, OP, Manzo, R: On an optimal -control problem in coefficients for linear elliptic variational inequality. Abstr. Appl. Anal. (2013). Publisher Full Text

20. D’Apice, C, De Maio, U, Kogut, PI, Manzo, R: On the solvability of an optimal control problem in coefficients for ill-posed elliptic boundary value problems. Adv. Differ. Equ. (to appear)

21. D’Apice, C, De Maio, U, Kogut, PI: On shape stability of Dirichlet optimal control problems in coefficients for nonlinear elliptic equations. Adv. Differ. Equ.. 15(7-8), 689–720 (2010)

22. D’Apice, C, De Maio, U, Kogut, PI: Optimal control problems in coefficients for degenerate equations of monotone type: shape stability and attainability problems. SIAM J. Control Optim.. 50(3), 1174–1199 (2012). Publisher Full Text

23. Kogut, PI, Kupenko, OP, Leugering, G: Optimal control problems in coefficients for nonlinear Dirichlet problems of monotone type: optimality conditions. Z. Anal. Anwend. (to appear)

24. Kupenko, OP: Optimal control problems in coefficients for degenerate variational inequalities of monotone type. I. Existence of solutions. J. Comput. Appl. Math.. 106(3), 88–104 (2011)

25. Serovajskiy, SY: Variational inequalities in non-linear optimal control problems. Methods Means Math. Model.. 1, 156–169 (1977)

26. Lions, JL: Some Methods of Solving Non-Linear Boundary Value Problems, Gauthier-Villars, Paris (1969)

27. Ju, E-Y, Jeong, J-M: Optimal control problems for hyperbolic equations with damping terms involving p-Laplacian. J. Inequal. Appl.. 2013, (2013) Article ID 92

28. Casado-Diaz, J, Couce-Calco, J, Martín-Gomez, J: Optimality conditions for non-convex multistate control problems in the coefficients. SIAM J. Control Optim.. 43(1), 216–239 (2004). Publisher Full Text

29. Casas, E, Fernandez, LA: Optimal control of quasilinear elliptic equations with non differentiable coefficients at the origin. Rev. Mat. Univ. Complut. Madr.. 4(2-3), 227–250 (1991)

30. Krumbiegel, K, Rehberg, J: Second order sufficient optimality conditions for parabolic optimal control problems with pointwise state constraints. SIAM J. Control Optim.. 51(1), 304–331 (2013). Publisher Full Text

31. Evans, LC, Gariepy, RF: Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton (1992)

32. Gajewski, H, Gröger, K, Zacharias, K: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie Verlag, Berlin (1974)

33. Ivanenko, VI, Mel’nik, VS: Variational Methods in Control Problems for Systems with Distributed Parameters, Naukova Dumka, Kiev (1988)

34. Al’shin, AB, Korpusov, MO, Sveshnikov, AG: Blow-up in Nonlinear Sobolev Type Equations, de Gruyter, Berlin (2011)

35. Fursikov, AV: Optimal Control of Distributed Systems. Theory and Applications, Am. Math. Soc., Providence (2000)

36. Ioffe, A, Tikhomirov, V: Extremal Problems, North-Holland, Amsterdam (1979)

37. Manfredi, JJ: p-Harmonic functions in the plane. Proc. Am. Math. Soc.. 103, 473–479 (1988)

38. Drabek, P, Milota, Y: Methods of Nonlinear Analysis. Applications to Differential Equations, Birkhäuser, Berlin (2007)