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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 821964, 13 pages
On an Optimal -Control Problem in Coefficients for Linear Elliptic Variational Inequality
1Department of System Analysis and Control, National Mining University, Karl Marx Avenue 19, Dnipropetrovsk 49005, Ukraine
2The Research Laboratory for Nonlinear Analysis of Differential-Operator Systems of Institute for Applied and System Analysis in National Technical University of Ukraine “Kiev Polytechnical Institute”, Peremogy Avenue 37, building 35, Kiev 03056, Ukraine
3Dipartimento di Ingegneria dell’Informazione, Ingegneria Elettrica e Matematica Applicata, Università degli Studi di Salerno, Via Giovanni Paolo II 132, 84084 Fisciano, Italy
Received 27 January 2013; Accepted 24 June 2013
Academic Editor: Matti Lassas
Copyright © 2013 Olha P. Kupenko and Rosanna Manzo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider optimal control problems for linear degenerate elliptic variational inequalities with homogeneous Dirichlet boundary conditions. We take the matrix-valued coefficients in the main part of the elliptic operator as controls in . Since the eigenvalues of such matrices may vanish and be unbounded in , it leads to the “noncoercivity trouble.” Using the concept of convergence in variable spaces and following the direct method in the calculus of variations, we establish the solvability of the optimal control problem in the class of the so-called -admissible solutions.
The aim of this work is to study the existence of optimal controls in the matrix-valued coefficients associated with a linear degenerate elliptic variational inequality with homogeneous Dirichlet boundary conditions. The controls are taken as the matrix of the coefficients in the main part of the elliptic operator. The most important feature of such controls is the fact that the eigenvalues of the matrix may either vanish on subsets with zero Lebesgue measure or be unbounded. In this case, the precise answer for the question of existence or nonexistence of optimal solutions heavily depends on the class of chosen admissible controls. Using the direct method in the calculus of variations, we discuss the solvability of this optimal control problem in the class of the so-called -admissible solutions (see, for instance, [1, 2]). It should be emphasized that in contrast to the paper , we do not make use of any relaxations for the original optimal control problem. We note that there are many physical phenomena related to the mathematical theory of cloaking, design of advanced materials, behaviour of the optical meta-materials in the context of cloaking, and others which lead to the appearance of variational inequalities and matrices in the main part of elliptic operators with degenerate spectrum (see, for instance, [4–8]). In particular, the mathematical models of equilibrium of continuous media which are “perfect” insulators or “perfect” conductors (see ) need eigenvalues of the matrix either to vanish somewhere or to be unbounded.
Also similar matrices with degenerate eigenvalues arise in an intensively studied nowadays area of complex dynamical networks [10–12]. Namely, such matrices describe an inner configuration of networks; that is, they can be interpreted as connection matrices among nodes [10, 11]. In fact, many properties of complex networks are mainly determined by their configuration matrices, in particular, such as network synchronizability as well as robustness and fragility of a network in synchronization. This fact could be a good motivation to work out techniques for achieving synchronization for dynamical networks which cannot synchronize, taking configuration matrices as controls. Such situation would be similar to the one described in the present paper, when varying a chosen control leads to changing a control object’s structure. Hence, one of the main goals of the paper is to construct a mathematical framework for dealing with problems of this kind.
It is worth noticing that even though numerous articles (see, for instance, [13–18] and references therein) are devoted to variational and nonvariational approaches to problems related to linear and nonlinear elliptic systems, only few deal with optimal control problems for degenerate partial differential equations and variational inequalities (see e.g., [1, 2, 19–23]).
In this paper, we deal with an optimal control problem in coefficients for the following linear degenerate elliptic variational inequality: Here,is a given distribution,is a convex closed subset of the space, andis a measurable nonnegative square symmetric matrix on a bounded open domain() such that.
We furnish this control object by the following cost functional:
The characteristic feature of this optimal control problem is the fact that the eigenvalues of admissible matricesmay vanish and be unbounded on subdomains of with zero Lebesgue measure. Indeed, we assume that for every admissible matrixthere exist two nonnegative -functions and such that , , and Because of this, such matrices are sometimes referred to as matrices with degenerate spectrum. It is clear that conditions (3) lead us to the so-called “noncoercivity trouble.” It means that boundary value problem (1) for some locally integrable matrix-valued functionsmay exhibit the Lavrentieff phenomenon, the nonuniqueness of weak solutions, as well as other surprising consequences. So, in general, the mappingcan be multivalued. On the other hand, this problem is ill-posed, in general. It means that there are no reasons to suppose that for everyand, the problem (1) admits at least one weak solution in. Thus, it makes it impossible to apply classical theorems (see [24, 25]) to establish solvability of the corresponding variational inequality and, hence, the regularity of the optimal control problem associated with it.
Besides, we show that for every admissible control functionit is plausible to consider weak solutions of problem (1) as elements of the corresponding weighted Sobolev spaceswhich are constructed as sets of functionsfor which the norm is finite. In addition, we note that even if the original elliptic operator is nondegenerate, that is, admissible controlsare such that with, the majority of optimal control problems in coefficients have no solution in general (see for instance [15, 26–28]).
Since for atypical matrices with degenerate spectrum, namely, such that , the space of smooth compactly supported functions is not dense in in general, we discuss the solvability of this optimal control problem in the class of the so-called-admissible solutions. To this end, we consider, as the main solution space, the closure ofin-norm and denote this space by . In order to specify the setin (1), we assume that there exists a closed convex subset of such that . In this case, we set as the closure of with respect to -norm (4).
At the same time, having formulated the optimal control problem in coefficients in terms of the corresponding weighted Sobolev spaces, we are facing another kind of challenge: changing the control matrix changes not only the solution space but also the control object itself. In fact, it means that the set of admissible solutions to the previous optimal control problem is a family of pairs, each of which belongs to the correspondent space. Therefore, in this situation there are several possible settings of the variational inequality and optimal control problem associated with it, which depend on the choice of solution space. The main questions are what is the right setting of the optimal control problem with -controls in coefficients, and what is the right class of admissible solutions to the considered problem? As we show in this paper, the precise answer for the question of existence or nonexistence of optimal solutions heavily depends on the class of chosen admissible controls.
The paper is organized as follows. Section 2 concerns some notation and preliminaries. In Section 3, we describe the main notion of weak and strong convergence in variable-spaces. Section 4 gives a collection of auxiliary results that we will use in the sequel. In particular, following Kogut and Leugering , we introduce the concept of-convergence in variable space and give sufficient conditions of sequential-compactness. In Section 5, we introduce the admissible control constrains, the class of -optimal admissible solutions, and discuss the regularity of the corresponding optimal control problem. In Section 6, using the direct method in the calculus of variations, we prove the existence of the so-called -optimal solutions to the original problem.
2. Notation and Preliminaries
Let be a bounded open subset of () with Lipschitz boundary. We assume that the boundary of , denoted by , has positive -dimensional measure.
By , we denote the set of functions from , compactly supported in . We define the Banach space as the closure of in the classical Sobolev space . For any subset , we denote by its -dimensional Lebesgue measure .
Symmetric Matrices with Degenerate Eigenvalues. We denote by the set of all symmetric matrices , (). We suppose that is endowed with the Euclidian scalar product and with the corresponding Euclidian norm . Let be the space of integrable functions whose values are symmetric matrices.
Let be a fixed positive value. Let be a given function satisfying the properties Let be a nonempty compact subset of such that for any the following conditions hold true: By, we denote the set of all matricessuch that Here, is a given function such that a.e. in and such that ,is the identity matrix in , and (8)-(9) should be considered in the sense of quadratic forms. Therefore, (8)-(9) imply the following inequalities:
Remark 1. Since every measurable matrix-valued functioncan be associated with the collection of its eigenvalues, where eachis counted with its multiplicity, (9), in view of the properties of the class, means that eigenvalues of matricesmay vanish and be unbounded on subdomains ofwith zero Lebesgue measure. Because of this, these matrices are sometime referred to as matrices with degenerate spectrum.
Weighted Sobolev Spaces. To each matrix , we will associate two weighted Sobolev spaces: whereis the set of functionsfor which the norm, given by (4), is finite, andis the closure ofin. Note that due to inequality (11) and estimates the space is complete with respect to the norm . It is clear that , and , are Hilbert spaces. If the eigenvalues of are bounded between two positive constants, then it is easy to verify that . However, for a “typical” matrix the space of smooth functions is not dense in . Hence, the identity is not always valid (for the corresponding examples in the case when , we refer the reader to [30, 31]).
Weak Compactness Criterion in . Throughout the paper we will often use the concept of weak and strong convergence in. Letbe a bounded sequence of matrices in. We recall thatis called equi-integrable on, if for anythere is asuch thatfor every measurable subsetof Lebesgue measure. Then the following assertions are equivalent for-bounded sequences:(i)the sequence is weakly compact in ; (ii)the sequence is equi-integrable.
Lemma 2 (Lebesgue’s Theorem). If a sequenceis equi-integrable andalmost everywhere in , then in .
Functions with Bounded Variation. Letbe a function of. Define where.
According to the Radon-Nikodym theorem, if , then the distribution is a measure and there exist a vector-valued function and a measure , singular with respect to the -dimensional Lebesgue measure restricted to , such that
Definition 3. A function is said to have a bounded variation in if . By , we denote the space of all functions in with bounded variation.
Under the norm,is a Banach space. The following compactness result for-functions is well-known.
Proposition 4. Uniformly bounded sets in-norm are relatively compact in.
Definition 5. A sequenceweakly converges to some, and we writeif and only if the following two conditions hold:strongly in, andweakly-in the space of Radon measures; that is,
In the following proposition, we give a compactness result related to this convergence, together with lower semicontinuity (see ).
Proposition 6. Letbe a sequence instrongly converging to someinand satisfying. Then (i) and ;(ii) in .
Elliptic Variational Inequalities. Following Lions , let us cite a well-known result concerning solvability and solution uniqueness for variational inequalities which will be useful in the sequel.
Lemma 7 (see [24, Theorem 8.3]). Letbe a Banach space andbe a closed convex subset. Letbe a strictly monotone operator andbe a given element of the dual space. Then the following variational problem: to find an elementsuch that admits a unique solution provided operator is corcive in the following sense: there exists an element such that
3. -Valued Radon Measures and Weak Convergence in Variable -Spaces
By a nonnegative Radon measure onwe mean a nonnegative Borel measure which is finite on every compact subset of. The space of all nonnegative Radon measures onwill be denoted by. According to the Riesz theory, each Radon measurecan be interpreted as an element of the dual space to spaceof all continuous functions with compact support. Letdenote the space of all-valued nonnegative Radon measures. Then,.
Let and be matrix-valued nonnegative Radon measures. We say that weakly-converges to in if A typical example of such measures is where or As we will see later (see Lemma 15), the setsare sequentially closed with respect to strong convergence in.
In this section, we suppose that the measures and are defined by (21) and in . Further, we will use to denote the set of measurable vector-valued functions on such that As follows from estimate (14), any vector-valued function ofis Lebesgue integrable on.
We say that a sequence is bounded if
Definition 8. Let and be matrices satisfying conditions (22). A bounded sequence is weakly convergent to a function in the variable space if
The main property concerning the weak convergence incan be expressed as follows (see for comparison ).
The next property of weak convergence inis the lower semicontinuity of the variable-norm.
Proposition 10 (see ). If the sequence converges weakly to , then
Definition 11. A sequence is said to be strongly convergent to a function if whenever in as . We have the following property of strong convergence in the variable -spaces.
Proposition 12 (see ). Weak convergence of a sequence to and is equivalent to strong convergence of in to .
4. Auxiliary Results
Lemma 13 (see ). Letbe any sequence in. Then, there is an elementsuch that, within a subsequence of, we have
For our further analysis, we make use of the following concept.
Definition 14. We say that a bounded sequence -converges toasif therefore,
In order to motivate this definition, we give the following result.
Lemma 15. Let be a sequence such that(i)the sequence is bounded; that is, (ii)and there exists a matrix-valued functionsuch that Then, ; and the original sequence is relatively compact with respect to-convergence. Moreover, each-limit pairbelongs to the space.
Proof. We note that (36)-(37), (4), and (13)-(14) immediately imply the boundedness of the original sequence in. Moreover, due to (37), we have
Thus, the compactness criterium for weak convergence in variable spaces (see Proposition 9) and (36) imply the existence of a pair such that, within a subsequence of, Our aim is to show that,, and. It is clear thatand this matrix satisfies (8). Sincefor all, it follows that there is a sequenceinsuch that Then, by-compactness of the set, there exists an elementsuch thatinas. Moreover, Lemma 13 implies strong convergence and (7). Hence, passing to the limit in (41) as, we come to (9). Thus,and the limit matrixsatisfies (10)-(11).
For our further analysis, we fix any test function and make use of the following equality: which is obviously true for eachand for all. Since it follows that the sequence is bounded. Consequently, combining this fact with (43), we conclude that in the variable space (see Definition 8). At the same time, strong convergence in (37) implies the relation Hence (see Proposition 12), Further, we note that for every measurable subset , the estimate implies equi-integrability of the family . Hence, is weakly compact in , which means the weak compactness of the vector-valued sequence in . As a result, by the properties of the strong convergence in variable spaces, we obtain Thus, in view of the weak compactness property ofin, we conclude that Since for all and the Sobolev space is complete, (39) and (49) imply , and consequently . It should be also observed that (39)-(40) guarantee the finiteness of the norm (see (4)), hence (to elaborate Propositions 9, 10, 12 and Lemma 13, for the sake of reader’s convenience, we closely followed the idea of ). However, to show that , we must be sure that there exists a sequence of smooth functions from strongly converging in to this element.
Since for all , , there exist sequences , such that strongly in as . Let us show that the sequence , chosen with respect to diagonalization procedure, converges weakly to in . To this end, it is enough to prove that in and in .
Indeed, for an arbitrary , we have since in , as and for any there exists such that for all . Similarly, for an arbitrary , we have It is only the convergence to zero of the third summand that should be explained here: Obviously,as. It is left to prove, that. Indeed, since as and is a bounded sequence in .
Hence, we proved the existence of the sequence from which is weakly convergent in , to the element . However, it is well known that due to Mazur’s lemma in there exists a sequence, strongly convergent to , which is constructed as a certain linear combinations of elements from . Linear combinations of smooth functions are smooth as well. Hence, as the strong limit of the sequence from , the element belongs to the space .
5. Setting of the Optimal Control Problem
Letbe a given constant matrix satisfying the condition
Letbe a convex closed subset of the space such that, and letbe a given distribution. Having fixed, we denote bythe closure ofwith respect to the norm, and consider the following elliptic variational inequality: to find an elementsuch that
Remark 16. The distinctive feature of this variational problem is the following fact: choosing a matrix of coefficients, we choose a space in which the problem is considered; or, speaking in terms of optimization problem in coefficients for given inequality, changing a control matrix, we change not only the state or even the space of states but also the control object itself.
To introduce the class of admissible controls in coefficients, we adopt the following concept.
Definition 17. We say that a matrix-valued functionis an admissible control for variational inequality (55) (it is written as) if Hereinafter, we assume that the setis nonempty.
The optimal control problem we consider here is to minimize the discrepancy (tracking error) between a given distribution and the solution of elliptic variational inequality (55)-(56) by choosing an appropriate coefficients matrix. More precisely, we are concerned with the following optimal control problem: subject to the constraints (55)-(56).
Remark 18. The second term in (57) plays a special role in this problem. Its appearance in the cost function (57) is motivated by the fact that there are no appropriate a priori estimates in the-norm for solutions of problem (55)-(56). Hence, the term together with the first one in (57) ensures the coercivity of the cost function.
We can indicate the set of admissible solutions to the original optimal control problem as follows:
To discuss solvability of variational inequality (55) and uniqueness of its solution, we make use of the following results.
Lemma 20. For every fixed admissible control, the operator, defined as is strictly monotone and coercive.
Proof. Letbe a fixed matrix. Then the relation implies the strict monotonicity and coercivity property of the operator. Indeed, by setting in (60),, we immediately obtain
Proof. Taking into account Lemmas 7 and 20, it is enough to show that there exists an elementsuch that In order to do this, we fix an arbitrary elementand a matrix. Then for all, in view of estimate (60), we have where Combining (63) and (64) we come to the inequality Hence, problem (55)-(56) satisfies all conditions of Lemma 7. Therefore, for every control and every , the considered problem has a unique solution.
Hypothesis 1. For a given set , we have the following: for all , for all , and for all there exists an element such that
Then, as a direct consequence of Theorem 8.7 in  and Proposition 21, we come to the following result.
As an example of the setsatisfying the Hypothesis 1, we propose the following one.
Example 23. Letbe a given function such that
Let us define the closed convex setas follows.
Then, for an arbitrary matrix, we have
Let us show that for all , for all there exists an element satisfying the following conditions:
As a result, we would have
and having set , we come to the equation
Obviously, the mappingis strictly monotone and coercive (see Proposition 21). Hence, there exists a unique solutionto (73). Moreover, in this case.
Let us show that, namely,almost everywhere in. In order to do this, we set. Then it is enough to prove that. From (72) it follows that Having noticed, we conclude that Sinceandby the initial assumptions, we have ; that is, At the same time, since, thenand, hence,on. As a result, we haveon. Therefore, Thus, we arrive at the following conclusion: eitherora.e. on. Hence,and this concludes the proof.
6. Existence of Optimal Solutions
Since our prime interest is the solvability of optimal control problem (55)-(56), (57), we begin with the study of the topological properties of the set of admissible solutions. To do so, we give some auxiliary results.
Definition 24. We say that a sequenceis bounded if
Lemma 25. Letbe a bounded sequence in the sense of Definition 24. Then there exists a pairsuch that, up to a subsequence,
Proof. By the compactness of -functions (see Proposition 6), there exist a subsequence of , still denoted by the same indices, and a matrix such that in . Thus,
and the condition (32) of Definition 14 holds true. In order to check the remaining conditions (33)-(34) of this definition and to show that, we make use of the following observation.
We have for all . Hence, there is a sequence in such that (see Lemma 13 for the details) and in as . Moreover, by properties of the set , the -limit element satisfies (7). Then, in view of -convergence , we may assume that almost everywhere in . Since a.e. in , it follows that for any subset. Hence, due to the strong-convergence, the sequenceis equi-integrable. Then, by Lebesgue’s Theorem (see Lemma 2) we obtaininas. As a result, by Lemma 15. Combining this fact with properties (80), we conclude.
It remains to notice that conditions (33)-(34) of Definition 14 and for the -limiting component of the sequence are ensured by Lemma 15. This concludes the proof.
Theorem 26. For any admissible initial data, the set of admissible solutionsis sequentially closed with respect to-convergence.
Proof. Let be a bounded -convergent sequence of admissible solutions to optimal control problem (55)-(56), (57). Let be its -limit. Our aim is to prove that . By Lemma 25, we have and .
Further, let us show that . Indeed, the condition implies for all , whereas relations (33)-(34) imply the weak convergence of to in within a subsequence (see (13)-(14)). Since the set is closed in and convex, then, in view of Mazur’s lemma, is weakly closed in . Therefore, and . Combining these facts, we conclude that .
It remains to show that the pair satisfies relation (55) for all . In order to do this, we divide the rest of the proof into three steps.
Step 1. Let us show that for every, there exists a sequence, such that Indeed, since, it follows that there exists a sequenceinsuch that Moreover, by the initial assumptions, contains only smooth functions and their limits with respect to the strong convergence in . Hence, for all indices large enough. Let us show that converges to weakly in . To begin with, we show the boundedness of the sequence in . Indeed, In view of Proposition 9, the sequence , within a subsequence, is weakly convergent in the variable space . Let us show that is its weak limit. Let be an arbitrary smooth function. Then Obviously,asprovided by the strong convergencein. Further, using the Cauchy-Bunjakowsky inequality, we have as the product of a bounded sequence and a sequence converging to zero. It follows that
Step 2. Here we have to substantiate the correctness of the limit passage in the following variational inequality: whereis its solution. To do so, it is enough to show that Since for every relation (89) holds true for every, we may consider only those elementswhich form convergent sequences to elementsin a way that was established in Step 1.
Taking into account Hypothesis 1 and Theorem 22, we obtain and, moreover, for an arbitrary , we have Hence, the assertion (90) is valid.
Step 3. In view of the initial assumptions, we have(1)strongly in;(2)in;(3)in;(4)in;(5) there exists a sequence , such that strongly in and in .Hence, using this, Proposition 10 and the lower semicontinuity property of-norm with respect to the weak convergence, we obtain As a result, we come to the desired relation: which means that the limit pair belongs to the set of admissible solutions of problem (55)-(56), (57). Hence, set is sequentially closed with respect to -convergence in , and this concludes the proof.
Proof. Since the cost functional is bounded below and , it provides the existence of a minimizing sequence to the problem (66). Then, implies the existence of a constant such that Hence, the minimizing sequence is bounded in the sense of Definition 24. Then, by Lemma 25, there exist functions and such that, up to a subsequence, . Since the set is sequentially closed with respect to the-convergence (see Theorem 26), it follows that the -limit pair is an admissible solution to (55)-(56), (57) (i.e., ). To conclude the proof, it is enough to observe that the cost functional is sequentially lower -semicontinuous. Hence, that is, is an optimal solution. The proof is complete.
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