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
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:
subject to the constraints
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 , Zolezzi . Tartar and Murat [5-7] showed examples of the non-existence for such problems (see e.g. 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  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  for a scalar problem, as one of the first papers in that direction, and later by Haslinger et al. 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  and Kogut and Leugering . 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  this problem has been considered in the context of linear problems (see also ). 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  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)
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:
where the degeneration occurs in a natural way with respect to the states.
This concept was proposed for linear problems by Serovajskiy , 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 , 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
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 ). 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
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
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 ).
(i) Let be a sequence in strongly converging to somefin 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 Generalizedp-Laplacian. Let α, β, γ, and m be given positive constants such that and . We define the class of admissible controls as follows:
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
admits a unique weak solution in . Let us recall that a function 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 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 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 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 3.2If and in , then in for any and in .
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 3.3 is a sequentially compact subset of for 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 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 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
Taking into account the inequality (see )
and definition of the class of admissible controls , we conclude: there exists a constant independent of such that
Since , , and in , it follows that
Hence, passing to the limit in (3.7) as , we arrive at a conclusion (see ):
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. □
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 .
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 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 [, 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 ).
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:
We have the following result.
Proposition 4.2Assume thatyis 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 . 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 ), the assumptions of Proposition 4.2 appears natural and it is not a restrictive supposition in practice. Indeed, following , 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 4.3Let be a given element, and let be a regular point of the Lagrangian (4.1). Then the mapping
is Gâteaux differentiable atyand its Gâteaux derivative
exists and takes the form
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
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
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
we arrive at the following obvious consequence of Lemma 4.3.
Corollary 4.4Let and be given elements, and let be a regular point of the Lagrangian (4.1). Then the mapping
is Gâteaux differentiable atyand 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 4.5Let , , and be given distributions. Assume that each point of the segment is regular for the mapping . Then there exists a positive value such that
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
Hence, splitting it in two terms, we obtain
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:
Using (4.7), we obtain
Now we introduce the concept of quasi-adjoint states that was first considered for linear problems by Serovajskiy .
Definition 5.1 We say that, for given and , a distribution is the quasi-adjoint state to if satisfies the following integral identity:
or in the operator form
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
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 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, in as , we immediately arrive at the following consequence of Lemma 6.1.
(i) in as ;
(ii) in as .
Further, we note that
(A1) if , where , and , then
and, hence, by the Hölder Inequality,
(A2) If and then
Then, in view of Corollary 6.2 and estimates (6.2) and (6.4), we can give the following obvious conclusion.
Corollary 6.3For 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
To begin with, we make use of the following observation.
Proposition 6.4Let 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 and the following chain of estimates:
(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 6.6Assume that, for given and , . Then each elementψof can be represented in a unique way as follows:
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