On optimality conditions for optimal control problem in coefficients for -Laplacian
© Kupenko and Manzo; licensee Springer. 2014
Received: 9 January 2014
Accepted: 11 March 2014
Published: 28 March 2014
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.
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.
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.
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.
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
Let be the characteristic function of a set and let be its N-dimensional Lebesgue measure.
where denotes the duality pairing between and , and denotes the scalar product of two vectors in .
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 some f in and satisfying condition . Then
- (ii)for every , , there exists a sequence such that
for every bounded sequence there exist a subsequence, still denoted by , and a function such that in .
It is clear that is a nonempty convex subset of with empty topological interior.
3 Setting of the optimal control problem
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).
Now to get the desired estimate we divide each part of the obtain relation into and raise each side to the power . □
Proposition 3.2 If and in , then in for any and in .
implies that in .
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 3.4 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 .
that is, the limit pair is an admissible to optimal control problem (3.1)-(3.3). The proof is complete. □
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 .
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 ).
We have the following result.
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.
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). □
we arrive at the following obvious consequence of Lemma 4.3.
Before deriving the optimality conditions, we need the following auxiliary result.
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 .
for all and such that .
Now we introduce the concept of quasi-adjoint states that was first considered for linear problems by Serovajskiy .
Here, is the identity matrix, is the solution of problem (3.2)-(3.3), , and is a constant coming from equality (5.3).
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 6.1 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, in as , we immediately arrive at the following consequence of Lemma 6.1.
in as ;
in as .
Further, we note that
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 .
To begin with, we make use of the following observation.
, where denotes the set of all symmetric matrices, which are obviously determined by scalars.
for all .
in as ;
in for any and in as ;
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.
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.
Notice that, in general, the adjoint operator is not densely defined.
and demi-continuous. However, because of the multiplier , this operator can lose the coercivity property.
We are now in a position to give an important property of the operator .
and the linear operator defines an isomorphism from into its dual .
where and .
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 6.9 Assume that the Hypothesis (H2) is valid. Then the Dirichlet boundary value problem (6.5) has a unique solution for every , , and .
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.
Lemma 6.10 Assume that Hypotheses (H2)-(H3) are valid. Then there exists a constant independent of θ such that , i.e. the sequence of quasi-adjoint states to is relatively compact with respect to the weak convergence of .