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
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)
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.
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
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:
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 .
Under the norm , is a Banach space. For our further analysis, we need the following properties of BV-functions (see ).
or via the pairing
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:
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
3 Setting of the optimal control problem
We consider the following optimal control problem:
subject to the constraints
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).
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
by Proposition 2.2(i) it follows that
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.
Taking into account the inequality (see )
Hence, passing to the limit in (3.7) as , we arrive at a conclusion (see ):
As a result, we finally have
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:
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.
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 .
is Gâteaux differentiable atyand its Gâteaux derivative
exists and takes the form
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
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.
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.
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:
Hence, splitting it in two terms, we obtain
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 .
or in the operator form
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.
Further, we note that
and, hence, by the Hölder Inequality,
Then, in view of Corollary 6.2 and estimates (6.2) and (6.4), we can give the following obvious conclusion.
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.
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.
where the constant C comes from the Poincaré Inequality. Using the evident equality
Taking this fact into account, it is plausible to introduce the following linear mapping:
Let us consider the following linear operator:
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
Thus, in view of (6.18)-(6.19),
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.
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
In view of Lemma 6.9, it makes a sense to accept the following hypothesis.
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.
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
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
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
(A3) up to a subsequence
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:
Using the Hölder Inequality and estimate (2) in Proposition 6.4, we have
Further it remains to repeat the trick like in (7.7). As a result, we obtain
Thus, summing up the results given above, we finally obtain
As for the term
we see that
As a result, we finally get
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. □
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.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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
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
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)
Casas, E: Optimal control in the coefficients of elliptic equations with state constraints. Appl. Math. Optim.. 26, 21–37 (1992). Publisher Full Text
Haslinger, J, Kocvara, M, Leugering, G, Stingl, M: Multidisciplinary free material optimization. SIAM J. Appl. Math.. 70(7), 2709–2728 (2010). Publisher Full Text
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
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
Kupenko, OP, Manzo, R: On an optimal -control problem in coefficients for linear elliptic variational inequality. Abstr. Appl. Anal. (2013). Publisher Full Text
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
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
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