On Henig Regularization of Material Design Problems for Quasi-Linear p-Biharmonic Equation

We study a Dirichlet optimal design problem for a quasi-linear monotone p-biharmonic equation with control and state constraints. We take the coefficient of the p-biharmonic operator as a design variable in ( ) BV Ω . In this article, we discuss the relaxation of such problem.


Introduction
The aim of this article is to analyze the following optimal design problem (OCP), which can be regarded as an optimal control problem, for quasi-linear partial differential equation (PDE) with mixed boundary conditions

( )
BV Ω stands for the control space, d y , f, and max ζ are given distributions.Problems of this type appear for p-power-like elastic isotropic flat plates of uniform thickness, where the design variable u is to be chosen such that the deflection of the plate matches a given profile.The model extends the classical weighted biharmonic equation, where the weight 3 u a = involves the thickness a of the plate, see e.g.[1]- [3], or u can be regarded as a rigidity parameter.The OCP ( 1)-( 4) can be considered as a prototype of design problems for quasilinear state equations.For an interesting exposure to this subject we can refer to the monographs [4]- [6].
A particular feature of OCP ( 1)-( 4) is the restriction by the pointwise constraints (4) in ( ) L Γ -space.In fact, the ordering cone of positive elements in p L -spaces is typically non-solid, i.e. it has an empty topological interior.Following the standard multiplier rule, which gives a necessary optimality condition for local solutions to state constrained OCPs, the constraint qualifications such as the Slater condition or the Robinson condition should be applied in this case.However, these conditions cannot be verified for cones such as int A stands for the topological interior of the set A. Therefore, our main intention in this article is to propose a suitable relaxation of the pointwise state constraints in the form of some inequality conditions involving a so-called Henig approximation .As a result, it leads to some relaxation of the inequality constraints of the considered problem, and, hence, to the approximation of the feasible set to the original OCP.Hence, the solvability of a given class of OCPs can be characterized by solving the corresponding Henig relaxed problems in the limit 0 ε → .
As was shown in the recent publication [7], the proposed approach is numerically viable for state-constrained optimal control problems with the state equation given by linear partial differential equations.In particular, using the finite element discretization of the Henig dilating cone of positive functions, it has been shown in [7] that the above approximation scheme, called conical regularization, where the regularization is done by replacing the ordering cone with a family of dilating cones, leads to a finite-dimensional optimization problem which can conveniently be treated by known numerical techniques.The non-emptiness of the feasible set for the stateconstrained OCPs is an open question even for the simplest situation.Therefore, we consider a more flexible notion of solution to the boundary value problem (2)- (3).With that in mind we discuss a variant of the penalization approach, called the "variational inequality (VI) method".Following this approach we weaken the requirements on admissible solutions to the original OCP and consider instead the family of penalized OCPs for appropriate variational inequalities , , , , , where the sets K ε are defined in a special way.As a result, we show that each of new penalized OCP is solvable for each 0 ε > and their solutions can be used for approximation of optimal pairs to the original problem.
The outline of the paper is the following.In Section 2 we report some preliminaries and notation we need in the sequel.In Sections 3, we give a precise statement of the state constrained optimal control (or design) problem and describe the main assumptions on the initial data and control functions.In Section 4, we provide the results concerning solvability of the original problem with control and state constraints.We show that this problem admits at least one solution if and only if the corresponding set of feasible solutions is nonempty.In Section 5 we show that the pointwise state constraints can be replaced by the weakened conditions coming from Henig relaxation of ordering cones.As a result, we give a precise definition of the relaxed optimization problems and show that the solvability of the original OCP can be characterized by the associated relaxed problems.In particular, we prove that the optimal solution to the original problem can be attained in the limit by the optimal solution of the relaxed problem.We consider in Section 6 the "variational inequality method" as an approximation of the OCPs.Following this approach, we weaken the requirements on feasible solutions to the original OCP.In contrast to the Henig relaxation approach, the penalized optimal control problem for indicated variational inequality has a non-empty feasible set and this problem is always solvable.In conclusion, we show that some of the optimal solutions to the original problem can be attained in the limit by optimal solutions of the penalized problem.However, it is unknown whether the entire set of the optimal solutions can be attained in such way.

Definitions and Basic Properties
Let Ω be a bounded open connected subset of N  ( 2 N ≥ ).We assume that the boundary ∂Ω is Lip- schitzian so that the unit outward normal ( ) is well-defined for a.e.x ∈ ∂Ω , where the abbreviation 'a.e.' should be interpreted here with respect to the ( ) W Ω is a Banach space with respect to the norm , and .We set : , : as closed subspaces of ( ) ; : 0 and 0 on . We define the Banach space ( ) as the closure of ( ) with respect to the norm is a uniformly convex Banach space [10].Moreover, the norm to the usual norm of ( ) : Ω is well defined and satisfies the following elliptic regularity estimate [11] ( ) ( ) and y is a solution of (8), then ( ) Ω , 0 y = on the boundary ∂Ω , and, therefore, ( ) , , for a suitable positive constant p C independent of f.On the other hand, it is easy to see that W Ω (for the details we refer to [12] [13]).

( )
BV Ω we denote the space of all functions in ( ) L Ω for which the norm BV Ω if and only if the two following conditions hold (see [14]): ( ) It is well-known also the following compactness result for BV-spaces (Helly's selection theorem, see [15]).

Setting of the Optimal Control Problem
Let 1 ξ , 2 ξ be fixed elements of ( ) ( ) where α is a given positive value.Let : F Ω × →   be a nonlinear mapping such that F is in the space
1) the function In addition, the following conditions of subcritical growth, monotonicity, and non-negativity are fulfilled: ( )  ; ∈ Ω , and ∂Ω be given distributions.The optimal control pro- blem we consider in this paper is to minimize the discrepancy between d y and the solutions of the following state-constrained boundary valued problem , : is the operator of fourth order called the generalized p-biharmonic operator, and the class of admissible controls ad A we define as follows | a.e. in .
It is clear that ad A is a nonempty convex subset of ( ) L Ω with an empty topological interior.
More precisely, we are concerned with the following optimal control problem Moreover, taking into account the growth condition (12) and the compactness of the Sobolev imbedding it is easy to show that operator ( ) ( ) ( ) is the weak solution (in the sense of Minty) to the boundary value problem (15) -( 16), for a given admissible control  ,  ,  ; .
Remark 3.1.Since the set ( ) , and we arrive at the standard definition of weak solution to the boundary value problem ( 15)- (16).However, in order to avoid some mathematical difficulties, we will mainly use the Minty inequality in our further analysis.It is worth to note that having applied Green's formula twice to operator ( ) we arrive at the identity is the weak solution of the boundary value problem ( 15) -( 16) in the sense of Definition 3.1, then relations ( 15)-( 16) are fulfilled as follows (for the details, we refer to ([16], Section 2.4.4) and ( In particular, taking w y = in (22), this yields the relation .
As a result, conditions (11), (18), and inequalities ( 14) and ( 9) lead us to the following a priori estimate The existence of a unique weak solution to the boundary value problem ( 15)-( 16) in the sense of Definition 3.1 follows from an abstract theorem on monotone operators.

Theorem 2 ([17]
) Let V be a reflexive separable Banach space.Let V * be the dual space, and let : be a bounded, hemicontinuous, coercive and strictly monotone operator.Then the equation Ay f = has a unique solution for each f V * ∈ .Here, the above mentioned properties of the strict monotonicity, hemicontinuity, and coercivity of the operator A have respectively the following meaning: ; , 0, , ; ; , 0 ; ; the function , is continuous for all , , ; ; , lim .
In our case, we can define the operator ( ) In view of the properties ( 12)-( 14) and compactness of the Sobolev imbedding and ( ) , A u ⋅ satisfies all assumptions of Theorem 2 (for the details we refer to [16] [17]).Hence, the variational problem For a given , find such that , , , , for which ( ) is its operator form, has a unique solution . We note that the duality pairing in the right hand side of (30) makes a sense for any distribution ; : ; . It remains to show that the solution y of (30) satisfies the Minty relation (21).Indeed, in view of the monotonicity of A, we have and, hence, in view of Remark 3.1, the Minty relation ( 21) holds true.
Taking this fact into account, we adopt the following notion.
Definition 3.2.We say that ( ) the pair ( ) , u y is related by the Minty inequality (21), ( ) , , where stands for the natural ordering cone of positive elements in ( ) We denote by Ξ the set of all feasible pairs for the OCP (19).We say that a pair ( )  16).In fact, one needs the feasible set Ξ to be nonempty.But even if we are aware that Ξ ≠ ∅ , this set must be suf- ficiently rich in some sense, otherwise the OCP (19) becomes trivial.From a mathematical point of view, to deal directly with the control and especially state constraints is typically very difficult [18]- [20].Thus, the nonemptiness of feasible set for OCPs with control and state constraints is an open question even for the simplest situation.
It is reasonably now to make use of the following Hypothesis.

Existence of Optimal Solutions
In this section we focus on the solvability of optimal control problem ( 15)- (19).Hereinafter, we suppose that the space ( ) ( ) Let τ be the topology on the set which we define as the product of the weak- * topology of ( ) BV Ω and the weak topology of ( ) We begin with a couple of auxiliary results.
Hence, it is immediate to pass to the limit and to deduce (33).
As a consequence, we have the following property.
Our next step concerns the study of topological properties of the feasible set Ξ to problem (19).
The following result is crucial for our further analysis.
. Then there is a pair ( ) Then by Lemma 1, we have ( ) .
It remains to show that the limit pair ( ) , u y is related by inequality (21) and satisfies the state constraints (31).With that in mind we write down the Minty relation for ( ) .
In view of (34) and Lemma 1, we have lim d d , Moreover, due to the compactness of the Sobolev imbedding where Hölder's inequality yields We, thus, can pass to the limit in relation (35) as k → ∞ and arrive at the inequality (21), which means that is a weak solution to the boundary value problem ( 15)-( 16).Since the injections (6) This fact together with ad u ∈ A leads us to the conclusion: ( ) , u y ∈ Ξ , i.e. the limit pair ( ) , u y is feasible to optimal control problem (19).The proof is complete. In conclusion of this section, we give the existence result for optimal pairs to problem (19).Theorem 4. Assume that, for given distributions ;  (19).Then the inequality Hence, in view of the definition of the class of admissible controls ad A and a priori estimate (24), the sequence ( ) . Therefore, by Theorem 3, there exist functions lim inf by (10). is an optimal pair, and we arrive at the required conclusion.

Henig Relaxation of State-Constrainted OCP (19)
The main goal of this section is to provide a regularization of the pointwise state constraints by replacing the ordering cone ( ) 32)) by its solid Henig approximation ( ) ε Λ (see [21]- [24]) and show that the conical regularization approach leads to a family of optimization problems such that their solutions can be obtained by solving the corresponding optimality system and the regularized solution τ-converge in the limit as 0 ε → to a solution of the original problem.
We begin with some formal descriptions and abstract results.Let Z be a real normed space, and let Z Λ ⊂ be a closed ordering cone in Z.

{ }
, where there is a unique repre- In what follows, we always assume that the ordering cone Λ has a closed base B ⊂ Λ .We note that, in general, bases are not unique.We denote the norm of Z by , Z ⋅ and for arbitrary elements 1 2 , z z Z ∈ we define as well as \ 0 .
In order to introduce a representation for a base of Λ , let * Z be the topological dual space of Z, and let we define the dual cone and the quasi-interior of the dual cone of Λ , respectively.Using the definition of the dual cone, the ordering cone Λ can be characterized as follows (see [25], Lemma 3.21): Due to Lemma 1.28 in [25], we can give the following result.Lemma 2. Let Z Λ ⊂ be a nontrivial ordering cone in a Banach space Z.Then the set is a closed base of ordering cone ( ) Now, we are prepared to introduce the definition of a so-called Henig dilating cone (see Zhuang, [24]) which is based on the existence of a closed base of ordering cone Λ .
Definition 5.2.Let Z be a normed space, and let Z Λ ⊂ be a closed ordering cone with a closed base B. Choosing 0 ε > arbitrarily, the corresponding Henig dilating cone is defined by ( ) where ( ) . .
Then we obtain ) which completes the proof.In order to show (40), let ( ) As a result, (40) is satisfied. Remark 5.2.The following property, coming from Proposition 6, turns out rather useful: in order to prove . The following result shows that Henig dilating cones ( ) possess good approximation properties.Proposition 7. Let Λ be a closed ordering cone in a normed space Z, and let B be an arbitrary closed base of Λ .Let parameter δ be defined as in (37), and let ( ) ( ) with respect to the norm topology of Z as k tends to infinity, that is | for all neighborhoods of there is such that , | for all neighborhoods of and every there is such that .
Let z ∈ Λ be chosen arbitrarily.Then N ∩ Λ ≠ ∅ holds true for every neighborhood N of z, and due to the inclusions Taking into account the inclusion (41) and the fact that ( ) ( ) To show that the sequence However, the inclusion (43) is equivalent to Λ be an arbitrarily element.Since Λ is closed, there is an open neighborhood N of z with respect to the norm topology of Z such that N ∩ Λ = ∅ .By Proposition 5 (see item (4)), there is a sufficiently large index  such that ( ) Combining (42), (43), and (44), we arrive at the relation or in a more compact form each of these problems can be stated as follows , .
L B ε + Γ stands for the corresponding Henig dilating cone.Since, by Proposition 6, the inclusion ε Ξ ⊆ Ξ holds true for all ε > 0, it is reasonable to call the OCP (47) a Henig relaxation of OCP (19).Moreover, as obviously follows from Proposition convergence 0 ε ε → Ξ  → Ξ in Kuratowski sense holds true with respect to the τ-topology on ( ) ( ) . We are now in a position to show that using the relaxation approach we can reduce the main suppositions of Theorem 4. In particular, we can characterize Hypothesis ; ∈ is τ-compact and each of its τ-cluster pairs is a feasible solution to the original OCP (19).
It remains to establish the inclusions , , By contraposition, let us assume that ( ) ( ) ( ) by Proposition 7 and definition of the Kuratowski limit, it is easy to conclude the existence of an index 0 However, in view of the strong convergence property (51), there is an index Combining ( 53) and (54), we finally obtain This, however, is a contradiction to In the same manner it can be shown that ( ) ( )  H ) is satisfied.The next result is crucial in this section.We show that some optimal solutions for the original OCP ( 19) can be attained by solving the corresponding Henig relaxed problems (45)-( 46).However, we do not claim that the entire set of the solutions to OCP ( 19) can be restored in such way.
Theorem 9. Let ( ) ∈ Ω , and ( )  be a sequence of optimal solutions to the Henig relaxed problems (45)-( 46) such that ( ) Then there is a subsequence ( ) ∈ and a pair ( ) Proof.In view of a priory estimate (24), the uniform boundedness of optimal controls with respect to BV-norm (55) implies the fulfilment of condition (50) 2 .Hence, the compactness property (56) and the inclusion ( ) 0 0 , u y ∈ Ξ are a direct consequence of Theorem 8.It remains to show that the limit pair ( ) 0 0 , u y is a solution to OCP (19).Indeed, the condition ( ) 0 0 , u y ∈ Ξ implies the fulfilment of Hypothesis On the other hand, by Proposition 5 (see property ( 4)), we have ( ) ∈ are the solutions to the corresponding relaxed problems (47), it follows that   , As a result, (60) leads to the estimate As was mentioned at the beginning of this section, the main benefit of the relaxed optimal control problems (45)-( 46) comes from the fact that the Henig dilating cone ) ( ) has a nonempty topological interior.Hence, it gives a possibility to apply the Slater condition or the Robinson condition in order to characterize the optimal solutions for the state constrained OCP (19).On the other hand, this approach provides nice convergence properties for the solutions of relaxed problems (45)-(46).However, as follows from Theorems 8 and 9 (see also Remark 5.5), the most restrictive assumption deals with the regularity of the relaxed problems (45)-( 46) for all ( ) So, if we reject the Hypothesis ( 1 H ), it becomes unclear, in general, whether the relaxed sets of feasible solutions ε Ξ are nonempty for all 0 ε  .In this case it makes sense to provide further relaxation for each of Henig problems (45)-(46).In particular, using the methods of variational inequalities, we show in the next section that original OCP (19) may admit the existence of the so-called weakened approximate solution which can be interpreted as an optimal solution to some optimization problem of a special form.

Variational Inequality Approach to Regularization of OCP (19)
As follows from Theorem 4, the existence of optimal solutions to the problem (19) can be obtained by using compactness arguments and the Hypothesis ( 1 H ).However, because of the state constraints (17) the fulfilment of Hypothesis ( 1H ) is an open question even for the simplest situation.Nevertheless, in many applications it is an important task to find a feasible (or at least an approximately admissible, in a sense to be made precise) solution when both control and state constraints for the OCP are given.Thus, if the set of feasible solutions is rather "thin", it is reasonable to weaken the requirements on feasible solutions to the original OCP.In particular, it would be reasonable to assume that we may satisfy the state equation ( and the corresponding state constraint with some accuracy.Here, the operator ( ) ( , : is defined by the left-hand side of relation (29).For this purpose, we make use of the following observation: If a pair ( ) , , , , for each 0 ε > , where K ε is defined as follows is the corresponding Henig dilating cone.Note that the reverse statement is not true in general.In fact, we discuss a variant of the penalization approach, called the "variational inequality (VI) method".This idea was first studied in [27].Thus, if a pair ( ) is related by variational inequality (61), then it is not necessary to suppose that ( ) , , d , where N * is the norm of the embedding operator ; ; , , lim inf , , ; Definition 6.2.We say that an operator ( ) ( ) ( ) , given by formula (29), is quasi-monotone provided assumptions (11)-( 14 With that in mind, we set and divide our proof onto several steps. Step 1.We show that, for each ( ) for all k ∈  by the initial assumptions.Hence, ( ) by the Lebesgue Dominated Theorem.Since the sequence { } 2 : L Ω .Therefore, the first term in (70) tends to zero as k → ∞ as the product of strongly and weakly convergent sequences.Combining this fact with (71), we arrive at the desired property (70).
Step 2. Let us show that ( . This means that ( ) .By monotonicity property (13), it follows that for every z ∈  and every positive function  .Thus, the relation (75) is a direct consequence of the convergence (76).
Step 3.This is the final step of our proof.As follows from (69), for every element ( ) be a fixed element.We put ( ) Passing to the limit in (79) as k → ∞ , we obtain

A u y y v A u y y y A u y y v A u y y y A u y y v B u y y B u y y y y B u y y y y
that is, the inequality (68) is valid. Remark 6.1.In fact (see [19], Remark 3.13), we have the following implication:      As a result, using the τ-lower semicontinuity property of the cost functional (63), we finally obtain Γ and S Γ are the disjoint part of the boundary ∂Ω ( D S ∂Ω = Γ ∪ Γ ), . Here, B is a fixed closed base of 9], Theorem 8.3), these linear operators can be extended continuously to the whole of space appropriate function ad u ∈ A as control.Here, constraints (15)-(18).
we will discuss the question of existence of admissible pairs to the problem(19), we note that the function

Remark 3 . 2 .
Before we proceed further, we need to make sure that minimization problem(19) is meaningful, i.e. there exists at least one pair ( ) , u y such that ( ) , u y satisfying the control and state constraints (16)-(18), ( ) , I u y < +∞ , and ( ) , u y would be a physically relevant solution to the boundary value problem (15)-( Ξ .Proof.By Theorem 1 and compactness properties of the space ( ) denoted by the same indices, and functions ( ) To conclude the proof, it is enough to show that the cost functional I is lower semicontinuous with respect to the τ-convergence.

3 .
and if B is a base of Λ , then there is an element * # z ∈ Λ satisfying As follows from Lemma 2, the set the closed unit ball in Z centered at the origin.particular choice of B. As follows from this definition, i.e.Henig dilating cone is proper solid.Moreover, we have the following properties of such cones (see[24] [26]).Proposition 5. Let Z be a normed space, and let Z Λ ⊂ be a closed ordering cone with a closed base B. Choosing of constraint qualifications problem, the following result plays an important role.Proposition 6.Let Z be a normed space, and let Z Λ ⊂ be a closed ordering cone with a closed base B. Choosing δ is defined by (37), the inclusion be chosen arbitrarily.By the definition of a base there is a unique representation z b true.Let's assume for a moment that and the proof is complete.Taking these results into account, we associate with OCP(19) the following family of Henig relaxed problems base B takes the form (36), and the feasible set if and only if ad u ∈ A , ( ) , I u y < +∞ , ( ) p y ∈ Ω  , the pair ( ) , u y is related by the Minty inequality (21), and

( 1 H
) by the non-emptiness properties of feasible sets ε Ξ for the corresponding Henig relaxed problems.decreasing sequence converging to 0 as k → ∞ .Then, obvious by Proposition 7, we concentrate on the proof of the inverse statement-property (50) implies the existence of at least one pair ( ) all k ∈  .Since the set ad A and a priory estimate(24) do not depend on parameter k ε and the condition (50) 2 implies sup k compactness arguments (see the proof of Theorem 4) that there exist a subsequence of ( )Closely following the proof of Theorem 3, it can be shown that the limit pair ( ) solution to the boundary value problem (

( 1 H
).Hence, by Theorem 4, the original OCP (19) has a nonempty set of solutions.Let ( ) , u y * * be one of them.Then the following inequality is obvious , taking into account the relations (58) and (59), and the lower semicontinuity property of the cost functional I with respect to the τ-convergence, we finally get following sense (because of the property (2) of Proposition 5) 1 2 u y is feasible to the original problem, i.e. ( ) , u y ∈ Ξ , then this pair satisfies the relation( coercive.Moreover, it is shown in [16, Proposition 2.42], the properties (11)-(14) ensure the following implication pseudo-monotone for each ad u ∈ A .Hence, following the well-know existence result (see, for instance,[28] [29]), there exists at least one solution ( )y y u = of vari-ational inequality (61) such that y K ε ∈ .As an obvious consequence of Lemma 3, we have the following noteworthy property of penalized OCP (63ˆε Ξ is nonempty.To proceed further, we introduce the following notion. We assume that inequality (67) holds true.Our aim is to establish the relation (68).
74) by definition of the weak convergence in ( ) p L Ω .Thus, in order to conclude the equality (73), it remains to show that the subcritical growth condition (12), we have the following estimate

∫
So, taking into account (76) and the fact that k y y → strongly in ( ) r L Ω by Sobolev embedding theorem, we can pass to the limit in this inequality as k → ∞ .As a result, we get

4
view of Theorem 10, we can claim that the operator defined by relation(29), possesses the property ( )M .We are now in a position to show that the penalized optimal control problem in the coefficient of variational inequality (63)-(64) is solvable for each value If the assumptions (11)-(14) are valid, then the OCP (63)-(64) admits at least one solution be a minimizing sequence to problem (63)-(64).The coerciveness property (66) and estimate So, comparing these two chains of relations, we arrive at the existence of a constant 0 C > such that C is independent of ad u ∈ A and V y C ≤ as far as y K ε ∈ is a solution to (63).Since ad K ε × Ais sequentially closed with respect to the τ-convergence, we may assume by Theroem 1 that there exists a pair ( ) the quasi-monotonicity property of the operator A. Combining this inequality with (84), we come to the relation pair to this problem.As follows from (83), the sequence Since the quasi-monotone operator possesses the ( ) M -property (see Remark 6.6), it follows that pair to the penalized problem (63)-(64). The next step of our analysis is to consider a sequence of optimal pairs ( ) of optimal pairs to penalized problems (63) -(64).In addition to the assumptions of Lemma 4, assume that there exists a constant 0 C > such that with respect to the τ-convergence and each of its τ-cluster pair ( ) Since the set Ξ is nonempty and the cost functional is bounded from below on Ξ , it follows that there exists a minimizing sequence ( )