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

(1)

http://dx.doi.org/10.4236/am.2016.714134

On Henig Regularization of Material

Design Problems for Quasi-Linear

p

-Biharmonic Equation

Peter Kogut1, Günter Leugering2, Ralph Schiel2

1_{Department of Differential Equations, Dnipropetrovsk National University, Dnipro, Ukraine}

2_{Department Mathematik, Lehrstuhl II Universität Erlangen-Nürnberg Cauerstr, Erlangen, Germany}

Received 25 June 2016; accepted 21 August 2016; published 24 August 2016

This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract

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 de-sign variable in BV

( )

Ω . In this article, we discuss the relaxation of such problem.

Keywords

p-Biharmonic Problem, Optimal Design, Relaxation, Henig Dilating Cone, Existence Result

1. 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

(

)

{

2

}

Minimize I u y, y y_d dx Du

Ω Ω

=

_∫

− +

_∫

(1) subject to the quasi-linear equation

(

2

)

(

)

, in ,

p

u y − y F x y f

∆ ∆ ∆ + = Ω (2)

0 on D, 0 on S,

y

y y y

ν ∂

= = Γ = ∆ = Γ

∂ (3)

(2)

( )

_{( )}

0 max a.e. on S,

y s

s ζ ν ∂

≤ ≤ Γ

∂ (4) and the design (control) constraints

( )

and 1

( )

2

( )

a.e. in .

u∈BV Ω ξ x ≤u x ≤ξ x Ω (5) Here, Γ_D and Γ_S are the disjoint part of the boundary ∂Ω (∂Ω = Γ ∪ Γ_D _S), BV

( )

Ω stands for the control space, y_d, f, and ζmax

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 _u=_a3_{involves the thickness}_a_{of the plate, see e.g.}_[1]-[3]_{, or}_u_{can be regarded as a rigidity} parame-ter. 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 Lp

( )

Γ_S -space. In fact, the ordering cone of positive elements in Lp-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

( )

{

( )

| 0 a.e. in

}

p p

S S

L+ Γ = ∈v L Γ v≥ Ω due to the fact that int

(

Lp+

( )

ΓS

)

= ∅, where 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 approxima-tion

(

p

( )

)

( )

S

L B

ε

+ Γ of the ordering cone of positive elements L+p

( )

ΓS . Here, B is a fixed closed base of

( )

p S

L+ Γ . Due to fact that L+p

( )

Γ ⊂S

(

L+p

( )

ΓS

)

_ε

( )

B for all ε >0, we can replace the cone

( )

p

S

L₊ Γ by its approximation

(

L+p

( )

ΓS

)

_ε

( )

B . As a result, it leads to some relaxation of the inequality constraints of the con-sidered 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, us-ing the finite element discretization of the Henig dilatus-ing 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 state- constrained OCPs is an open question even for the simplest situation. Therefore, we consider a more flexible no-tion of soluno-tion to the boundary value problem (2)-(3). With that in mind we discuss a variant of the penalizano-tion 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

(

)

(

)

( )

( ) ( ) ( ( )) ( )

2

; ;

, , , , ,

p p

p

u y y F x y ζ y f ζ y ∗ ζ K_ε

∗

−

Ω Ω

∆ ∆ ∆ + − ≥ − _ _ ∀ ∈

 

where the sets Kε are defined in a special way. As a result, we show that each of new penalized OCP is

(3)

contrast to the Henig relaxation approach, the penalized optimal control problem for indicated variational in-equality 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.

2. Definitions and Basic Properties

Let Ω be a bounded open connected subset of N (N≥2). We assume that the boundary ∂Ω is Lip- schitzian so that the unit outward normal ν ν=

( )

x is well-defined for a.e. x∈ ∂Ω, where the abbreviation ‘a.e.’ should be interpreted here with respect to the

(

N−1

)

-dimensional Hausdorff measure. We also assume that the boundary ∂Ω consists of two disjoint parts ∂Ω = Γ ∪ ΓD S, where the sets ΓD and ΓS have posi-tive

(

N−1

)

-dimensional measures, and ΓS is of

2 C .

Let p be a real number such that 2≤ < ∞p . By W2,p

( )

Ω we denote the Sobolev space as the subspace of

( )

p

L Ω of functions y having generalized derivatives D yk up to order k=2 in Lp

( )

Ω . We note that thanks to interpolation theory, see ([8], Theorem 4.14), W2,p

( )

Ω is a Banach space with respect to the norm

( )

(

_{( )}

)

(

)

2,

1 1

2 2

( ) d ,

p p p

p p

W L L

y _Ω y _Ω D y _Ω y D y x

Ω

= + =

_∫

+

where

(

)

1 2 1 2 1 2

2 2

1 2

2 2 2 2 2

, 1

, and .

N

i i i i i i

y v

D y D y D y D y D v

x x x x

=

∂ ∂

= ⋅ ⋅ =

∂ ∂ ∂ ∂

∑

For any y∈C1

( )

Ω we define the traces

( )

0 , and 1 .

y

y y y

γ γ

ν

∂Ω

∂

= =

∂

By ([9], Theorem 8.3), these linear operators can be extended continuously to the whole of space W2,p

( )

Ω . We set

( )

2 1 , 2, 1 1 , 2,

0 1

: , :

p p p p p p

W − ∂Ω =γ _W Ω _ W − ∂Ω =γ _W Ω _

as closed subspaces of W1,p

( )

∂Ω and Lp

( )

∂Ω , respectively. Moreover, the injections

( )

2 1 , 1, 1 1 ,

and

p p p p p p

W − ∂Ω W ∂Ω W − ∂Ω L ∂Ω (6) are compact.

Let 0

(

;

)

0

( )

: 0 and 0 on

N N

D D

C ϕ C ϕ ϕ

ν

∞ _{Γ =} _∈ ∞ ₌ ∂ ₌ _Γ 

 _∂ 

 

  . We define the Banach space ₀2,p

(

;

)

D

W Ω Γ as the closure of 0

(

;

)

N D

C∞  Γ with respect to the norm y_W2,p_{( )}_Ω . Let

(

)

2,

; q

D

W− Ω Γ be the dual space to

(

)

2,

0 ;

p D

W Ω Γ , where q= p

(

p−1

)

is the conjugate of p. We also define the space 01,

( )

p

W Ω as the closure of C0

( )

∞ _Ω

with respect to the norm 1,_{( )}

(

)

0

1

d p

p p W

y _Ω =

∫

_Ω∇y x .

Throughout this paper, we use the notation

( )

: 02,

(

;

)

01,

( )

p p

p Ω =W Ω Γ ∩D W Ω

 . Let us notice that p

( )

Ω

equipped with the norm

( )

(

)

1 2 1

2 ,

1

: p d d

p p N p

p

p L

i i

y

y y y x x

x

∆ Ω Ω Ω ₌

 _∂ 

 

= ∆ = ∆ =

 ∂ 



∑



∫

(7)

is a uniformly convex Banach space [10]. Moreover, the norm ⋅ _p_,_∆ is equivalent on _p

( )

Ω to the usual norm of W2,p

( )

Ω . Indeed, since the Laplace operator −∆ acts from _p

( )

Ω in Lp

( )

Ω and the Dirichlet boundary value problem

in , 0 on

y f y

(4)

is uniquely solvable in p

( )

Ω for all

( )

p

f∈L Ω , it follows that the inverse operator T:= −∆

( )

−1:

( )

2,

( )

1,

( )

0

p p p

L Ω →W Ω ∩W Ω is well defined and satisfies the following elliptic regularity estimate [11]

( )

2,p p p_{( )}.

W L

Tf _Ω ≤C f _Ω

This allows us to conclude the following. If f∈Lp

( )

Ω and 01,

( )

p

y W∈ Ω are such that y 0

ν ∂ ₌

∂ on ΓD

and y is a solution of (8), then −∆ ∈y Lp

( )

Ω , y=0 on the boundary ∂Ω, and, therefore, y∈p

( )

Ω .

Hence,

( )

(

)

( ) ( )

2,p 2,p p p p _, ,

W W L p

y _Ω = T −∆y _Ω ≤C ∆y _Ω =C y _∆ (9) for a suitable positive constant C_p independent of f. On the other hand, it is easy to see that

( )

2,

, p .

p W

y _∆≤ y _Ω

Thus, by the Closed Graph Theorem, we can conclude that ⋅ _p_,_∆ is equivalent to the norm induced by

( )

2,p

W Ω (for the details we refer to [12][13]).

By BV

( )

Ω we denote the space of all functions in L1

( )

Ω for which the norm

( ) ( )

( )

{

(

)

( )

}

1

1 0

sup div d : ; , 1 for

BV L

N L

f f Df

f f ϕ x ϕ C ϕ x x

Ω Ω Ω

Ω Ω

= +

= + ∈ Ω ≤ ∈ Ω

∫



is finite.

We recall that a sequence

{ }

fk k₁

∞

= converges weakly-* to f in BV

( )

Ω if and only if the two following

conditions hold (see [14]): f_k → f strongly in L1

( )

Ω and DfkDf weakly-* in the space of Radon measures 

( )

Ω , i.e.

( )

0

lim k .

k→∞

∫

ΩϕDf =

∫

ΩϕDf ∀ ∈ϕ C Ω

It is well-known also the following compactness result for BV-spaces (Helly’s selection theorem, see [15]).

Theorem 1. If

{ }

fk k₁ BV

( )

∞

= ⊂ Ω and supk∈ fk BV( )Ω < +∞, then there exists a subsequence of

{ }

fk k1 ∞

=

strongly converging in L1

( )

Ω to some f∈BV

( )

Ω such that Dfk Df

∗

 weakly-∗ in the space of Radon measures 

( )

Ω . Moreover, if

{ }

fk k₁ BV

( )

∞

= ⊂ Ω strongly converges to some f in

( )

1

L Ω and satisfies

supk∈

∫

_ΩDfk < +∞, then

( )

(i) and lim inf ;

(ii) in .

k k

k

f BV Df Df

f f BV

Ω →∞ Ω

∗

∈ Ω ≤

Ω

∫



(10)

3. Setting of the Optimal Control Problem

Let ξ1, ξ2 be fixed elements of L

( )

BV

( )

∞ _{Ω ∩} _Ω

satisfying the conditions

( )

1 2

0< ≤α ξ x ≤ξ x a.e. inΩ, (11) where α is a given positive value.

Let F:Ω × →  be a nonlinear mapping such that F is in the space Car

(

Ω ×

)

of Carathéodory func-tions on Ω ×, i.e.

1) the function F x

( )

,⋅ is continuous in  for almost all x∈ Ω; 2) the function F

( )

⋅,y is measurable for each y∈.

In addition, the following conditions of subcritical growth, monotonicity, and non-negativity are fulfilled:

( )

1

, r for a.e. and all ,

(5)

( )

(

)

(

F x,η −F x,η′

)

(

η η− ′

)

> 0 for a.e.x∈ Ωand all ,η η′∈,η η≠ ′, (13)

( )

, 0 for a.e. and all .

F xη η≥ x∈ Ω η∈ (14) for some r∈

( )

1,p∗ , where

(

2

)

, 2 ,

, 2

Np N p p N

p

p N

∗_{= } − <

+∞ ≥



is the critical exponent for the Sobolev imbedding W2,p

( )

Ω Lr

( )

Ω , and C₁>0. In particular, conditions (13) - (14) imply that

F

(

x

,

⋅

)

is monotonically increasing on  and F x

( )

, 0 =0 for almost all x∈ Ω.

Let f ∈W−2,q

(

Ω Γ; _D

)

, yd∈L2

( )

Ω , and

( )

max p

L

ζ ∈ ∂Ω be given distributions. The optimal control problem we consider in this paper is to minimize the discrepancy between y_d and the solutions of the following state-constrained boundary valued problem

( )

2

, , in ,

p u y F x y f

∆ + = Ω (15)

0 on _D, 0 on _S,

y

y y y

ν ∂

= = Γ = ∆ = Γ

∂ (16)

( )

_{( )}

0 max a.e. on S

y s

s ζ ν ∂

≤ ≤ Γ

∂ (17) by choosing an appropriate function u∈A_ad as control. Here,

( )

(

2

)

2

, : p

p u y u y y

−

∆ = ∆ ∆ ∆

is the operator of fourth order called the generalized p-biharmonic operator, and the class of admissible controls

ad

A we define as follows

( ) ( )

( )

{

| 1 2 a.e. in

}

.

ad = u∈BV Ω ξ x ≤u x ≤ξ x Ω

A (18)

It is clear that A_ad is a nonempty convex subset of L1

( )

Ω with an empty topological interior. More precisely, we are concerned with the following optimal control problem

( )

{

2

}

Minimize , d

subject to the constraints (15)-(18). d

I u y =

∫

_Ω y−y x+

∫

_ΩDu

(19)

Before we will discuss the question of existence of admissible pairs to the problem (19), we note that the function F∈Car

(

Ω ×

)

can be associated with operator F: p

( )

(

p

( )

)

∗

Ω → Ω

  defined by the rule

( )

, ₍ _{( )}₎_; _{( )}

( )

, d ,

( )

.

p p p

F y v _Ω ∗ _Ω F x y v x v

Ω

=

_∫

∀ ∈ Ω

   (20)

Moreover, taking into account the growth condition (12) and the compactness of the Sobolev imbedding

(

)

( )

2,

0 ;

p r

D

W Ω Γ L Ω for r< p∗ it is easy to show that operator F: p

( )

(

p

( )

)

∗

Ω → Ω

  is compact.

Definition 3.1. We say that an element y∈p

( )

Ω is the weak solution (in the sense of Minty) to the

boundary value problem (15) - (16), for a given admissible control u∈Aad, if

(

)

( )

₍ _{( )}₎ _{( )}

( ) 2, ( )

(

)

2, 0

2

;

0

; ; ;

d ,

, , ; .

p p

p q

D D

p

D

W W

u y x F y

f y C

ϕ ϕ ϕ ϕ ϕ

ϕ ϕ

∗ −

−

Ω Ω

Ω

∞

Ω Γ Ω Γ

∆ ∆ ∆ − ∆ + −

≥ − ∀ ∈ Ω Γ

∫

 

(21)

Remark 3.1. Since the set 0

(

;

)

N D

C∞  Γ is dense in p

( )

Ω , it follows that the element ϕ = +y tw with

an arbitrary w∈p

( )

Ω and t>0 can be taken as a test function in (21). As a result,(21) implies that

(

)

(

)

( ) 2, ( )

( )

2, 0

2

; ; ;

d , d

, q p , .

D D

p

W W

u y t w y t w w x F x y tw w x

f w − w

−

Ω Ω

Ω Γ Ω Γ

∆ + ∆ ∆ + ∆ ∆ + +

≥ ∀ ∈ Ω

∫

(6)

Passing to the limit as t→0 (because F∈Car

(

Ω ×

)

), we get

( )

2,₍ ₎ 2, ₍ ₎

( )

0

2

; ; ;

d , d , q p , .

D D

p

W W

u y − y w x F x y w x f w − _{Ω Γ} _{Ω Γ} w

Ω ∆ ∆ ∆ + Ω ≥ ∀ ∈ Ω

∫



Hence,

(

)

2, ₍ ₎ 2, ₍ ₎

0 2

; ; ;

d , d , q p ,

D D

p

W W

u y y w x F x y w x f w −

−

Ω Γ Ω Γ

Ω ∆ ∆ ∆ + Ω =

∫

(22)

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 ∆

(

u ∆yp−2∆y

)

tested by v C0

(

; D

)

∞

∈ Ω Γ ,

we arrive at the identity

(

)

(

)

(

)

(

)

(

)

2

2 2 1

2 2 1 2 1

2 2 ₁

0 d

, d d

d d d

d d ; .

N

D S

S

p

p p N

p p N p N

p p _N

D

u y y v x

u y y v x u y y v

v v

u y y v x u y y u y y

v

u y y v x u y y v C

ν

ν ν

ν

− Ω

− − ₋

Ω ∂Ω

− − ₋ − ₋

Ω Γ Γ

− − − ∞

Ω Γ

∆ ∆ ∆

∂

= − ∇ ∆ ∆ ∇ + ∆ ∆

∂

∂ ∂

= ∆ ∆ ∆ − ∆ ∆ − ∆ ∆

∂ ∂

∂

= ∆ ∆ ∆ − ∆ ∆ ∀ ∈ Ω Γ

∂

∫

 

 



Hence, if y as an element of

( )

: 02,

(

;

)

01,

( )

p p

p Ω =W Ω Γ ∩D W Ω

 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 ([4], Section 2.4.2))

(

)

( )

(

)

( )

(

( )

)

2

0

2 1 , 0

1 , 1

1 , 1 ,

0

, in ; ,

0 in ,

0 in : .

p D

p p

q p D

p p p q

S S

u y F y f C

y W

y W W

γ γ γ

∗ ∞

−

∗ −



∆ + = Ω Γ _



= ∂Ω _



= Γ _



∆ = Γ = Γ _

In particular, taking w=y in (22), this yields the relation

( )

2,₍ ₎ 2, ₍ ₎

0

; ; ;

d , d , q p .

D D

p

W W

u y x F x y y x f y − _{Ω Γ} _{Ω Γ}

Ω ∆ + Ω =

∫

(23)

As a result, conditions (11), (18), and inequalities (14) and (9) lead us to the following a priori estimate

(

)

(

2,( )

)

1 1

, : d q ; D .

p

p p

p ad

p W

y _∆ =

∫

_Ω∆y x ≤ α−C f − _{Ω Γ} − ∀ ∈u A (24) 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 A V: →V∗

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 opera-tor A have respectively the following meaning:

;

, _V _V 0, , ;

Ay−Av y−v ∗ ≥ ∀y v∈V (25)

;

, _V _V 0 ;

Ay−Av y−v ∗ = ⇒ =y v (26)

(

)

_;

the function , is continuous for all , , ; V V

t A y+tv w ∗ y v w V∈ (27)

; ,

lim .

V

V V

y

V Ay y

y

∗

(7)

In our case, we can define the operator A u

( )

,⋅ as a mapping p

( )

(

p

( )

)

∗

Ω → Ω

  by

(

)

₍ _{( )}₎ _{( )} 2

(

)

;

, , : d , d .

p p

p

A u y w ∗ u y − y w x F x y w x

Ω Ω =

∫

Ω ∆ ∆ ∆ +

∫

Ω

  (29)

In view of the properties (12)-(14) and compactness of the Sobolev imbedding 02,

(

;

)

( )

p r

D

W Ω Γ →L Ω for

r< p∗, it is easy to show that A u y

( )

, = ∆2_p

( )

u y, +F y

( )

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

, , , ,

p p

ad p

p

u y

A u y ϕ ∗ f ϕ _Ω ∗ _Ω ϕ

Ω Ω

∈ ∈ Ω

= _ _ ∀ ∈ Ω

 



A

(30)

for which A u y

( )

, = f is its operator form, has a unique solution y=y u

( )

∈p

( )

Ω . We note that the

duali-ty pairing in the right hand side of (30) makes a sense for any distribution 2,q

(

;

)

D

f ∈W− Ω Γ because

(

)

(

)

(

( )

)

2, 2,

0

; : ;

q p

D D p

W− Ω Γ = W Ω Γ ∗⊂  Ω ∗. 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

( )

(

)

₍ _{( )}₎ _{( )}

( )

₍ _{( )}₎ _{( )}

(

)

₍ _{( )}₎ _{( )}

( )

₍ _{( )}₎ _{( )} ₍ _{( )}₎ _{( )}

;

; ;

by (30)

; ;

0 , , ,

, , , ,

, , , .

p p

p p p p

p p

A u v A u y v y

A u v v y A u y v y

A u v v y f ϕ

∗

∗ ∗

Ω Ω

Ω Ω Ω Ω

Ω Ω

≤ − −

= − − −

= − −

 

   

 

Thus,

( )

₍ _{( )}₎ _{( )} 2,₍ ₎ 2, ₍ ₎

( )

0

; ; ;

;

, , , q p , ,

D D

p p W W p

A u v v−y _ _Ω ∗_ _Ω ≥ f ϕ − _{Ω Γ} _{Ω Γ} ∀ ∈v  Ω

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

(

u y,

)

is a feasible pair to the OCP (19) if u∈Aad ⊂BV

( )

Ω , y∈p

( )

Ω ,

the pair

(

u y,

)

is related by the Minty inequality (21), I u y

(

,

)

< +∞, and

( )

1 , 1 ,

p max p

S S

y L y L

γ ∈ ₊ Γ ζ −γ ∈ ₊ Γ (31) where L+p

( )

ΓS stands for the natural ordering cone of positive elements in

( )

p S

L Γ , i.e.

( )

{

( )

1

}

: | 0 -a.e. on .

p p N

S S S

L+ Γ = v∈L Γ v≥  − Γ (32)

We denote by Ξ the set of all feasible pairs for the OCP (19). We say that a pair

(

0 0

)

1

( )

, p

u y ∈BV Ω × Ω

is an optimal solution to problem (19) if

(

0 0

)

(

0 0

)

_{( )}

( )

,

, and , inf , .

u y

u y I u y I u y

∈Ξ

∈ Ξ =

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)-(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 non- emptiness 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.

(H1) There exists at least one pair

(

u y,

)

∈BV

( )

Ω ×p

( )

Ω such that

(

u y,

)

∈ Ξ.

4. Existence of Optimal Solutions

(8)

space BV

( )

Ω ×p

( )

Ω is endowed with the norm

( )

u y, BV( )Ω ×_p( )Ω := u BV( )Ω + y p,∆. Let τ be the to-

pology on the set Ξ ⊂BV

( )

Ω ×p

( )

Ω which we define as the product of the weak-∗ topology of BV

( )

Ω

and the weak topology of 02,

(

;

)

p D

W Ω Γ . We begin with a couple of auxiliary results.

Lemma 1. Let

{

(

uk,yk

)

∈ Ξ

}

_k_∈_ be a sequence such that

(

u yk, k

)

( )

u y,

τ

→ in BV

( )

Ω ×p

( )

Ω . Then

we have

(

)

0

lim k k d d ; D .

k u y ϕ x u y ϕ x ϕ C

∞

Ω Ω

→∞

∫

∆ ∆ =

∫

∆ ∆ ∀ ∈ Ω Γ (33)

Proof. Since u_k →u in L1(Ω) and

{ }

uk _k_∈_ is bounded in L

( )

∞ _Ω

, we get that u_k →u strongly in Lr(Ω) for every 1≤ < +∞r . In particular, we have that u_k →u in Lq

( )

Ω and ∆ ∆y_k ϕ∆ ∆y ϕ in Lp

( )

Ω . Hence, it is immediate to pass to the limit and to deduce (33).

As a consequence, we have the following property.

Corollary 1. Let

{

(

uk,yk

)

∈ Ξ

}

_k_∈_ and

{

(

)

}

2,

0 ;

q

k W D _k

ζ ∈ Ω Γ _∈

 be sequences such that

(

u yk, k

)

( )

u y,

τ

→

in BV

( )

Ω ×p

( )

Ω and ζk →ζ in

(

)

2,

0 ;

q D

W Ω Γ . Then

lim k k kd d .

k→∞

∫

Ωu∆ ∆y ζ x=

∫

Ωu y∆ ∆ζ x

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.

Theorem 3. Let

{

(

uk,yk

)

}

_k_∈_⊂ Ξ be a bounded sequence in BV

( )

Ω ×p

( )

Ω . Then there is a pair

(

u y,

)

∈BV

( )

Ω ×_p

( )

Ω such that, up to a subsequence,

(

u y_k, _k

)

τ→

( )

u y, and

(

u y,

)

∈ Ξ.

Proof. By Theorem 1 and compactness properties of the space p

( )

Ω , there exists a subsequence of

(

)

{

uk,yk ∈ Ξ

}

_k_∈_, still denoted by the same indices, and functions u∈BV

( )

Ω and y∈p

( )

Ω such that

( )

(

)

( )

1 2, 1,

0 0

in , in p ; , and, therefore, in p .

k k D k

u →u L Ω y y W Ω Γ y y W Ω (34) Then by Lemma 1, we have

(

)

0

lim k kd d , ; D .

k u ϕ y x u ϕ y x ϕ C

∞

Ω Ω

→∞

∫

∆ ∆ =

∫

∆ ∆ ∀ ∈ Ω Γ

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

(

u_k,y_k

)

:

(

)

(

)(

)

( ) 2, ( )

(

)

2, 0 2

0

; ; ;

d , d

, q p , ; .

D D

p

k k k

k W W D

u y x F x y x

f y C

ϕ ϕ ϕ ϕ ϕ

ϕ − ϕ

−

Ω Ω

∞

Ω Γ Ω Γ

∆ ∆ ∆ − ∆ + −

≥ − ∀ ∈ Ω Γ

∫

(35)

In view of (34) and Lemma 1, we have

lim p _kd p d , k→∞

∫

Ω∆ϕ u x=

∫

Ω∆ϕ u x

2 2

lim k p kd p d .

k u ϕ ϕ y x u ϕ ϕ y x

− −

Ω Ω

→∞

∫

∆ ∆ ∆ =

∫

∆ ∆ ∆

Moreover, due to the compactness of the Sobolev imbedding W₀2,p

(

Ω Γ; _D

)

Lr

( )

Ω for r< p∗, we have

(

,

)(

k

)

d

(

,

)(

)

d k,

F xϕ ϕ y x F xϕ ϕ y x J

Ω − = Ω − +

∫

where Hölder’s inequality yields

(

)(

)

by (12)

(

)

1 _{( )}

1

: , d d r 0 as .

r

r _r

k k k L

J F xϕ y y x C ϕ x y y k

−

Ω

Ω Ω

=

_∫

− ≤

_∫

− → → ∞

We, thus, can pass to the limit in relation (35) as k→ ∞ and arrive at the inequality (21), which means that

( )

p

y∈ Ω is a weak solution to the boundary value problem (15)-(16). Since the injections (6) are compact and the cone L+p

( )

ΓS is closed with respect to the strong convergence in

( )

p S

L Γ , it follows that yk y

ν ν

∂ ∂

(9)

strongly in Lp

( )

ΓS and, hence,

( )

1 1 1

lim k p S and max p S .

k→∞γ y =γ y ∈L+ Γ γ y ∈ζ −L+ Γ

This fact together with u∈A_ad 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 f ∈W−2,q

(

Ω Γ; _D

)

, yd∈L2

( )

Ω , and

( )

max p

L

ζ ∈ ∂Ω , the Hypothesis (H1) is valid. Then optimal control problem (19) admits at least one solution

(

uopt,yopt

)

∈BV

( )

Ω ×p

( )

Ω .

Proof. Since the set Ξ is nonempty and the cost functional is bounded from below on Ξ, it follows that there exists a minimizing sequence

{

(

uk,yk

)

∈ Ξ

}

_k_∈_ to problem (19). Then the inequality

( )

2 ,

inf , lim k d d k ,

u y∈ΞI u y k→∞ Ω y x y x x ΩDu

 

= __

_∫

− +

_∫

__< +∞

implies the existence of a constant C>0 such that

.

sup k

k

Du C

Ω ∈

≤

∫



Hence, in view of the definition of the class of admissible controls A_ad and a priori estimate (24), the sequence

{

(

uk,yk

)

∈ Ξ

}

_k_∈_ is bounded in BV

( )

Ω ×p

( )

Ω . Therefore, by Theorem 3, there exist functions

* ad

u ∈A and y*∈p

( )

Ω such that

(

)

* *

,

u y ∈ Ξ and, up to a subsequence, uk →u* weakly-∗ in BV(Ω)

and yk y* weakly in

(

)

2,

0 ;

p D

W Ω Γ . To conclude the proof, it is enough to show that the cost functional I is lower semicontinuous with respect to the τ-convergence. Since yk →y* strongly in

( )

p

L Ω by Sobolev em-bedding theorem, it follows that

( )

2 _*

( )

2

*

lim d d and,

lim inf by (10).

k d d

k

k k

y x y x x y x y x x

Du Du

Ω Ω

→∞

Ω Ω

→∞

− = −

≥

∫

Thus,

(

* *

)

(

)

₍ ₎

(

)

,

, lim inf _k, _k inf , .

k u y

I u y I u y I u y

→∞ ∈Ξ

≤ =

Hence,

(

* *

)

,

u y is an optimal pair, and we arrive at the required conclusion. 

5. 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 Λ =: L₊p

( )

Γ_S (see (32)) by its solid Henig approximation

( )

Λ _ε (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 Λ ⊂Z be a closed ordering cone in Z.

Definition 5.1. A nonempty convex subset B of a nontrivial ordering cone Λ ⊂Z (i.e. Λ ≠

{ }

0Z , where 0_Z is the zero element in Z) is called base of Λ if for each element z∈ Λ\ 0

{ }

Z there is a unique repre-

sentation z=µb where µ>0 and b∈B.

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 z z1, 2∈Z we

de-fine

{ }

1 2 2 1 as well as 1 2 2 1 \ 0Z .

z ≤_Λ z ⇔z − ∈ Λz z <_Λ z ⇔ − ∈ Λz

(10)

*_, , _{Z Z}

⋅ ⋅ be the dual pairing. Moreover, by

{

*

}

* * * *

,

: | , 0

Z Z

z Z z z z

Λ = ∈ ≥ ∀ ∈ Λ

and

{ }

{

*

}

# * * *

,

: z Z | z z, _{Z Z} 0 z \ 0Z

Λ = ∈ > ∀ ∈ Λ

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):

{

*

}

* * *

,

| , 0

Z Z

z Z z z z

Λ = ∈ ≥ ∀ ∈ Λ

Due to Lemma 1.28 in [25], we can give the following result.

Lemma 2. Let Λ ⊂Z be a nontrivial ordering cone in a Banach space Z. Then the set

{

*

}

* ,

: | , 1

Z Z

B = z∈ Λ z z = is a base of Λ for every * #

z ∈ Λ . Moreover, if Λ is reproducing in Z, i.e. if

Z

Λ − Λ = , and if B is a base of Λ, then there is an element z*∈ Λ# satisfying

{

*

}

* ,

: | , 1

Z Z

B = z∈ Λ z z = .

Remark 5.3. As follows from Lemma 2, the set

( )

{

1

}

: | d 1

S

p N

S

B = ξ∈L+ Γ

∫

_Γξ  − = (36)

is a closed base of ordering cone Λ =: Lp₊

( )

Γ_S .

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 Λ ⊂Z be a closed ordering cone with a closed base B.

Choosing ε >0 arbitrarily, the corresponding Henig dilating cone is defined by

( )

: cl

(

cone

(

( )

0

)

: cl

(

{

| 0,

( )

0

}

)

,

Z Z Z Z

B B B z z B B

ε ⋅ ε ⋅ µ µ ε

Λ = + = ≥ ∈ +

where ε1Bε

( )

0Z := ∈

{

v Z| vZ ≤1

}

is the closed unit ball in Z centered at the origin.

It is clear that Λ_ε

( )

B depends on the particular choice of B. As follows from this definition, int

( ) ( )

Λ _ε B ≠ ∅ for every ε>0, 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 Λ ⊂Z be a closed ordering cone with a closed base B.

Choosing ε∈

(

0,δ

)

, where

{

}

: inf b_Z |b B 0,

δ = ∈ > (37) the following statements hold true.

1) Λ_ε

( )

B is pointed, i.e. Λε

( )

B ∩ −Λ

(

ε

( )

B

)

=

{ }

0Z ; 2) Λ_ε

( )

B ⊂ Λ_{ε γ}₊

( )

B ∀ >γ 0;

3)

( )

cone cl

(

( )

0

)

Z Z

B B B

ε ⋅ ε

Λ = + ;

4)

( )

0

B

ε ε δ

< <

Λ =

_

Λ ; 5) the implication

( ) ( )

( )

,

Z

B

ε

ξ ξ ξ

κ ε

∈ Λ ⇒ ₊ + ∈ −Λ/ (38)

. . _Z

i e ξ ε ξ κ ε

Λ − ₊



(11)

In the context of constraint qualifications problem, the following result plays an important role.

Proposition 6. Let Z be a normed space, and let Λ ⊂Z be a closed ordering cone with a closed base B.

Choosing ε∈

(

0,δ

)

arbitrarily, where δ is defined by (37), the inclusion

{ }

0Z int

(

ε

( )

B

)

Λ ⊂ ∪ Λ (39) holds true.

Proof. Let z∈ Λ\ 0

{ }

_Z be chosen arbitrarily. By the definition of a base there is a unique representation

z=λb with λ>0 and b∈B. Obviously,

{ }

( )

(

)

(

( )

)

int 0Z int

z∈ λb +B_λε = B_λε λb

holds true. Let’s assume for a moment that

( )

cone

(

{ }

( )

0Z

)

.

Bλε λ ⊆b b +Bε (40)

Then we obtain

{ }

( )

(

)

(

)

(

( )

)

(

( )

)

int cone 0Z int cone 0Z int ,

z∈ b +Bε ⊆ B+Bε = Λε B

which completes the proof. In order to show (40), let x∈B_λε

( )

λb be chosen arbitrarily, i.e. .

Z x−λb ≤λε

Then

1

Z Z

x

b x λb λε ε

λ − =λ − ≤ λ =

yields

{

| Z , 0

}

cone

(

{ }

( )

0Z

)

.

x∈ µy y−b ≤ε µ≥ = b +B_ε

As a result, (40) is satisfied. 

Remark 5.2. The following property, coming from Proposition 6, turns out rather useful: in order to prove

( )

(

)

int

z∈ Λ_ε B , it is sufficient to check whether z∈ Λ\ 0

{ }

_Z .

The following result shows that Henig dilating cones Λ_ε

( )

B possess good approximation properties.

Proposition 7. Let Λ be a closed ordering cone in a normed space Z, and letB be an arbitrary closed base of Λ. Let parameter δ be defined as in (37), and let

( )

_k

( )

0,

k

ε _∈ ⊂ δ

 be a monotonically decreasing sequ-

ence such that lim k 0

k→∞ε = . Then the sequence of cones

{

Λεk

( )

B

}

k∈ converges to Λ in Kuratowski sense

with respect to the norm topology of Z as k tends to infinity, that is

( )

lim inf _k lim sup _k ,

k _k

K _→∞ ε B K ε B

→∞

− Λ = Λ = − Λ

where

( ) {

( )

}

0 0

lim inf : | for all neighborhoods of there is

such that ,

k

K B z Z N z a

k N B k k

ε

→∞

− Λ = ∈

∈ ∩ Λ ≠ ∅ ∀ ≥

( ) {

( )

}

0

lim sup : | for all neighborhoods of and every

there is such that .

k

k k

K B z Z N z k

a k k N B

ε

ε →∞

− Λ = ∈ ∈

≥ ∩ Λ ≠ ∅



Proof. Let z∈ Λ be chosen arbitrarily. Then N∩ Λ ≠ ∅ holds true for every neighborhood N of z, and due to the inclusions

k

ε

Λ ⊂ Λ ∀ ∈k , we see that

k

N∩ Λ ≠ ∅_ε for all k∈. Hence,

( )

lim inf _k . k

K _→∞ ε B

(12)

( )

lim inf lim sup ,

k k

k _k

K _ε B K _ε B

→∞ _→∞

− Λ ⊆ − Λ

we get

( )

lim inf lim sup .

k k

k _k

K _ε B K _ε B

→∞ _→∞

Λ ⊆ − Λ ⊆ − Λ (42) To show that the sequence

{

( )

}

k B k

ε _∈

Λ

 converges to Λ in Kuratowski sense, it remains to show

( )

lim sup .

k

K _ε B

→∞

− Λ ⊆ Λ (43) However, the inclusion (43) is equivalent to

(

\

)

\ lim sup

( )

. k k

Z Z K _ε B

→∞

 

Λ ⊆_ − Λ _

  (44) Let z∈Z \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 k0∈ such that

( )

0.

k

N∩ Λ_ε B = ∅ ∀ ≥k k

This implies

( )

\ lim sup .

k k

z Z _ε B

→∞

∈ Λ

Combining (42), (43), and (44), we arrive at the relation

( )

lim inf lim sup .

k k

k _k

K _ε B K _ε B

→∞ _→∞

Λ ⊆ − Λ ⊆ − Λ ⊆ Λ

Thus, lim

( )

k k

K _ε B

→∞

Λ = − Λ and the proof is complete.

Taking these results into account, we associate with OCP (19) the following family of Henig relaxed problems

(

)

{

2

}

Minimize I u y, =

∫

_Ω y−y_d dx+

∫

_ΩDu (45) subject to the constraints

( )

(

)

( )

(

( )

)

( )

(

( )

)

( )

2

0 2 1 , 0

1 , 1

1 , 0

1

, in ; ,

0 in ,

,

p D

p p

q p D p p

S p

S

max p

S

ad

u y F y f C

y W

y L B

u

ε

γ γ γ γ

ζ γ

∗ ∞

−

+



∆ + = Ω Γ _



= ∂Ω _



= Γ



∆ = Γ _



∈ Γ _



− ∈ Γ 



∈A _

(46)

or in a more compact form each of these problems can be stated as follows

( )u y,inf∈ΞεI u y

( )

, , ∀ ∈ε

( )

0,δ , (47)

where

( )

{

}

inf p : ,

S

L B

δ = ξ _Γ ξ∈ (48) the base B takes the form (36), and the feasible set Ξ ⊂ε BV

( )

Ω ×p

( )

Ω we define as follows:

( )

u y, ∈ Ξε

if and only if u∈Aad, I u y

(

,

)

< +∞, y∈p

( )

Ω , the pair

(

u y,

)

is related by the Minty inequality (21),

(13)

( )

(

( )

)

( )

(

( )

)

( )

1 , 1 .

p max p

S S

y L B y L B

ε ε

γ ∈ + Γ ζ −γ ∈ + Γ (49)

Here,

(

Lp+

( )

ΓS

)

_ε

( )

B stands for the corresponding Henig dilating cone.

Since, by Proposition 6, the inclusion Ξ ⊆ Ξ_ε 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 7, the convergence

0

ε

ε →

Ξ →Ξ in Kuratowski sense holds true with respect to the τ-topology on BV

( )

Ω ×p

( )

Ω .

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 (H₁) by the non-emptiness properties of feasible sets

ε

Ξ for the corresponding Henig relaxed problems.

Theorem 8. Let

{ }

_k

( )

0, k

ε _∈_⊂ δ be a monotonically decreasing sequence converging to 0 as k→ ∞. Then, for given distributions f ∈W−2,q

(

Ω Γ; _D

)

, yd∈L2

( )

Ω , and

( )

max p

L

ζ ∈ ∂Ω , the Hypothesis (H1) implies that

the Henig relaxed problem (47) has a nonempty set of feasible solutions Ξε for all ε ε= k, k∈. And vice

versa, if there exists a sequence

{

(

k, k

)

}

k

u y

∈_ satisfying conditions

(

,

)

for all , and sup

(

,

)

,

k

k k k k

k

u y _ε k I u y

∈

∈ Ξ ∈ < +∞



 (50) then the sequence s

{

(

k, k

)

}

k

u y

∈ is τ-compact and each of its τ-cluster pairs is a feasible solution to the original OCP (19).

Proof. Since the implication

(

Ξ ≠ ∅ ⇒ Ξ ≠ ∅

) (

_ε for allε >0

)

is obvious by Proposition 7, we concentrate on the proof of the inverse statement—property (50) implies the existence of at least one pair

(

u y,

)

such that

(

u y,

)

∈ Ξ. Let

{

(

k, k

)

}

k

u y

∈ be an arbitrary sequence with property:

(

,

)

k k k

u y ∈ Ξ_ε for all k∈. Since the set A_ad

and a priory estimate (24) do not depend on parameter ε_k and the condition (50)2 implies supk∈

∫

_Ω Duk ≤ ∞, it follows by compactness arguments (see the proof of Theorem 4) that there exist a subsequence of

{

(

k, k

)

}

k

u y

∈ (still denoted by the same index) and a pair

(

u y,

)

ad p

( )

∗ ∗ _∈_A _×_ _Ω

such that

(

k, k

)

(

,

)

as .

u y τ→ u∗ y∗ k→ ∞

Closely following the proof of Theorem 3, it can be shown that the limit pair

(

u∗,y∗

)

is such that u ad

∗_∈

A ,

(

,

)

J u y∗ ∗ < +∞, and function y p

( )

∗_∈_ _Ω

is a weak solution to the boundary value problem (15) - (16). Moreover, in view of the compactness properties of injections (6), we may suppose that

( )

1 1 strongly in as .

k p

S

y y L k

γ →γ Γ → ∞ (51) It remains to establish the inclusions

( )

1 , 1 ,

p max p

S S

y L y L

γ ∗ ζ γ ∗

+ +

∈ Γ − ∈ Γ (52) By contraposition, let us assume that : 1

( )

\

( )

max p p

S S

y L L

ξ∗ ζ γ ∗

+

= − ∈ Γ Γ . Since the cone Lp+

( )

ΓS is closed, it follows that there is a neighborhood 

( )

ξ∗ of ξ∗ in Lp

( )

ΓS such that

( )

p S

L

ξ∗ +

∩ Γ = ∅

 .

Using the fact that

( )

(

( )

)

( )

(

( )

)

( )

, ,

k l

p p p

S S S

L L B L B k l

ε ε

+ Γ ⊂ + Γ ⊆ + Γ ∀ ≥

by Proposition 7 and definition of the Kuratowski limit, it is easy to conclude the existence of an index k₀∈ such that

( ) (

( )

)

( )

, 0. k

p S

L B k k

ε

ξ∗ +

∩ Γ = ∅ ∀ ≥

 (53)

However, in view of the strong convergence property (51), there is an index k1∈ satisfying

( )

, 1.

k

k k

ξ _∈_ ξ∗ _{∀ ≥}

(54) Combining (53) and (54), we finally obtain

( )

(

( )

)

( )

{

}

1 \ , max 0, 1 .

k

k max k p p

S S

y L L B k k k

ε