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5 ``Directional Stability'' of Weighted Sobolev Spaces

# 5 ``Directional Stability'' of Weighted Sobolev Spaces

Tải bản đầy đủ - 0trang

24 Optimality Conditions for L 1 -Control in Coefficients …

443

Therefore, in order to prove the inclusion y ∈ Hu , it is enough to show that

∇yε ∈ L p (Ω, u dx)N for all ε > 0 and ∇yε

∇y in L p (Ω, u dx)N .

(24.45)

The first assertion in (24.45) is obvious because ∇yε ∈ L p (Ω, uε dx)N and uε ≥ u

almost everywhere in Ω for all ε > 0. As for the weak convergence property (24.45)2 ,

we note that

(1) uε − u → 0 almost everywhere in Ω (by the initial assumptions);

∇y in L 1 (Ω)N by Lemma 24.2.

(2) ∇yε

Hence, ∇yε (uε − u) → 0 almost everywhere in Ω and in view of estimate

Ω

|∇yε |u dx ≤

Ω

|∇yε |uε dx ≤ ∇yε

L p (Ω,uε dx)N

(p−1)/p

L 1 (Ω) ,

the sequence {∇yε (uε − u)}ε>0 is equi-integrable. Therefore, Lebesgue theorem

implies that

(24.46)

∇yε (uε − u) → 0 in L 1 (Ω)N as ε → 0.

Taking (24.46) into account the fact that the smooth compactly supported functions

are dense in L p (Ω, u dx)N , for every ϕ ∈ C0∞ (Ω), we get

Ω

(∇yε , ∇ϕ)RN u dx −

+

Ω

(∇y, ∇ϕ)RN u dx ≤

Ω

Ω

(∇yε , ∇ϕ)RN uε dx −

(∇yε (uε − u), ∇ϕ)RN dx

Ω

(∇y, ∇ϕ)RN u dx = I1 + I2 ,

where I1 tends to zero as ε → 0 by (24.46) and I2 → 0 by (24.44). Thus, ∇yε

weakly in L p (Ω, u dx)N , and hence, y ∈ Hu .

∇y

Lemma 24.5 Under the assumptions of Lemma 24.4, we have the strong convergence property: yε → y in L p (Ω), ∇yε → ∇y in the variable space L p (Ω, uε dx)N .

Proof Taking into account the result of Lemma 24.4 and the following arguments of

Remarks 24.4 and 24.5, for each ε > 0, we have the energy equalities

Ω

uε (x)|∇yε |p dx =

Ω

fyε dx,

Ω

u(x)|∇y|p dx =

fy dx.

(24.47)

Ω

Since ∇yε

∇y in the variable space L p (Ω, uε dx)N , we derive from (24.47) the

following relation:

Ω

u(x)|∇y|p dx ≤ lim inf

ε→0

= lim inf

ε→0

Ω

Ω

uε (x)|∇yε |p dx

fyε dx =

Ω

fy dx =

u(x)|∇y|p dx.

Ω

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P.I. Kogut and O.P. Kupenko

Hence, lim

ε→0 Ω

uε (x)|∇yε |p dx =

u(x)|∇y|p dx and this implies the strong converΩ

gence ∇yε

∇y in the variable space L p (Ω, uε dx)N . Using the fact that Huε ⊂ Hu

(because uε ≥ u in Ω) and the embedding Hu → L p (Ω) is compact, we finally

conclude the strong convergence yε → y in L p (Ω). The proof is complete.

Lemma 24.6 Let u ∈ Aad and {uε }ε>0 ⊂ L 1 (Ω) be such that ξ1 (x) ≤ uε (x) ≤ u(x)

≤ ξ2 (x) almost everywhere in Ω for all ε > 0, and uε → u in L 1 (Ω) as ε → 0. For

each ε > 0, let yε = y(uε ) be Wuε -solutions to the boundary value problem (24.26).

τ

Then, up to a subsequence, we have (uε , yε ) −→ (u, y) as ε → 0, where y ∈ Wu

and y is a Wu -solution to the boundary value problem (24.26) for the given control

u. Moreover, in this case, we have

∇yε → ∇y in the variable space L p (Ω, uε dx)N .

Proof By the arguments of the proof of Lemma 24.4, we conclude that up to a

τ

subsequence, (uε , yε ) −→ (u, y) as ε → 0, where y ∈ Wu is a weak solution of

(24.26) in the sense of Minty. Taking into account the definition of Wuε -solution, for

each ε > 0, we have

Ω

uε (x)|∇ϕ|p−2 (∇ϕ, ∇ϕ − ∇yε )RN dx ≥

Ω

f (ϕ − yε ) dx, ∀ ϕ ∈ Wuε . (24.48)

However, for an arbitrary function ϕ ∈ Wu , because of inequality uε ≤ u, we have:

ϕ ∈ Wuε . Moreover, the strong convergence uε → u in L 1 (Ω) and Lebesgue theorem

imply

lim

ε→0 Ω

|∇ϕ|p uε dx =

Ω

|∇ϕ|p u dx,

i.e., ∇ϕ → ∇ϕ strongly in the variable space L p (Ω, uε dx)N . Therefore, passing to

the limit in (24.48) with an arbitrary ϕ ∈ Wu , we arrive at the relation

Ω

u(x)|∇ϕ|p−2 (∇ϕ, ∇ϕ − ∇y)RN dx ≥

Ω

f (ϕ − y) dx, ∀ ϕ ∈ Wu ,

(24.49)

i.e., y is the Wu -solution to the boundary value problem (24.26). The strong convergence properties for the sequence ∇yε ∈ L p (Ω; uε dx)N ε>0 can be established by

analogy with Lemma 24.5.

Remark 24.6 It is easy to show that Lemma 24.6 remains true if we replace the

conditions ξ1 (x) ≤ uε (x) ≤ u(x) ≤ ξ2 (x) almost everywhere in Ω for all ε > 0, and

uε → u in L 1 (Ω) as ε → 0 by the following stability property of the Sobolev space

Wu : if uε → u in L 1 (Ω), then Wuε ⊇ Wu for ε > 0 small enough. It should be emphasized that from an optimal control theory point of view, an L 1 -approximation of

element u ∈ Aad by the sequence {uε }ε>0 is meaningful if only {uε }ε>0 are admissible controls. However, the set Aad has an empty topological interior. Therefore, the

24 Optimality Conditions for L 1 -Control in Coefficients …

445

existence of L 1 -convergent sequences of admissible controls {uε }ε>0 ⊂ Aad , with

monotone property ξ1 (x) ≤ uε (x) ≤ u(x) ≤ ξ2 (x) or ξ1 (x) ≤ u(x) ≤ uε (x) ≤ ξ2 (x)

in Ω, is an unrealistic assumption.

Taking this observation into account, it is reasonable to introduce the following

concept.

Definition 24.5 Let u, u ∈ Aad be a given pair of admissible controls. Let uε :=

u + ε(u − u) for each ε ∈ [0, 1]. We say that the weighted Sobolev space Hu is

stable along the direction u − u if Hu = limε→0 Huε in the following sense:

(K1 )

(K2 )

τ

for every y ∈ Hu , there exists a sequence yε ∈ Huε ε>0 such that (uε , yε ) −→

(u, y) as ε → 0;

if {εk }k∈N is a sequence converging to 0, and {yk }k∈N is a sequence such that

τ

yk ∈ Huεk for every k ∈ N and (uεk , yk ) −→ (u, y), then yk → y strongly in

p

L (Ω) and y ∈ Hu .

The definition of the limit Wu = limε→0 Wuε can be done in a similar manner. As a

result, Lemmas 24.4–24.6 can be easily generalized to the following assertion.

Lemma 24.7 Assume that for a given u, u ∈ Aad , the weighted Sobolev space Hu

(respectively, Wu ) is stable along the direction u − u. For each ε > 0, let uε := u +

ε(u − u) and let yε = y(uε ) be Huε -solutions (resp., Wuε -solutions) to the boundary

value problem (24.26). Then, up to a subsequence, we have

uε → u in L 1 (Ω),

(24.50)

yε → y in L (Ω), ∇yε →∇y in variable space L (Ω, uε dx) ,

p

p

N

(24.51)

where y ∈ Hu is the Hu -solution (resp., (24.50) and (24.51) take place and y ∈ Wu

is the Wu -solution) to the boundary value problem (24.26) for the given control u.

As for the proof, it is enough to apply Definition 24.5 and repeat the main arguments of the proofs of Lemmas 24.4–24.6.

Our next observation deals with some specification of the set of admissible controls

Aad . Having supposed that the functions ξ1 and ξ2 are extended to the whole space

of RN such that

1

1

(RN ), 0 ≤ ξ1 (x) ≤ ξ2 (x) a.e. in Ω, and ξ1−ν ∈ Lloc

(RN ),

ξ1 , ξ2 ∈ Lloc

we assume that there exists a constant C > 0 such that

sup

B∈RN

1

|B|

ξ2 dx

B

1

|B|

−1/(p−1)

B

ξ1

p−1

dx

≤ C,

(24.52)

where B is a ball in RN . In this case, we have the following result.

Theorem 24.5 Assume the condition (24.52) holds true for some constant C > 0.

Then, boundary value problem (24.26) has a unique weak solution for each u ∈ Aad .

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P.I. Kogut and O.P. Kupenko

Proof The main idea of this proof is to show that Wu = Hu for each u ∈ Aad and this

relies on the fact that such u belongs to the class of Muckenhoupt weights Ap . We

omit the details.

Corollary 24.1 Let u, u ∈ Aad be a given pair of admissible controls. Let uε :=

u + ε(u − u) for each ε ∈ [0, 1]. Assume there exists a constant C > 0 such that the

estimate (24.52) is valid. For each ε > 0, let yε = y(uε ) be the corresponding weak

solutions to the boundary value problem (24.26). Then, up to a subsequence, the

properties (24.50) and (24.51) hold true, where y ∈ Hu is the weak solutions to the

boundary value problem (24.26) for the given control u.

Proof As follows from Theorem 24.5, the weak solutions yε = y(uε ) ∈ Wuε ε>0

can be defined in a unique way. Moreover, by a priori estimate (24.31), we see

that the sequence {(uε , yε )}ε>0 is bounded. Hence, by Theorem 24.3, this sequence is

relatively τ -compact and each of its τ -cluster pairs (u, y) belongs to the set Ξ . Since,

unique weak solution, it follows that the τ -cluster pair (u, y) is uniquely defined. In

order to show the strong convergence property (24.51), it remains to repeat the trick

coming from the proof of Lemma 24.5.

24.6 On Differentiability of Lagrange Functional

In this section, we discuss the differentiable properties of the Lagrange functional

associated with optimal control problem (24.26)–(24.28). Since the relations (24.26)

can be seen as constraints, we define the Lagrangian as follows:

(u, y, μ) = I(u, y) + au (y, μ) −

= y − yd

p

L p (Ω)

+ ∇y

Ω

f μ dx

p

L p (Ω,u dx)N

+ au (y, μ) −

Ω

f μ dx,

(24.53)

1,p

where μ ∈ Wu := W0 (Ω, u dx) is a Lagrange multiplier and

au (y, μ) = −Δp (u, y), μ

Wu∗ ;Wu

=

Ω

u(x) |∇y|p−2 ∇y, ∇μ

RN

dx.

In what follows, to each distribution y ∈ Wu , where u ∈ Aad , we associate the

following sets:

S0 (y) = {x ∈ Ω : |∇y(x)| = 0} , S1 (y) = int S0 (y).

For our further analysis, we adopt the following concept.

(24.54)

24 Optimality Conditions for L 1 -Control in Coefficients …

447

Definition 24.6 Let u ∈ Aad be a given control. We say that an element y ∈ Wu is a

regular point for the Lagrangian (24.53) if

Ω\S 1 (y) is a connected set with Lipschitz boundary,

and S0 (y)\S1 (y) has zero u-measure, i.e. u(S0 ) :=

(24.55)

u dx = 0,

(24.56)

S0

and we say that an element y ∈ Wu is a strongly regular point for the Lagrangian

(24.53) if y is its regular point and S1 (y) = ∅.

Remark 24.7 In the non-degenerate case, i.e., when u + u−1 ∈ L ∞ (Ω), due to the

results of Manfredi (see [27]), the notion of regularity is not too restrictive. In

this case, the set S0 := {x ∈ Ω : |∇y| = 0} for non-constant solutions of the pLaplace equation (a p-harmonic function) has zero Lebesgue measure (see [27]).

However, in the case of degenerate p-Laplacian Δp (u, y) with u ∈ Aad , the regularity assumption is not obvious. Therefore, we put forward a hypothesis that

if y ∈ Wu is a regular point of the functional (u, y, μ), then for every v ∈ Wu

there exists a positive number α ∈ R (α = 0) such that each point of the segment

[y, αv] = {y + t(αv − y) : ∀ t ∈ [0, 1]} ⊂ Wu is also regular for (u, y, μ).

We are now ready to study the differentiability properties of the Lagrangian

(u, y, λ). We begin with the following result.

Lemma 24.8 Let u ∈ Aad be the given control, and let y ∈ Wu be a regular point of

the Lagrangian (24.53). Then, the mapping

Wu

y → Δp (u, y) = −div u(x)|∇y|p−2 ∇y ∈ Wu∗

is Gâteaux differentiable at y and its Gâteaux derivative −Δp (u, y)

exists and takes the form:

G

∈ L Wu , Wu∗

⎨ −div u(x)|∇y|p−2 ∇h

Δp (u, y) G [h] = −(p − 2) div u(x)|∇y|p−4 (∇y, ∇h)RN ∇y , in Ω\S1 (y),

0,

in S1 (y);

(24.57)

for p > 2, and

Δp (u, y)

G

[h] = −div (u(x)∇h) in Ω, if p = 2.

(24.58)

Proof Let y ∈ Wu be a regular point for the Lagrangian (24.53), and let h ∈ Wu be

an arbitrary distribution. Following the definition of Gâteaux derivative, we have to

deduce the following equality:

lim

λ→+0

Δp (u, y + λh) − Δp (u, y)

− Δp (u, y)

λ

G

[h]

Wu∗

= 0,

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P.I. Kogut and O.P. Kupenko

where Δp (u, y) G [h] is defined by (24.57). With that in mind, let us consider the

vector-valued function g(λ) := |∇y + λ∇h|p−2 (∇y + λ∇h) for which the Taylor’s

expansion with the remainder term in the Lagrange form leads to the relation

|g(λ) − g(0)| ≤ |g (θ )|λ, θ ∈ (0, λ),

where g(0) = |∇y|p−2 ∇y and

g (θ ) = |∇y + θ ∇h|p−2 ∇h

+ (p − 2)|∇y + θ ∇h|p−2 θ |∇h|2 + (∇y, ∇h)RN

=|∇y + θ ∇h|p−2 ∇h + (p − 2)|∇y + θ ∇h|p−2

∇y + θ ∇h

(∇y + θ ∇h, ∇h)RN

|∇y + θ ∇h|

1

|∇y + θ ∇h|2

∇y + θ ∇h

|∇y + θ ∇h|

Let δ > 0 be an arbitrary value. Let us consider the following decomposition:

Ω = S1 (y) ∪ S0 (y)\S1 (y) ∪ Ωδ ∪ Ωδ ,

where the sets S0 (y) and S1 (y) are defined in (24.54), and Ωδ and Ωδ are u-measurable

subsets of Ω such that

Ωδ = {x ∈ Ω\S1 (y) : |∇y(x)| ≥ δ} , Ωδ = {x ∈ Ω\S1 (y) : 0 < |∇y(x)| < δ} .

Closely following [1, p.598] (see also [11]), it can be shown that for every ε > 0,

there exists a positive value δ0 > 0 such that

g (θ ) − (p − 2)|∇y|p−4 (∇y, ∇h)RN ∇y − |∇y|p−2 ∇h

g (θ ) − (p − 2)|∇y|

p−4

(∇y, ∇h)RN ∇y − |∇y|

p−2

∇h

L q (Ω

δ

;u dx)N

L q (Ωδ ;u dx)N

ε

, (24.59)

2

ε

(24.60)

<

2

<

for all δ ∈ (0, δ0 ), θ ∈ (0, λ), and λ > 0 small enough. Moreover, as immediately

follows from (24.54), we have the following relations:

g (θ)

L q (S1 (y);u dx)N

g (θ) − ∇h

= (p − 1)θ p−2 ∇h

L q (S1 (y);u dx)N

p−1

→ 0 as θ → +0 if p > 2,

L p (S1 (y);u dx)N

= ∇h − ∇h Lq (S1 (y);u dx)N = 0 if p = 2.

Since u S0 (y)\S1 (y) = 0 by the initial assumptions, it follows from (24.59) and

(24.60) that the vector-valued function |∇y|p−2 ∇y is Gâteaux differentiable. Hence,

the operator Δp (u, y) = −div u(x)|∇y|p−2 ∇y is Gâteaux differentiable at each regular point y ∈ Wu and its Gâteaux derivative takes the form (24.57).

24 Optimality Conditions for L 1 -Control in Coefficients …

449

As an obvious consequence of this result and the fact that Gâteaux differentiability

of operator y → Δp (u, y) implies existence of Gâteaux derivative for the functional

ϕ : Wu → R, where

ϕ(y) = −Δp (u, y), μ

=

W ∗ ;Wu

and ϕG (y), h Wu∗ ;Wu = −Δp (u, y)

lowing obvious assertion.

G

Ω

u(x) |∇y|p−2 ∇y, ∇μ

[h], μ

Wu∗ ;Wu ,

RN

dx

∀ μ ∈ Wu , we arrive at the fol-

Corollary 24.2 Let u ∈ Aad be a given element, and let y ∈ Wu be a regular point

of the Lagrangian (24.53). Then, the mapping

y→

Wu

(u, y, μ) = I(u, y) + au (y, μ) − f , μ

is Gâteaux differentiable at y and its Gâteaux derivative

and takes the form:

G (u, y, μ), h Wu∗ ;Wu

+p

Ω

|y − yd |

=p

p−2

+ (p − 2)

Ω\S 1 (y)

Ω

|y − yd |

p−2

Ω

G (u, y, μ)

∈ Wu∗ exists

Ω\S 1 (y)

u(x)|∇y|p−2 ∇μ, ∇h

u(x)|∇y|p−4 (∇y, ∇μ)RN ∇y, ∇h

RN

RN

dx

dx

|∇y|p−2 (∇y, ∇h)RN u dx

(y − yd ) h dx +

+ (p − 2)

∈R

|∇y|p−2 (∇y, ∇h)RN u dx

(y − yd ) h dx +

=p

+p

Ω\S 1 (y)

Wu∗ ;Wu

Ω

Ω

u(x)|∇y|p−2 ∇μ, ∇h

u(x)|∇y|p−4 (∇y, ∇μ)RN ∇y, ∇h

RN

RN

dx

dx.

(24.61)

Remark 24.8 Taking into account the equality (∇y, ∇μ)RN ∇y = ∇y ⊗ ∇y ∇μ,

the last term in (24.61) can be rewritten as follows:

(p − 2)

u(x)|∇y|p−4

Ω

∇y ⊗ ∇y ∇μ, ∇h

RN

dx.

Before deriving the optimality conditions, we need the following auxiliary result.

Lemma 24.9 Let u ∈ Aad , y ∈ Wu , and v ∈ Wu be the given distributions. Assume

that each point of the segment [y, v] = {y + α(v − y) : ∀ α ∈ [0, 1]} ⊂ Wu is regular for the mapping v → (u, v, μ). Then, there exists a positive value ε ∈ (0, 1)

such that

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P.I. Kogut and O.P. Kupenko

(u, v, μ) −

=

Ω

+p

+p

(u, y, μ) =

G (u, y

|∇y + ε∇h|p−2 ∇μ, ∇h

Ω

Ω

+ εh, μ), h

RN

Wu∗ ;Wu

u dx

|∇y + ε∇h|p−2 ∇y + ε∇h, ∇h

RN

u dx

|y + εh − yd |p−2 y + εh − yd h dx

+ (p − 2)

Ω

|∇y + ε∇h|p−4

(∇y + ε∇h) ⊗ (∇y + ε∇h) ∇μ, ∇h

RN

u dx

(24.62)

with h = v − y.

Proof For given u, μ, yd , y, and v, let us consider the scalar function ϕ(t) =

(u, y + t(v − y), μ). Since by Corollary 24.2, the functional (u, ·, μ) is Gâteaux

differentiable at each point of the segment [y, v], it follows that the function ϕ = ϕ(t)

is differentiable on (0, 1) and

ϕ (t) =

G (u, y

+ t(v − y), μ), v − y

Wu∗ ;Wu ,

∀ t ∈ (0, 1).

To conclude the proof, it remains to take into account (24.61) and apply the mean

value theorem: ϕ(1) − ϕ(0) = ϕ (ε) for some ε ∈ (0, 1).

24.7 Formalism of the Quasi-adjoint Technique

Let (u0 , y0 ) ∈ Ξ be an optimal pair for problem (24.26)–(24.28). Let Astab

A

,

of L 1 (Ω) such that Astab

∇y0

p

L p (Ω,u dx)N

:=

Ω

|∇y0 |p u dx < +∞ and u/u0 ∈ L ∞ (Ω), for all u ∈ Astab

(24.63)

and the weighted Sobolev space Hu0 is stable along the direction u − u0 for each

stab

stab

u ∈ Astab

of this set.

We begin with the following assumption:

(H0)

(H1)

(H2)

y0 is an Hu0 -solution to the boundary value problem (24.26).

For a given distribution f ∈ L q (Ω), the optimal state y0 ∈ Hu0 is a regular

point of the mapping y → (u, y, λ) in the sense of Definition 24.6.

The set Astab

24 Optimality Conditions for L 1 -Control in Coefficients …

451

Then,

Δ

=

(u0 , y0 , λ) ≥ 0, ∀ (u, y) ∈ Ξ, ∀ λ ∈ C0∞ (Ω).

(u, y, λ) −

(24.64)

Since the set of admissible controls Aad ⊂ BV (Ω) has an empty topological

interior, we justify the choice of perturbation for an optimal control as follows: uθ :=

u0 + θ (u − u0 ), where u ∈ Astab

ad and θ ∈ [0, 1]. As was indicated in Remark 24.3, for

each θ ∈ [0, 1], there exists a unique Huθ -solution yθ := y (uθ ) = y (u0 + θ (u − u0 ))

to boundary value problem (24.34) and (24.35). Then, due to Hypotheses (H0)–(H2),

we can suppose that the segment [y0 , yθ ] belongs to Hu0 for θ small enough (by the

directional stability property). We also assume that

for each u ∈ Astab

ad , there exists a numerical sequence {θk }k∈N ⊂ (0, 1] such

that θk → 0 as k → ∞, and yθk := y uθk k∈N are strongly regular points

for the mappind v → (u, v, λ).

(H3)

We note that if y0 ∈ Hu0 is a strongly regular point of the mapping y →

then fulfillment of Hypothesis (H3) is obvious. As a result, we obtain

Δ

(u, y, λ),

= (uθ , yθ , λ) − (u0 , y0 , λ) = (uθ , yθ , λ) − (uθ , y0 , λ)

+ (uθ , y0 , λ) − (u0 , y0 , λ) = (uθ , yθ , λ) − (uθ , y0 , λ)

+ (uθ − u0 , y0 , λ) = Δyθ (uθ , y0 , λ) + θ (ˆu − u0 , y0 , λ) ≥ 0. (24.65)

Hence, by Lemma 24.9, there exists a value εθ ∈ (0, 1) such that condition (24.65)

can be represented as follows:

Δ

(uθ , yθ , λ) −

=

=

G (uθ , y0

(u0 , y0 , λ)

+ εθ (yθ − y0 ), λ), yθ − y0

Hu∗0 ;Hu0

+ θ (u − u0 , y0 , λ) ≥ 0.

(24.66)

Using (24.61), we obtain

Δ

=p

Ω

|∇yεθ ,θ |p−2 ∇yεθ ,θ , ∇yθ − ∇y0

+p

+

Ω

Ω

|∇yεθ ,θ |p−2 ∇λ, ∇yθ − ∇y0

Ω

uθ dx + θ

Ω

(u − u0 ) |∇y0 |p dx

|yεθ ,θ − yd |p−2 yεθ ,θ − yd (yθ − y0 ) dx

+ (p − 2)

RN

Ω

|∇yεθ ,θ |p−4

uθ dx

∇yεθ ,θ ⊗ ∇yεθ ,θ ∇λ, ∇yθ − ∇y0

(u − u0 ) |∇y0 |p−2 ∇y0 , ∇λ

where yεθ ,θ = y0 + εθ (yθ − y0 ).

RN

RN

dx ≥ 0, ∀ u ∈ Astab

RN

uθ dx

(24.67)

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P.I. Kogut and O.P. Kupenko

Now, we introduce the concept of quasi-adjoint states that were first considered

for linear problems by Serovajskiy [31].

Definition 24.7 We say that for given θ ∈ [0, 1] and u ∈ Aad , a distribution ψθ is a

quasi-adjoint state to y0 ∈ Hu0 if ψθ satisfies the following integral identity:

Ω

|∇yεθ ,θ |p−2

I + (p − 2)

+p

+p

Ω

Ω

∇yεθ ,θ

∇yεθ ,θ

∇ψθ , ∇ϕ

|∇yεθ ,θ | |∇yεθ ,θ |

|∇yεθ ,θ |p−2 ∇yεθ ,θ , ∇ϕ

RN

RN

uθ dx

uθ dx

|yεθ ,θ − yd |p−2 yεθ ,θ − yd ϕ dx = 0, ∀ϕ ∈ Huθ ,

(24.68)

or in terms of distributions, ψθ is a solution to the following degenerate boundary

value problem

− div(ρθ Aθ ∇ψθ ) = gθ in Ω,

ψθ = 0 on ∂Ω.

(24.69)

Here,

ρθ = uθ |∇yεθ ,θ |p−2 ,

∇yεθ ,θ

∇yεθ ,θ

Aθ = I + (p − 2)

,

|∇yεθ ,θ | |∇yεθ ,θ |

(24.70)

gθ = p div uθ |∇yεθ ,θ |p−2 ∇yεθ ,θ − p |yεθ ,θ − yd |p−2 yεθ ,θ − yd ,

(24.72)

(24.71)

I ∈ L (RN ; RN ) is the identity matrix, yθ := y (uθ ) = y (u0 + θ (u − u0 )) is the Huθ solution of problem (24.34) and (24.35), yεθ ,θ = y0 + εθ (yθ − y0 ), and εθ = ε(uθ ) ∈

(0, 1) is a constant coming from equality (24.66).

A crucial point of this definition is the choice of the class of test functions in

integral identity (24.68) (ϕ ∈ Huθ ). At the end of this section, it will be shown that

Definition 24.7 makes a sense, and moreover, under some additional assumptions,

the quasi-adjoint states {ψθ }θ→0 can be defined in a unique way for each θ ∈ [0, 1]

in spite of the fact that boundary value problem (24.69) is degenerate in general.

Remark 24.9 If we assume that the quasi-adjoint state ψθ is defined by Definition 24.7 and the integral Ω (u − u0 ) |∇y0 |p−2 ∇y0 , ∇ψθ RN dx exists for all

u ∈ Astab

ad , then as (yθ − y0 ) ∈ Huθ for each θ > 0 (see (24.81) and Hypothesis (H2)),

the element λ in (24.67) can be defined as the quasi-adjoint state. As a result, having

put λ = ψθ in (24.67), the increment of the Lagrangian (24.67) can be simplified as

Ω

(u − u0 ) |∇y0 |p + |∇y0 |p−2 ∇y0 , ∇ψθ

RN

dx ≥ 0, ∀ u ∈ Astab

(24.73)

Thus, in order to derive the necessary optimality conditions, it remains to prove

the existence and the compactness properties of the sequence of quasi-adjoint states

24 Optimality Conditions for L 1 -Control in Coefficients …

453

{ψθ }θ→0 (with respect to some appropriate topology) and pass to the limit in (24.69)–

(24.73) as θ → +0.

To begin with, we establish a few auxiliary results. The characteristic feature of the

class of admissible controls Aad is the fact that strong convergence uk → u in L 1 (Ω)

implies weak convergence in variable space L p (Ω, uk dx)N of ∇y(uk ) → ∇y(u) as

k → ∞ (see, for instance, Theorem 24.3). However, we infer from Lemma 24.7

that the mapping u0 → y(u0 ) enjoys stronger properties provided some “directional

stability assumptions” on the space Hu0 hold. In particular, in this case, we have the

following result (for the details, see the proof of Lemma 24.5).

Lemma 24.10 Assume that for a given u ∈ Aad , Hypotheses (H0)–(H3) are valid.

Let θ ∈ [0, 1], uθ := u + θ (u − u), and let yθ = y(uθ ) be the corresponding Huθ solutions to the boundary value problem (24.26). Then, uθ → u0 in L 1 (Ω),

yθ → y0 in L p (Ω), ∇yθ → ∇y0 in the variable space L p (Ω, uθ dx)N as θ → 0.

(24.74)

Taking this fact into account, we arrive at the following properties of the sequence

yεθ ,θ = y0 + εθ (yθ − y0 )

θ→0

.

(24.75)

Proposition 24.2 Assume that for a given u ∈ Aad , Hypothesis (H2) is valid. Then,

yεθ ,θ ∈ Huθ , ∀ θ ∈ [0, 1],

(24.76)

|∇yεθ ,θ | uθ → |∇y0 | u0 in L (Ω) as θ → 0,

p

|yεθ ,θ − yd |

p−2

p

1

yεθ ,θ − yd → |y0 − yd |

p−2

(24.77)

(y0 − yd ) in L (Ω) as θ → 0.

(24.78)

q

Proof By definition of the functions yθ and y0 , we have

∇yθ

L p (Ω,uθ dx)N

< +∞, and ∇y0

Using the convexity of the norm ·

∇yεθ ,θ

L p (Ω,uθ dx)N

L p (Ω,uθ dx)N

≤ (1 − εθ ) ∇y0

p

L p (Ω,uθ dx)N

= (1 − θ ) ∇y0

L p (Ω,u0 dx)N

< +∞.

(24.79)

and representation (24.75), we get

L p (Ω,uθ dx)N

+ εθ ∇yθ

L p (Ω,uθ dx)N .

(24.80)

Since

∇y0

p

L p (Ω,u0 dx)N

+ θ ∇y0

p

,

L p (Ω,u dx)N

(24.81)

by Hypothesis (H2) and (24.79)2 , it follows that ∇y0 Lp (Ω,uθ dx)N < +∞. Thus, the

inclusion yεθ ,θ ∈ Huθ is a direct consequence of the condition yθ ∈ Huθ and inequality

(24.80). As for the property (24.77), in view of Lemma 24.10, we have, within a

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