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
uε
(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
444
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 .
446
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,
for each admissible control u ∈ Aad , the boundary value problem (24.26) admits a
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,
448
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
450
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
ad be a subset
⊂
A
,
of L 1 (Ω) such that Astab
ad
ad
∇y0
p
L p (Ω,u dx)N
:=
Ω
|∇y0 |p u dx < +∞ and u/u0 ∈ L ∞ (Ω), for all u ∈ Astab
ad ,
(24.63)
and the weighted Sobolev space Hu0 is stable along the direction u − u0 for each
stab
stab
u ∈ Astab
ad . It is clear that Aad is always non-empty because u0 ∈ Aad by definition
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
ad is not a singleton.
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)
+θ
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
ad ,
RN
uθ dx
(24.67)
452
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
ad .
(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