On the structural representation of S-homogenized optimal control problems
The aim of this paper is an application of variational S-convergence [1-3] to homogenization theory of optimal control problems and to explore the structure of homogenized problems. We have proved the existence of strongly S-homogenized optimal control problems for a family of nonlinear systems with...
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Інститут прикладної математики і механіки НАН України
1999
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| Назва видання: | Нелинейные граничные задачи |
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| Цитувати: | On the structural representation of S-homogenized optimal control problems / P.I. Kogut // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 84-91. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-1692762020-06-10T01:26:28Z On the structural representation of S-homogenized optimal control problems Kogut, P.I. The aim of this paper is an application of variational S-convergence [1-3] to homogenization theory of optimal control problems and to explore the structure of homogenized problems. We have proved the existence of strongly S-homogenized optimal control problems for a family of nonlinear systems with distributed parameters, have derived some important properties which will be use in future and give the formula for representation of homogenized problems. 1999 Article On the structural representation of S-homogenized optimal control problems / P.I. Kogut // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 84-91. — Бібліогр.: 7 назв. — англ. 0236-0497 http://dspace.nbuv.gov.ua/handle/123456789/169276 en Нелинейные граничные задачи Інститут прикладної математики і механіки НАН України |
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The aim of this paper is an application of variational S-convergence [1-3] to homogenization theory of optimal control problems and to explore the structure of homogenized problems. We have proved the existence of strongly S-homogenized optimal control problems for a family of nonlinear systems with distributed parameters, have derived some important properties which will be use in future and give the formula for representation of homogenized problems. |
| format |
Article |
| author |
Kogut, P.I. |
| spellingShingle |
Kogut, P.I. On the structural representation of S-homogenized optimal control problems Нелинейные граничные задачи |
| author_facet |
Kogut, P.I. |
| author_sort |
Kogut, P.I. |
| title |
On the structural representation of S-homogenized optimal control problems |
| title_short |
On the structural representation of S-homogenized optimal control problems |
| title_full |
On the structural representation of S-homogenized optimal control problems |
| title_fullStr |
On the structural representation of S-homogenized optimal control problems |
| title_full_unstemmed |
On the structural representation of S-homogenized optimal control problems |
| title_sort |
on the structural representation of s-homogenized optimal control problems |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/169276 |
| citation_txt |
On the structural representation of S-homogenized optimal control problems / P.I. Kogut // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 84-91. — Бібліогр.: 7 назв. — англ. |
| series |
Нелинейные граничные задачи |
| work_keys_str_mv |
AT kogutpi onthestructuralrepresentationofshomogenizedoptimalcontrolproblems |
| first_indexed |
2025-07-15T04:02:02Z |
| last_indexed |
2025-07-15T04:02:02Z |
| _version_ |
1837684095663996928 |
| fulltext |
ON THE STRUCTURAL REPRESENTATION OF
S-HOMOGENIZED OPTIMAL CONTROL PROBLEMS
c© P.I. Kogut
The aim of this paper is an application of variational S-convergence [1-3] to ho-
mogenization theory of optimal control problems and to explore the structure of ho-
mogenized problems. We have proved the existence of strongly S-homogenized optimal
control problems for a family of nonlinear systems with distributed parameters, have
derived some important properties which will be use in future and give the formula for
representation of homogenized problems.
Let us consider the following family of optimal control problems
inf Iε(u, y), (1)
Aε(u, y) = fε, Fε(u, y) ≥ 0, u ∈ U∂ , y ∈ Kε, (2)
where U = V ? – control space, which is dual of a separable Banach space V , U∂ –
admissible class of control with U , Kε – weakly closed subsets in a separable Banach
space Y , Y ⊂ X with continuous and dense injection, where X is a reflexive Banach
space, Z – semi-ordered by reproducing cone L a Banach space, ε denotes a ”small”
multiparameter with a set E, partialy ordered by decreasing (0 ≤ ε for every ε ∈ E and
0 is the minimal element in E), fε is a fixed element with Y ∗, Aε : U × (D (Aε) ⊂ X) →
Y ∗, Fε : U × Y → Z – are nonlinear operators, which may arbitrarily depend on ε,
Iε : U×X → R – is a cost function. In so doing, we will assume that Kε ⊂ (D (Aε) ∩ Y ).
Definition 1. The pair ”control-state” (u, y) ∈ U × Y will be called an admissible if
(u, y) satisfies restrictions (2). We denote by Ξε the set of all admissible pairs for the
fixed ε.
Definition 2. The admissible pair
(
u0
ε, y
0
ε
) ∈ Ξε, which gives the least possible value
to the functional Iε we shall call an optimal pair, i.e. Iε
(
u0
ε, y
0
ε
)
= inf
(u,y)∈Ξε
Iε(u, y).
In the future we assume that: (a) the cost function Iε is sequentially lower semicon-
tinuous in the ∗-weak topology of U and weak topology of Y , i.e. from un → u ∗-weakly
in U and yn → y weakly in Y it follows that lim inf
n→∞
Iε (un, yn) ≥ Iε(u, y); (b) Aε is
a ∗-demicontinuous operator, i.e. Aε is a continuous operator from U × (D (Aε) ⊂ X)
with the ∗-weak topology of U and the weak topology of Y in Y ∗ with the ∗-weak
topology; (c) for every ε ∈ E the operator Aε is coercive on Y , i.e. for every bounded
set G of U we have inf
u∈G
〈Aε(u,y),y〉Y
‖y‖Y
as ‖y‖Y →∞; (d) the operator Fε : U × Y → Z is
a weakly continuous; (e) U∂ is a nonempty, bounded and ∗-weakly closed subset of U ;
(f) the injection Y ⊂ X is a compact; (g) Kε ⊂ (D (Aε) ∩ Y ) is nonempty and weakly
closed subsets of Y .
By analogy with [4] it is possible to prove the following result.
Theorem 1. Assume that conditions (a)-(g) are satisfyed. Then for fixed ε the optimal
control problem (1)-(2) has a solution if and only if this problem is a regular (i.e. the
set Ξε ⊂ U∂ ×Kε is nonempty).
Let τ be the topology of U × Y wich equal to the product of ∗-weak topology of U
and weak topology of Y . For any x = (u, y) belonging to U ×Y , let us denote by Nτ (x)
the filter of neighbourhoods of x with respect to topology τ . Let τ−Li Ξε, τ−Ls Ξε are
lower and upper topological limits of the generalized sequence {Ξε}ε∈E (see [5,6]). If
Ξ = τ−Li Ξε = τ−LsΞε then the sequence {Ξε}ε∈E is said to be topological convergent
to Ξ and the limit set Ξ will be denoted by τ−LmΞε.
Before introducing the formal axiomatics for homogenization process we rewrite the
problems (1)-(2) in another form:
{〈
inf
(u,y)∈Ξε
Iε(u, y)
〉
, ε ∈ E
}
, (3)
Thus the homoginization of family problems (1)-(2) consists in studing of limitary
and variational properties of generalized sequence (3).
Definition 3. The lower S-limit of the generalized sequence
{
Iε : Ξε → R
}
ε∈E
, de-
noted by Is (or τ−lisIε), is the functional from τ−Ls Ξε to R defined by
Is(u, y) = sup
W∈Nτ (u,y)
lim inf
(ε∈E,W∩Ξε 6=∅)
inf
(v,p)∈W∩Ξε
Iε(v, p).
Definition 4. The upper S-limit of the generalized sequence
{
Iε : Ξε → R
}
ε∈E
, de-
noted by Is (or τ−lssIε), is the functional from τ−Li Ξε to R defined by
Is(u, y) = sup
W∈Nτ (u,y)
lim sup
ε∈E
inf
(v,p)∈W∩Ξε
Iε(v, p).
Definition 5. The generalized sequence
{
Iε : Ξε → R
}
ε∈E
is said to be S-convergent
if the equality Is(u, y) = Is(u, y) holds for every (u, y) ∈ τ−Li Ξε. This common value
is then denoted as τ−lmsIε. If the foregoing identity is true for every (u, y) ∈ τ−Ls Ξε,
then this common value, denoted by τ−lma
sIε, will be called the absolute S-limit of such
sequence.
The techniques of S-convergence and basic variational and topological properties of
S-limits are discussed in more detail in [1-3].
Definition 6. The lower homogenized (respectively, upper homogenized) optimal con-
trol problem for the family problems (1)-(2) is defined by inf
(u,y)∈τ−Ls Ξε
Is(u, y), ( respec-
tively, inf
(u,y)∈τ−Li Ξε
Is(u, y)). If directedness
{
Iε : Ξε → R
}
ε∈E
S-converges to
τ−lmsIε(u, y) (respectively, absolutely S-convergence to τ−lma
sIε(u, y)), then the mi-
nimization problem inf
(u,y)∈τ−Li Ξε
(τ−lmsIε) (u, y) (respectively, inf
(u,y)∈τ−Lm Ξε
(τ−lma
sIε)
(u, y)) is called S-homogenized (respec- tively, strong S-homogenized) optimal control
problem.
Let us denote by
M (Iε, Ξε) =
{(
u0
ε, y
0
ε
) ∈ Ξε
∣∣∣∣Iε
(
u0
ε, y
0
ε
)
= inf
(u,y)∈Ξε
Iε(u, y)
}
,
Mα (Iε,Ξε) = {(uε, yε) ∈ Ξε |Iε (uε, yε) ≤
sup
(
inf
(u,y)∈Ξε
Iε(u, y) + α,−1/α
)}
, ∀α > 0
the sets of all optimal and α-optimal solution of (1)-(2).
Definition 7. The generalized sequence
{
Iε : Ξε → R
}
ε∈E
will be called an equi-
coercive if exists a τ -lower semicontinuous and τ -lower semicompactness function Ψ :
U × Y → R such that Iε(u, y) ≥ Ψ(u, y) ∀(u, y) ∈ Ξε ∀ε ∈ E.
Let us assume that hypotheses (a)–(g) are true. Then the main results of [1-3] can
be extended on the case of homogenized optimal control problem.
Proposition 1. The functionals cost Is : τ−LsΞε → R and Is : τ−Li Ξε → R are
τ -lower semicontinuous on τ−Ls Ξε and τ−LiΞε. Furthermore, the domaines of its
functionals are τ -closed sets.
Proposition 2. The following inequalities and inclutions hold:
τ−Li Ξε ⊆ τ−LsΞε ⊆ U∂ × τ−Ls Kε;
Is(u, y) ≥ Is(u, y) for every (u, y) ∈ τ−Li Ξε.
Proposition 3. Suppose that τ−Li Ξε 6= ∅ and
{
Iε : Ξε → R
}
ε∈E
is equi-coercive.
Then the S-limits Is and Is are τ -lower semicompactness and the sets of solutions for
lower and upper S-homogenized optimal control problems are nonempty and τ -compact.
Proposition 4. Let {(uα
ε , yα
ε ) ∈ Mα (Iε,Ξε) ⊂ U∂ ×Kε}ε∈E be a generalized sequence
of α-optimal pairs of (1)-(2) and (uα
ε , yα
ε ) τ−→ (
u0, y0
)
. Then
Is
(
u0, y0
)
= lim inf
ε∈E
inf
(u,y)∈Ξε
Iε(u, y);
(
u0, y0
) ∈ M (Is, τ−LsΞε);
Is
(
u0, y0
)
= lim sup
ε∈E
inf
(u,y)∈Ξε
Iε(u, y);
(
u0, y0
) ∈ M (Is, τ−Li Ξε).
Proposition 5. Assume that
{
Iε : Ξε → R
}
ε∈E
S-converges to a functional
I : τ−Li Ξε → R and that I is a not identicaly +∞ on τ−Li Ξε 6= ∅. Then: (i) for the
family problems (1)-(2) there exists of a strong S-homogenized optimal control problem;
ii) M (I, τ−li Ξε) =
⋂
α>0
τ−LiMα (Iε, Ξε) ⊂ U∂ × τ−Li Kε;
iii) min
(u,y)∈τ−Li Ξε
I(u, y) = lim
ε∈E
inf
(u,y)∈Ξε
Iε(u, y).
Morever, if
{
Iε : Ξε → R
}
ε∈E
is equi-coercive, then M (I, τ−Li Ξε) is nonempty and
τ -compact.
We now turn to studing the compactness properties of the class of optimal control
problems (1)-(2) with respect to S-homogenization. The main compactness theorem
is founded on the following abstract result (7): from each directedness of functions{
Gε : Wε → R
}
ε∈E
defined on the subsets of a second countable topological space (W, τ)
and for which , one τ−LiWε 6= ∅, can extract an absolute S-convergence sequence{
Gn : Wn → R
}
n∈N
.
Remark 1. It may be noted that the τ -topology on U × Y is a separable since the
control space U is separable in ∗-weak topology and Y is weakly separable by initial
assumptions. Let us assume that the hypothesis (a)-(g) are true. Consider the subset
U∂ ×B of U × Y , where B is bounded and weakly closed subset of Kε. Since the space
Y is nonreflexive, the product of sets U∂ × B, generally is not τ -compact. Hence τ -
topology on U∂ ×B is nonmetrizable. Let us now make use of the fact that any weakly
closed subset B of a weakly compact set Kε is also weakly compact. As it appears from
the above mentioned, the τ -topology on U∂×B is metrizable if the sets Kε are compact
in weak topology on Y .
Theorem 2. Assume that: (1) the generalized sequence of the sets of all admissible
pairs {Ξε}ε∈E satisfies the condition τ−LiΞε 6= ∅; (2) the sets Kε are compact in
weak topology of Y ; (3) the initial assumptions (a)-(g) are true. Then: (i) from each
directedness of optimal control problems (1)-(2) one can extract the sequence
inf Iεn(u, y), (4)
Aεn(u, y) = fεn , Fεn(u, y) ≥ 0, u ∈ U∂ , y ∈ Kεn , εn −→
n→∞
0, (5)
for wich a strong homogenized optimal control problem in sense of definition 6 exists;
(ii) any sequence of optimal pairs
{(
u0
εn
, y0
εn
)}
n∈N
for the family of problems (4)-(5) is
compact in τ -topology;
(iii) if (u, y) is τ -limit of sequence
{(
u0
εn
, y0
εn
)}
n∈N
then
(u, y) ∈ U∂ × τ (Yw)−LiKε, lim
n→∞
Iεn
(
u0
εn
, y0
εn
)
= (τ−lma
sIε) (u, y), where
(τ−lma
sIε) (u, y) = inf
(u,y)∈τ−Lm Ξε
(τ−lma
sIε) (u, y), i.e. (u, y) is optimal solution in
strong S-homogenized optimal control problem. (Here we denote by τ (Yw) a weak topol-
ogy on Y ).
In order to study a structure of S-homogenized problems we consider the family of
µ-approximatical minimum problems
{〈
inf
(u,y)∈U∂×Kε
Iµ
ε (u, y)
〉
, ε ∈ E, µ > 0
}
, (6)
where
Iµ
ε (u, y) = Iε(u, y) + µ−1 (‖Aε(u, y)− fε‖Y ∗
+ sup
φ∈S∗1∩L∗
[ν (〈φ, Fε(u, y)〉Z)]2
)
,
S∗1 is a unit sphere in Z∗, L∗ is a dual cone of L, i.e.
L = {ξ |ξ ∈ Z , 〈φ, ξ〉Z ≥ 0 ∀φ ∈ L∗} .
Let Gε
1, G
ε
2 be are functions difined by Gε
1(u, y) = ‖Aε(u, y) − fε‖Y ∗ , Gε
2(u, y) =
supφ∈S∗1∩L∗ [ν (〈φ, Fε(u, y)〉Z)]2. By analogy with [4] we can prove that under initial
assumptions of Theorem 1 the problem (6) has solutions for every ε ∈ E and µ > 0.
Furthermore, if {(uµ
ε , yµ
ε ) ∈ U∂ ×Kε}µ>0 be an arbitrary sequence of ”µ-optimal pair”,
i.e. solutions of problems (6), then we can extract a subsequence which τ -convergence
to some optimal pair in problem (1)-(2).
In the future by [F |E ] we denote the restriction of any function F on a set E.
Proposition 6. Assume that conditions (a)-(g) are satisfied and the family of optimal
control problems (1)-(2) are uniformly regular (i.e. Ξε 6= ∅ ∀ε ∈ E). Then
τ−Li Ξε = A, τ−Ls Ξε = B,
where τ -closed sets A ⊂ U × Y and B ⊂ U × Y are defined by
A =
(u, y)
∣∣∣∣∣∣∣
sup
µ>0
(
τ−lss
[
µ−1 (Gε
1 + Gε
2) |U∂ ×Kε
])
≡ τ−lss
[
lim
µ↓0
µ−1 (Gε
1 + Gε
2) |U∂ ×Kε
]
< +∞
,
B =
(u, y)
∣∣∣∣∣∣∣
sup
µ>0
(
τ−lis
[
µ−1 (Gε
1 + Gε
2) |U∂ ×Kε
])
≡ τ−lis
[
lim
µ↓0
µ−1 (Gε
1 + Gε
2) |U∂ ×Kε
]
< +∞
.
Proof. We shall prove only the second equality, the proof of the other one being analo-
gous. Define
Ωε = {(u, y) ∈ U × Y |Aε (u, y) = fε, Fε (u, y) ≥ 0} .
Given ε ∈ E, by the definition of Ωε we can write lim
µ↓0
{
µ−1 (Gε
1+ Gε
2)} =
[χclτΩε |D (Aε) ∩ Y ], where χclτΩε denotes the indicator function of τ -closure of the set
Ωε. Taking into account that χclτΩε = χΩε , we have
τ−lis
[
µ−1 (Gε
1 + Gε
2) |U∂ ×Kε
]
= τ−lis [χΩε |U∂ ×Kε ] .
Note that, by definition 3, the function τ−lis [χΩε
|U∂ ×Kε ] there exists on the set
τ−Ls (U∂ ×Kε). But it is ibvious that
τ−Ls (U∂ ×Kε) = U∂ × τ (Yw)−LsKε, where τ (Yw) is a weak topology on Y . Then,
using the equality
[χΩε |U∂ ×Kε ] =
[
χΩε∩(U∂×Kε) |U∂ ×Kε
]
= [χΞε |U∂ ×Kε ]
and the standard properties of S-limits, we can conclude that
τ−lis
[
µ−1 (Gε
1 + Gε
2) |U∂ ×Kε
]
= [χτ−Ls Ξε |U∂ × τ (Yw)−LsKε ] .
Since Ξε ⊆ (Uε ×Kε) with the above equality we have
τ−Ls Ξε =
{
(u, y)
∣∣∣∣τ−lis
[(
lim
µ↓0
µ−1 (Gε
1 + Gε
2)
)
|U∂ ×Kε
]
<+∞
}
.
Now we need to show that
τ−Ls Ξε =
{
(u, y)
∣∣∣∣sup
µ>0
(
τ−lis
[
µ−1 (Gε
1 + Gε
2)|U∂ ×Kε
])
<+∞
}
.
For this purpose we fix a arbitrary pair (u, y) ∈ τ−Ls Ξε. Then for every V ∈ Nτ (u, y)
there exists subdirectedness {Θβ}β∈B , {Lβ}β∈B ,
{
Qβ
1 + Qβ
2
}
β∈B
of respectively direct-
edness {Ξε}ε∈E , {Kε}ε∈E , {Gε
1 + Gε
2}ε∈E such that V ∩Θβ 6= ∅ for every β ∈ B. Hence
inf
(v,x)∈V ∩(U∂×Lβ)
µ−1
(
Qβ
1 (v, x) + Qβ
2 (v, x)
)
= 0, β ∈ B. Since Gε
1 + Gε
2 ≥ 0 on the set
U × (D (Aε) ∩ Y ), we have (for every V ∈ Nτ (u, y))
0 ≤ inf
(ε∈E,V ∩(U∂×Kε)6=∅)
inf
(v,x)∈V ∩(U∂×Kε)
µ−1 (Gε
1(v, x) + Gε
2(v, x))
≤ lim inf
β∈B
inf
(v,x)∈V ∩(U∂×Lβ)
µ−1
(
Qβ
1 (v, x) + Qβ
2 (v, x)
)
= 0.
On the taking the supremum over all V ∈ Nτ (u, y), we see that
τ−lis
[
µ−1 (Gε
1 + Gε
2) |U∂ ×Kε
]
(u, y) = 0 ∀(u, y) ∈ τ−LsΞε
for every µ > 0. Thus
sup
µ>0
τ−lis
[
µ−1 (Gε
1 + Gε
2) |U∂ ×Kε
]
≤ [χτ−Ls Ξε |U∂ × τ (Yw)−LsKε ] . (7)
To prove the opposite inequality, we fix (u, y) with the set (U∂× τ (Yw)−LsKε) \
τ−LsΞε. Then there exists V ∈ Nτ (u, y) and ε0 ∈ E such that V ∩ Ξε 6= ∅ for every
ε ≤ ε0. Since Ξε = (U∂ ×Kε) ∩ Ωε, there exists γ > 0 such that, for every ε ≤ ε0,
inf
(v,x)∈V ∩(U∂×Kε)
µ−1 (Gε
1(v, x) + Gε
2(v, x)) ≥ µ−1γ. Then we have
inf
ε∈E
inf
(v,x)∈V ∩(U∂×Kε)
µ−1 (Gε
1(v, x) + Gε
2(v, x)) ≥ µ−1γ,
Hence τ−lis
[
µ−1 (Gε
1 + Gε
2) |U∂ ×Kε
]
(u, y) ≥ µ−1γ. Therefore, for every (u, y) ∈
(U∂ × τ (Yw)−LsKε) \ τ−LsΞε, we have
sup
µ>0
(
τ−lis
[
µ−1 (Gε
1 + Gε
2) |U∂ ×Kε
])
(u, y) ≥ +∞.
Thus, on the set U∂ × τ (Yw)−LsKε, will be true the next inequality
sup
µ>0
(
τ−lis
[
µ−1 (Gε
1 + Gε
2) |U∂ ×Kε
])
≥ [χτ−Ls Ξε |U∂ × τ (Yw)−LsKε ] ,
which, together with (7), concludes the prof of the proposition.
Corollary. The directedness of the sets of admissible pair {Ξε}ε∈E convergence in
topological sense to Ξ 6= ∅ iff B ⊆ Ξ ⊆ A 6= ∅.
By analogy with provious it may be proved the following results.
Proposition 7. Under initial conditions of proposition 6, suppose in addition that{
Iε : U ×X → R
}
ε∈E
is lower equi-bounded and there exists a constant γ > 0 such that
Iε(u, y) < γ for every (u, y) ∈ U∂ ×Kε and ε ∈ E.Then
Dom
(
sup
µ>0
(
τ−lss
[
Iε + µ−1 (Gε
1 + Gε
2) |U∂ ×Kε
]))
= τ−Li Ξε
= Dom
(
τ−lss
[
lim
µ↓0
(
Iε + µ−1 (Gε
1 + Gε
2)
) |U∂ ×Kε
])
,
Dom
(
sup
µ>0
(
τ−lis
[
Iε + µ−1 (Gε
1 + Gε
2) |U∂ ×Kε
]))
= τ−Ls Ξε
= Dom
(
τ−lis
[
lim
µ↓0
(
Iε + µ−1 (Gε
1 + Gε
2)
) |U∂ ×Kε
])
.
Proposition 8. Suppose that initial conditions of proposition 6 will be true and func-
tionals
{
Iε : U ×X → R
}
ε∈E
are lower equi-bounded. Then
sup
µ>0
(
τ−lis
[
Iε + µ−1Gε
1 + µ−1Gε
2 |U∂ ×Kε
])
= τ−lis [Iε |Ξε ]
= τ−lis
[
lim
µ↓0
(
Iε + µ−1Gε
1 + µ−1Gε
2
) |U∂ ×Kε
]
on τ−Ls Ξε,
sup
µ>0
(
τ−lss
[
Iε + µ−1Gε
1 + µ−1Gε
2 |U∂ ×Kε
])
= τ−lss [Iε |Ξε ]
= τ−lss
[
lim
µ↓0
(
Iε + µ−1Gε
1 + µ−1Gε
2
) |U∂ ×Kε
]
on τ−Li Ξε.
Theorem 3. (Main result). Let us assume that initial conditions of proposition 7
will be true. Then for the family of optimal control problems (1)-(2) there exists a strong
S-homogenized problem iff the quality
sup
µ>0
(
τ−lis
[
Iε + µ−1Gε
1 + µ−1Gε
2 |U∂ ×Kε
])
= sup
µ>0
(
τ−lss
[
Iε + µ−1Gε
1 + µ−1Gε
2 |U∂ ×Kε
]) 6≡ +∞
holds on the set U∂ × τ (Yw)−Ls Kε.
Corollary. Under initial conditions of main theorem for S-homogenized optimal con-
trol problem
〈
inf
(u,y)∈τ−Lm Ξε
(τ−lma
sIε) (u, y)
〉
will be holds the next representation
〈
inf
(u,y)∈
{
sup
µ>0
(τ−lss[Iµ
ε |U∂×Kε ])(u,y)<+∞
} sup
µ>0
(τ−lss [Iµ
ε |U∂ ×Kε ])
〉
.
References
1. P.I. Kogut, Variational S-convergence of Minimization Problems. Part I. Definitions and Basic
Properties, Problemy Upravlenia i Informatiki 5 (1996), 29–43. (Russian)
2. , Variational S-convergence of Minimization Problems. Part II. Topological Properties of
S-limits, Problemy Upravlenia i Informatiki 3 (1997), 78–90. (Russian)
3. , Variational Convergence of Minimum Problems and its Geometrical Interpretation, Dop.
NAN Ukrainy 6 (1997), 89–93. (Ukraine)
4. V.I. Ivanenko and V.S. Mielnik, Variational Methods in optimal Control Problems for Distributed
Parameter Systems, Naukova Dumka, Kyiv, 1988.
5. H. Attouch, Variational Convergence for Functions and Operatiors, Pitman, London, 1984.
6. G. Dal Maso, Introduction to Γ-convergence, Birkhauser, Boston, 1993.
7. P.I. Kogut, S-convergence in Homogenization Theory of Optimal Control Problems, Ukrainian Math-
ematical Journal (to appear). (Russian)
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