Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
4/2026
OPTIMAL
DISTRIBUTED
CONTROL
OF
SECOND-GRADE
FLUID
FLOWS
THROUGH
A
POROUS
MEDIUM
∗
Evgenii
S.
Baranovskii
†
With
profound
respect,
this
work
is
dedicated
to
Professor
Constantin
Fetecau
for
his
80th
birthday
DOI 10.56082/annalsarscimath.2026.4.205
Abstract
This
paper
deals
with
optimal
distributed
control
of
a
strongly
nonlinear
system
modeling
the
steady-state
flow
of
a
variable-viscosity
second-grade
fluid
through
a
porous
medium.
We
consider
the
internal
flow
within
a
bounded
Lipschitz
domain
(of
dimension
two
or
three)
under
the
no-slip
boundary
condition,
assuming
that
the
external
force
serves
as
the
control
parameter.
The
problem
is
rigorously
formulated
in
both
classical
and
weak
settings.
Our
main
result
establishes
the
existence
of
an
optimal
weak
solution
by
applying
a
known
solvability
principle
for
degenerate
nonlinear
operator
equations
involving
weakly
and
completely
continuous
mappings
in
Sobolev
spaces.
In
addition,
we
analyze
the
associated
value
function,
proving
its
lower
semicon-
tinuity
with
respect
to
the
one-sided
Hausdorff
distance,
and
present
a
necessary
condition
for
the
existence
of
a
classical
solution
to
the
considered
flow
problem.
Keywords:
second-grade
fluid,
porous
media,
optimization
problem,
distributed
control,
classical
solution,
weak
solution,
existence
theorem,
value
function,
energy
discrepancy
functional.
MSC:
49J20,
76D55,
76A05.
∗
Accepted
for
publication
on
August
04,
2026
†
esbaranovskii@gmail.com
,
Department
of
Applied
Mathematics,
Informatics
and
Mechanics,
Voronezh
State
University,
Russia
205
E.S.
Baranovskii
207
System
(1)
is
normalized
with
respect
to
the
constant
fluid
density,
so
that
the
density
does
not
appear
explicitly.
Moreover,
in
what
follows,
we
suppose
that
the
contributions
of
α
D
(
y
)
W
(
y
)
and
α
W
(
y
)
D
(
y
),
which
involve
products
of
first-order
spatial
derivatives
of
the
velocity,
are
negligi-
ble
relative
to
the
other
terms
appearing
in
the
third
equation
of
(1),
and
may
consequently
be
omitted.
This
reduction
is
analogous
to
a
first-order
linearization
when
all
terms
of
order
higher
than
one
are
discarded.
Sub-
stitution
of
the
simplified
extra
stress
tensor
into
the
momentum
equation
leads
to
the
following
system:
(
y
·
∇
)
y
−
∇
·
η
D
(
y
)
+
α
(
y
·
∇
)
D
(
y
)
+
∇
p
+
κ
y
+
β
|
y
|
y
=
f,
∇
·
y
=
0
,
(2)
which
is
commonly
regarded
as
a
suitable
model
for
the
flow
of
viscous
fluids
containing
polymer
additives,
in
particular
aqueous
solutions
of
polyacry-
lamide
and
polyethylene
oxide
[1,34].
It
should
be
noted
that
even
for
this
simplified
formulation,
the
issue
of
existence
and
uniqueness
of
solutions
is
nontrivial.
The
principal
difficulty
arises
from
the
term
∇
·
[
α
(
y
·
∇
)
D
(
y
)],
which
is
a
third-order
differential
operator,
whereas
the
viscous
term
∇
·
[
η
D
(
y
)]
is
merely
the
Laplacian.
A
possible
approach
to
overcoming
this
obstacle
is
based
on
a
modified
Faedo–Galerkin
scheme
involving
a
specially
constructed
basis
of
eigenfunc-
tions
[6].
The
mathematical
analysis
of
the
governing
equations
is
of
fundamental
importance
for
predicting
flow
behavior
and
for
the
design
of
optimal
flow
configurations
in
various
technological
contexts.
Following
the
pioneering
works
of
Oskolkov
[31,32],
a
substantial
number
of
studies
have
been
devoted
to
diverse
aspects
of
the
well-posedness
of
the
second-grade
fluid
model
and
its
modifications
[9,13,15,17,18,24,33,35].
A
discussion
of
some
open
problems
can
be
found
in
the
monograph
[16].
The
present
paper
is
concerned
with
an
optimal
control
problem
for
the
equations
describing
the
internal
flow
of
second-grade
fluid
through
a
porous
medium
in
a
bounded
domain
Ω
⊂
R
n
.
In
contrast
to
the
majority
of
earlier
contributions,
which
treat
the
vis-
cosity
as
constant,
we
consider
a
nonlocal
variable-viscosity
formulation
.
More
precisely,
the
viscosity
is
assumed
to
depend
on
the
shear
rate
accord-
ing
to
η
=
η
(
|
M
D
(
y
)
|
),
where
M
is
a
prescribed
linear
operator
such
that