Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 22, pp. 1–12.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

UNIQUENESS FOR OPTIMAL CONTROL PROBLEMS OF

TWO-DIMENSIONAL SECOND GRADE FLUIDS

ADILSON ALMEIDA, NIKOLAI V. CHEMETOV, FERNANDA CIPRIANO

Abstract. We study an optimal control problem with a quadratic cost func-

tional for non-Newtonian fluids of differential type. More precisely, we con-
sider the system governing the evolution of a second grade fluid filling a two-

dimensional bounded domain, supplemented with a Navier slip boundary con-

dition. Under certain assumptions on the size of the initial data and pa-
rameters of the model, we prove second-order sufficient optimality conditions.

Furthermore, we establish a global uniqueness result for the solutions of the

first-order optimality system.

1. Introduction

The optimization of evolutionary phenomena is crucial in several branches of the
knowledge, for instance in finance, biology, rcology, aviation etc. [4, 5, 14, 17]. The
optimal control of fluid flows is a major problem in mathematical physics, with
relevant consequences in industrial applications. In the past decades, extensive
research work has been carried out on the control of fluid flows described by the
Navier-Stokes equations. However, many incompressible viscous fluids present in
the nature and used in the industry do not satisfy the Newton’s law of viscosity,
and consequently cannot be described by the Navier-Stokes equations. Among
these fluids, called non-Newtonian fluids, we can find colloidal suspensions and
emulsions, some industrial oils, ink-jet prints, geological flows, biological fluids,
body care fluids, some materials arising in polymer processing as well as in food
processing, and many others.

In this article, we study second grade fluids, which belong to the class of non-
Newtonian complex viscoelastic fluids of differential type. To understand the phys-
ical principles associated to the second grade fluid equations, as well as the physical
properties of these fluids, we refer to [1, 20, 21, 22, 28].

From the mathematical point of view, the equations governing the evolution of
second grade fluids are strongly nonlinear partial differential equations. The exis-
tence and uniqueness problems with a Dirichlet boundary condition were established
in the pioneering works [6, 15, 16, 25] Despite the most usual boundary condition
to be the non-slip Dirichlet boundary condition, practical studies show that some
viscoelastic fluids slip against the boundary surface. Let us refer, for instance, [32]
on capillary flow of highly entangled polyethylene (PE) melts, and [23] on microgel

2020 Mathematics Subject Classification. 35R60, 49K20, 60G15, 60H15, 76D55.

Key words and phrases. Second grade fluids; optimal control; uniqueness.
©2022. This work is licensed under a CC BY 4.0 license.
Submitted September 6, 2021. Published March 18, 2022.

1



2 A. ALMEIDA, N. V. CHEMETOV, F. CIPRIANO EJDE-2022/22

pastes and concentrated emulsions exhibiting a generic slip behavior at low stresses
when sheared near smooth surfaces. Therefore, to accurately describe certain physi-
cal systems, a slip boundary condition should be considered [8, 9, 10, 11]. Article [7]
established the well-posedness for the second grade fluid equations under a Navier
slip boundary condition. Referring to the stochastic framework, the existence and
uniqueness results have been investigated in [26, 27, 12, 29, 30] under non-slip and
slip boundary conditions.

In the deterministic context, the optimal control problem for the second grade
fluid equations was studied in the articles [2, 3], while the optimal control of the
stochastic dynamic has been addressed in [13, 18]. The authors established the
existence of an optimal solution for the control problem, and by analyzing the
linearized state equation as well as the adjoint equation deduced the first-order
optimality conditions.

In this article, we perform a second-order analysis by studying the second-order
derivative of the objective functional, in addition, we obtain a global unique-
ness result for the optimal solution. Essentially, if the fluid material is filling a
bounded two-dimensional domain, the sufficient second-order optimality condition
is achieved for sufficiently elastic and viscous fluid material with small size ini-
tial data. Alternatively, the same result can be obtained by taking an objective
functional with strong intensity of the cost.

We should emphasize that the second-order sufficient optimality conditions are
crucial to numerically solve the control problem, being necessary for the stability of
the optimal solution, as well as to prove the convergence of the numerical approxi-
mations. As far as we know, the second-order analysis and the uniqueness problem
for the optimization of second grade fluids is being addressed here for the first time.

In Section 2, we formulate the problem and recall some known results in the
literature. We write key estimates for the solutions of the state equation, linearized
equation and adjoint equation that will be necessary in the following sections. Sec-
tion 3 establishes the second-order sufficient optimality conditions. Section 5 is
devoted to the global uniqueness problem.

2. Formulation of the problem and preliminary results

We consider an optimal control problem associated with a non-stationary viscous,
incompressible, second grade fluid. We assume that O ⊂ R2 is a bounded, simply
connected domain, having a sufficiently smooth boundary Γ. The fluid dynamic on
a time interval [0, T ] is described by the following state equations

∂

∂t
υ(y) = ν∆y − curl υ(y)× y −∇π + u,

div y = 0 in Q = (0, T )×O,
y · n = 0, (n ·Dy) · τ = 0 on Σ = (0, T )× Γ,

y(0) = y0 in O,

(2.1)

where y = (y1, y2) is the velocity field of the fluid,

Dy =
∇y + (∇y)T

2

corresponds to the symmetric part of the velocity gradient, and

υ(y) = y − α∆y,



EJDE-2022/22 UNIQUENESS FOR OPTIMAL CONTROL PROBLEMS 3

where α > 0 is a viscoelastic parameter. Moreover π denotes the hydrodynamic
pressure ν > 0 is the viscosity of the fluid, n = (n1, n2) is the unit normal to the
boundary Γ, τ = (−n2, n1) is the tangent vector to Γ and u represents an external
mechanical force, which acts on the system as the control variable. Here υ = υ(y)
and y are two-dimensional vectors. To perform a three-dimensional calculus in the
term curl υ(y)× y (or in a similar ones) we consider the usual identifications

curl v =
∂v2
∂x1
− ∂v1
∂x2

≡ (0, 0,
∂v2
∂x1
− ∂v1
∂x2

), y = (y1, y2) ≡ (y1, y2, 0).

To formulate the problem and establish the results, we introduce the convenient
functional spaces and some useful notation. We denote by Lp = Lp(O), 1 ≤ p ≤ ∞,
the Lebesgue spaces, endowed with their natural norms ‖·‖p and consider the usual
notations for the scalar products on the finite dimensional spaces R2 and R2×2,

u · v =

2∑
i=1

uivi, u, v ∈ R2; η : ζ =

2∑
i,j=1

ηijζij , η, ζ ∈ R2×2,

as well as for the scalar products in L2

(u, v) =

∫
O
u(x) · v(x) dx, (η, ζ) =

∫
O
η(x) : ζ(x) dx.

We consider the standard Sobolev spaces W k,p = W k,p(O), 1 ≤ p ≤ ∞, endowed
with their natural norms ‖ · ‖Wk,p . In the particular case p = 2, we set Hk = W k,2

and ‖ · ‖Hk = ‖ · ‖Wk,2 .
Let us introduce the following divergence-free spaces:

H = {v ∈ L2 : div v = 0 in D′(O) and v · n = 0 on H−1/2(Γ)},
V = {v ∈ H1 : div v = 0 in O and v · n = 0 on Γ},

W = {v ∈ V ∩H2 : (n ·Dv) · τ = 0 on Γ}.

On the space V , we consider the following scalar product and the corresponding
norm

(u, v)V = (u, v) + 2α(Du,Dv), ‖u‖V = (u, u)
1/2
V , ∀u, v ∈ V.

Throughout the article, we denote by C,C1, . . . , C4 and K, K̃, K̂, the constants
which depend only on the domain O.

Now, we recall the usual Korn inequality

‖y‖2H1 ≤ K(‖y‖22 + ‖Dy‖22), ∀y ∈ H1. (2.2)

Let us define the operator

Ay=P4y for y ∈W,

where P : L2 → H is the Helmholtz projector in L2. From [2, Lemma 2.3], we have
the inequality

‖y‖2H2 ≤ K̃(‖y‖22 + ‖Ay‖22), ∀y ∈W ∩H3. (2.3)

Considering the trilinear form

b(φ, z, y) = (φ · ∇z, y),

the nonlinear term of the equations can be written as

(curl υ(y)× z, φ) = b(φ, z, υ(y))− b(z, φ, υ(y)), ∀y, z ∈W ∩H3, φ ∈ V. (2.4)



4 A. ALMEIDA, N. V. CHEMETOV, F. CIPRIANO EJDE-2022/22

Now, let us state a property of the nonlinear term, which will be useful in the
following considerations.

Lemma 2.1. For all z, φ ∈ H2, we have

|(curl υ(z)× z, φ)| ≤ K̂‖φ‖H2‖z‖2H2 . (2.5)

Proof. We satisfy that

|(curl υ(z)× z, φ)| ≤ |b(φ, z, υ(z))|+ |b(z, φ, υ(z))|
≤ ‖φ‖4‖z‖W 1,4‖z‖H2 + ‖z‖4‖φ‖W 1,4‖z‖H2

≤ K̂‖φ‖H2‖z‖2H2 . �

In this article, we assume that the external mechanical force u and the initial
data y0 satisfy

u ∈ L2(0, T ;H1) and y0 ∈W ∩H3. (2.6)

Lemma 2.2. Under assumptions (2.6) there exists a unique solution

y ∈ L∞(0, T ;W ∩H3) with
∂y(t)

∂t
∈ L2(0, T ;V )

of the problem (2.1), which is understood in the distributional sense(∂y(t)

∂t
, φ
)

+ 2α
(
D
∂y(t)

∂t
,Dφ

)
+ 2ν(Dy(t), Dφ)

+ (curl υ(y(t))× y(t), φ) = (u(t), φ), ∀φ ∈ V.
(2.7)

Moreover, the solution satisfies

‖y‖2L∞(0,T ;H3) ≤
C2

1λ
2
1

α2
(2.8)

with

λ21 = (1 + 4Kα̂)(‖y0‖2H3 + ‖u‖2L1(0,T ;H1)),

where the constant K is defined by (2.2), and α̂ = max((2α)−1, 2α).

Proof. The solvability of the problem (2.1) is shown in [7]. Let us write in a
convenient form the estimates for the state variable y with respect to the initial
data y0 and the control variable u. The estimates (3.4) and (3.5) of Proposition
3.2 in [2] give

‖y‖2L∞(0,T ;H3) ≤
C2

α2

(
‖y‖2L∞(0,T ;H1) + ‖ curlσ(y0)‖2L2 + ‖ curlu‖2L1(0,T ;L2)

)
. (2.9)

In addition, relation (3.3) of Proposition 3.2 in [2] for y in H1, and the Korn’s
inequality (2.2) yield

‖y‖2L∞(0,T ;H1) ≤ 4K
max(1, 2α)

min(1, 2α)

(
‖y0‖2H1 + ‖u‖2L1(0,T ;L2)

)
. (2.10)

Combining inequalities (2.9) and (2.10) we derive (2.8). �

Let us introduce the so-called control-to-state mapping S : u → y, namely
y = S(u) is the solution of the equation (2.1) corresponding to the control u, and
consider the reference pair (u, y = S(u)). In the next lemma, we recall a stability
result for the solution of (2.1), which was proved in [2, Proposition 4.4].



EJDE-2022/22 UNIQUENESS FOR OPTIMAL CONTROL PROBLEMS 5

Lemma 2.3. Let us consider the initial data y0 and two different control variables
u1, u2 satisfying the assumptions (2.6). Let y1 = S(u1), y2 = S(u2) be the cor-
responding solutions of (2.1) with the same initial data y0. Then the difference
y = y2 − y1 satisfies the estimate

‖y‖2L∞(0,T ;H2) ≤ λ
2
2‖u‖2L2(Q) (2.11)

with u = u2 − u1 and

λ22 = K̃
[(

1 + C2T (1 + α−1)
C1λ1
α

)
eC2T (1+(1+α−1)C1λ1) + (αν)−1

]
for some positive constant C1, C2 depending only on O.

Proof. By the first estimate of [2, Proposition 4.4] and (2.8), we deduce

‖y‖2L∞(0,T ;L2) + 2α‖Dy‖2L∞(0,T ;H1) ≤ e
C2T (1+(1+α−1)C1λ1)‖u‖2L2(Q).

Using the second estimate of [2, Proposition 4.4] and (2.3), we obtain

‖y‖2L∞(0,T ;H2)

≤ K̃
[
‖y‖2L∞(0,T ;L2) +

1

α

(1

ν
‖u‖2L2(Q) + C2T (1 + α−1)C1λ1‖y‖2L∞(0,T ;L2)

)]
≤ K̃

[(
1 + C2T (1 + α−1)

C1λ1
α

)
‖y‖2L∞(0,T ;L2) + (αν)−1‖u‖2L2(Q)

]
.

These two inequalities imply (2.11). �

According to [2, Proposition 4.5], the first-order Gâteaux derivative z = S′(u)[w]
of the mapping S, at the point u, in the direction w, is given by the solution of the
linearized state equation at (u, y)

∂υ(z)

∂t
− ν∆z + curl υ(z)× y + curl υ(y)× z +∇π = w,

∇ · z = 0 in Q,

z · n = 0, (n ·Dz) · τ = 0 on Σ,

z(0) = 0 in O,

(2.12)

which is well-posed in the Sobolev space H2. The next lemma establishes a suitable
estimate for the solution z.

Lemma 2.4. Under assumptions (2.6) there exists a unique solution z ∈ L∞(0, T ;W )
of (2.12), such that

‖z‖2L∞(0,T ;H2) ≤ λ
2
3‖w‖2L2(Q), (2.13)

where

λ23 = K̃
[
(αν)−1eC3TC1λ1(

1+α

α2 )
(

1 +
2

α
C3TC1λ1(1 + α−1)eC3TC1λ1(

1+α

α2 )
)

× eC3T (1+C1λ1(1+α
−1))

]
with λ1 defined in (2.8).

Proof. From the first estimate of [2, Proposition 4.3] and (2.8), we have

‖z‖2L∞(0,T ;L2) ≤ ‖w‖
2
L2(Q)e

C3T (1+C1λ1(1+α
−1)). (2.14)



6 A. ALMEIDA, N. V. CHEMETOV, F. CIPRIANO EJDE-2022/22

In addition, the second estimate of [2, Proposition 4.3] and (2.8) yield

‖Az‖2L∞(0,T ;L2)

≤ 1

α

[1

ν
‖w‖2L2(Q) + 2C3TC1λ1(1 + α−1)‖z‖2L∞(0,T ;L2)

]
eC3TC1λ1(

1+α

α2 ) = R.
(2.15)

Therefore, taking into account (2.3) and (2.15), we deduce that

‖z‖2L∞(0,T ;H2) ≤ K̃[‖z‖2L∞(0,T ;L2) +R]

= K̃
[(

1 +
1

α
2C3TC1λ1(1 + α−1)eC3TC1λ1(

1+α

α2 )
)

× ‖z‖2L∞(0,T ;L2) + (αν)−1eC3TC1λ1(
1+α

α2 )‖w‖2L2(Q)

]
.

Using (2.14), we obtain (2.13). �

Now, we formulate the control problem (see [2]). The space Uad of the admissible
control variables is a bounded, closed and convex subset of L2(0, T ;H1). Namely,
there exists a constant L > 0 such that

‖u‖L2(0,T ;H1) ≤ L, ∀u ∈ Uad. (2.16)

The control on the evolution of the physical system is imposed through a distributed
mechanical force u ∈ Uad, aiming to match a desired target velocity profile

yd ∈ L2(Q).

The control u and the state y = S(u) are constrained to satisfy the system (2.1),
and the optimal control problem reads

minimizeu{J(u, y) : u ∈ Uad, y = S(u)}, (2.17)

where the objective functional is

J(u, y) =
1

2

∫ T

0

∫
O
|y − yd|2 dxdt+

λ

2

∫ T

0

∫
O
|u|2 dx dt, u ∈ Ubad, (2.18)

and λ ≥ 0 is a fixed cost coefficient.
Let us recall that the first-order optimality conditions, at a locally optimal pair

(u∗, y∗), can be formally deduced through the Lagrange’s multipliers method. More
precisely, considering the Lagrange function

L(u, y, p) = J(u, y)−
∫ T

0

(
p,− ∂

∂t
υ(y) + ν∆y − curl υ(y)× y −∇π + u

)
dt,

the optimal pair (u∗, y∗) should satisfy the relations

L′y(u∗, y∗, p)[z] = 0, (2.19)

L′u(u∗, y∗, p)[w] ≥ 0, ∀w ∈ Uad. (2.20)

Equation (2.19) yields the adjoint equation of the state equation linearized at the
point y = y∗, and the system composed by equations (2.19), (2.20) and the state
equations (2.1) for (u∗, y∗) constitute the so-called first-order optimality conditions.



EJDE-2022/22 UNIQUENESS FOR OPTIMAL CONTROL PROBLEMS 7

The adjoint equation of the state equation linearized at the point y = y∗ arising
from (2.19) is

∂

∂t
υ(p) = −ν∆p− curl υ(y)× p+ curl υ(y × p) +∇π − (y − yd),

div p = 0 in Q,

p · n = 0, (n ·Dp) · τ = 0 on Σ,

p(T ) = 0 in O.

(2.21)

Here, we recall the main result in [2], which shows the existence of an optimal
solution and establishes the first-order optimality conditions.

Theorem 2.5 ([2]). Under assumptions (2.6) and (2.16), problem (2.17) admits
at least one solution

(u, y) = (u, S(u)) ∈ Uad × L∞(0, T ;W ∩H3).

Furthermore, there exists a unique solution

p = S∗(u) ∈ L∞(0, T ;H2)

of the adjoint system (2.21), such that the following optimality condition holds∫ T

0

(λu+ p,Ψ− u) dt ≥ 0, ∀Ψ ∈ Uad. (2.22)

A triplet (u∗, y∗, p∗) obtained as a solution of the coupled system (constituted
by the state equation (2.1), the adjoint equation (2.21) and the variational inequal-
ity (2.22)) is a candidate for an optimal solution, but not necessarily an optimal
solution. The goal of this article is to analyze the conditions (on the initial data or
on the parameters of the model) which guarantee that the solution of the coupled
system is optimal and unique. This is a crucial step towards the implementation of
the numerical methods to approximate the optimal control. To perform this task,
we start by establishing convenient estimates for the adjoint state p.

Lemma 2.6. Under the assumptions of Theorem 2.5, the adjoint state p satisfies

‖p‖2L∞(0,T ;H2)

≤ λ24

= 2K̃
[(

1 +
1

α
C4TC1λ1(1 + α−1)eC4TC1λ1(

1+α

α2 )
)

× eC4T (1+C1λ1(1+α
−1)) + (αν)−1eC4TC1λ1

(1+α)2

α

](C2
1λ

2
1

α2
+ ‖yd‖2L2(Q)

)
,

(2.23)

where the constants K̃ and λ1 are defined by (2.3) and (2.8), respectively.

Proof. Taking into account the first estimate of [2, Proposition 5.4], we have

‖p‖2L∞(0,T ;L2) ≤ ‖y − yd‖
2
L2(Q)e

C4T (1+C1λ1(1+α
−1)). (2.24)

The second estimate of the same proposition gives

‖Ap‖2L∞(0,T ;L2)

≤ 1

α
(
1

ν
‖y − yd‖2L2(Q) + C4TC1λ1(1 + α−1)‖p‖2L∞(0,T ;L2))e

C4TC1λ1
(1+α)2

α .
(2.25)

Therefore, using the inequalities (2.24) and (2.25) and (2.3), we can apply the same
reasoning as in the proof of Lemma 2.4 to obtain the claimed result. �



8 A. ALMEIDA, N. V. CHEMETOV, F. CIPRIANO EJDE-2022/22

3. Uniqueness results for the control problem (2.17)

This section establishes the main results of the article. First, by analyzing the
second-order derivative of the control-to-state mapping, as well as the second-order
derivative of the cost functional, we deduce a sufficient second-order optimality
condition, which guarantees that any triplet (u∗, y∗, w) obtained as a solution of
the first-order coupled optimality system will produce a locally optimal pair (u∗, y∗).

Next, we will be able to prove that the solution of the first-order coupled op-
timality system is unique. This uniqueness result conjugated with the result of
Theorem 2.5 yields the uniqueness of the solution for the optimal control problem
(2.17). Our results will be achieved under some natural assumptions relying on
the size of the initial data and the parameters of the model, or on the intensity
of the cost. Essentially, if the fluid material is sufficiently viscous and elastic and
the initial condition is small enough, or instead if the intensity of the cost is big
enough, the solution of the first-order optimality system is unique, and corresponds
to the unique global solution of the optimal control problem (2.17).

3.1. Sufficient second-order optimality conditions. In this section, we per-
form a second-order analysis. Let y = S(u) be the solution of (2.1) for u (and y0)
satisfying (2.6). Let us consider some wi, i = 1, 2, satisfying (2.6). Denoting

zi = S′(u)[wi], i = 1, 2,

we can use standard arguments to show that the second-order Gâteaux derivative

z̃ = S′′(u)[w1, w2] =
d

dε

∣∣∣
ε=0

S′(u+ εw2)[w1]

solves the system

∂υ(z̃)

∂t
− ν4z̃ + curl υ(z̃)× y + curl υ(y)× z̃ +∇π̃

= − curl υ(z1)× z2 − curl υ(z2)× z1,
∇ · z̃ = 0 in Q,

z̃ · n = 0, (n ·Dz̃) · τ = 0 on Σ,

z̃(0) = 0 in O.

(3.1)

Definition 3.1. The control problem (2.17) is said to satisfies the second-order
optimality condition at an optimal pair (u, S(u)), if there exists δ > 0, such that
the coercivity condition

J ′′(u)[w,w] > δ‖w‖2L2(Q), ∀w ∈ Uad, (3.2)

holds.

Theorem 3.2. Assume the hypothesis of Theorem 2.5 hold. Then the control
problem (2.17) satisfies the second-order optimality condition at the optimal pair
(u, S(u)) under the assumption that

λ > 2K̂λ23λ4, (3.3)

where K̂, λ3, λ4 are defined in Lemmas 2.1, 2.4 and 2.6.

Proof. Denoting J(u) = J(u, S(u)), we have

J ′(u)[w1] =

∫ T

0

((z1, y − yd) + λ(u,w1))dt



EJDE-2022/22 UNIQUENESS FOR OPTIMAL CONTROL PROBLEMS 9

by [2, Proposition 4.5]. By calculations similar to the mentioned proposition, we
can satisfy that

J ′′(u)[w1, w2] =

∫ T

0

((z1, z2) + (z̃, y − yd) + λ(w2, w1))dt. (3.4)

Then system (3.1), and the duality relation proved in [2, Proposition 5.5] yield

J ′′(u)[w1, w2] =

∫ T

0

((z1, z2)− (p, curl υ(z1)× z2 + curl υ(z2)× z1) + λ(w2, w1))dt.

Hence taking w1 = w2, we have

J ′′(u)[w1, w1] =

∫ T

0

(‖z1‖22 + λ‖w1‖22 − 2(p, curl υ(z1)× z1))dt.

By Lemmas 2.1, 2.4 and 2.6, we deduce the estimate∫ T

0

|(p, curl υ(z1)× z1)|dt ≤ K̂
∫ T

0

‖z1‖2H2‖p‖H2dt

≤ K̂λ4
∫ T

0

‖z1‖2H2dt

≤ K̂λ23λ4
∫ T

0

‖w1‖22dt.

Therefore,

J ′′(u)[w1, w1] > (λ− 2K̂λ23λ4)

∫ T

0

‖w1‖22dt,

which implies the result of the theorem. �

3.2. Uniqueness of the global optimal solution. Now, we are able to show
that the solution of the coupled system is unique, and provides the unique global
optimal solution for the non-convex optimal control problem (2.17).

Theorem 3.3. For any λ > 2K̂λ22λ4, the optimal control problem (2.17) has a

unique global solution, where K̂, λ3, λ4 are defined in Lemmas 2.1, 2.3, and 2.6.

Proof. We assume that u1 and u2 are two optimal control variables for problem
(2.17). Let yi = S(ui), i = 1, 2, be the corresponding optimal states with the
adjoint states pi = S∗(ui), i = 1, 2. Let us set u = u2 − u1, π = π2 − π1. The
differences y = y2 − y1 and u = u2 − u1 solve the system

∂

∂t
υ(y) = ν∆y − curl υ(y)× y1 − curl υ(y1)× y

− curl υ(y)× y −∇π + u,

div y = 0 in Q,

y · n = 0, (n ·Dy) · τ = 0 on Σ,

y(0) = 0 in O.

(3.5)

The functions y1, y2 are the weak solutions of (2.1), which satisfies eq2.7). There-
fore, considering the test function φ = p1 in eq2.7), we show that y verifies the
variational equality(∂y(t)

∂t
, p1
)
V

= −2ν(Dp1(t), Dy)− b(p1, y1, υ(y)) + b(y1, p1, υ(y))



10 A. ALMEIDA, N. V. CHEMETOV, F. CIPRIANO EJDE-2022/22

− b(p1, y, υ(y1)) + b(y, p1, υ(y1))− (curl υ(y)× y, p1) + (u, p1).

Writing the adjoint system (2.21) for p1 in a respective variational form with the
test function φ = y, we obtain the equality(∂p1

∂t
, y(t)

)
V

= 2ν(Dp1(t), Dy)− b(y, p1, υ(y1)) + b(p1, y, υ(y1))

+ b(p1, y1, υ(y))− b(y1, p1, υ(y))− (y1 − yd, y).

Therefore summing the last two equalities, we obtain

∂

∂t
(y(t), p1)V = −(curl υ(y)× y, p1)− (y1 − yd, y) + (u, p1).

The integration over t ∈ (0, T ) and the initial and final conditions for y and p1 give

0 =

∫ T

0

−(curl υ(y)× y, p1)− (y1 − yd, y) + (u, p1) dt. (3.6)

By the symmetry, we can easily satisfy that the difference y1 − y2 = −y solves the
system

− ∂

∂t
υ(y) = −ν∆y + curl υ(y)× y2 + curl υ(y2)× y − curl υ(y)× y +∇π − u,

div y = 0 in Q,

y · n = 0, (n ·Dy) · τ = 0 on Σ,

y(0) = 0 in O.

Using the same reasoning as above, applied for y, we deduce that y1 − y2 satisfies
the relation

0 =

∫ T

0

−(curl υ(y)× y, p2) + (y2 − yd, y)− (u, p2) dt. (3.7)

The sum of the equalities (3.6) and (3.7) yields∫ T

0

‖y‖22 dt+

∫ T

0

(u, p1 − p2)dt =

∫ T

0

(curl υ(y)× y, p1 + p2) dt. (3.8)

The optimality condition (2.22) for the optimal control u1 with Ψ = u2 reads∫ T

0

{λ(u1, u) + (u, p1)} dt ≥ 0.

Analogously, the optimality condition (2.22) for the optimal control u2 with Ψ = u1
yields

−
∫ T

0

{(λu2, u) + (u, p2)} dt ≥ 0.

Adding the last two inequalities, we deduce that

λ

∫ T

0

‖u‖22 dt ≤
∫ T

0

(u, p1 − p2) dt. (3.9)

Introducing this relation in (3.8), we obtain

λ

∫ T

0

‖u‖22 dt+

∫ T

0

‖y‖22 dt ≤
∫ T

0

(curl υ(y)× y, p1 + p2) dt.



EJDE-2022/22 UNIQUENESS FOR OPTIMAL CONTROL PROBLEMS 11

By Lemmas 2.1, 2.3, and 2.6 we have∫ T

0

|(curl υ(y)× y, p1 + p2)|dt ≤
∫ T

0

‖y‖2H2(‖p1‖H2 + ‖p2‖H2)dt

≤ 2K̂λ4

∫ T

0

‖y‖2H2dt

≤ 2K̂λ22λ4

∫ T

0

‖u‖22dt.

Therefore,

(λ− 2K̂λ22λ4)

∫ T

0

‖u‖22 +

∫ T

0

‖y‖22 dt ≤ 0,

which implies the result of the theorem. �

Let us remark that in Theorem 3.3, the constants λ2, λ4 just depend on the
bounded set Uad of the admissible control variables and the target and initial ve-
locities yd, y0. Hence, from practical point of view, the motion of a second grade
fluid can be optimally controlled by taking a sufficiently large intensity of the cost
λ.

Acknowledgments. Nikolai V. Chemetov was supported by the project no. 2021
/ 03758-8 of Regular Research Grant - FAPESP, Brazil, and by the Program of
“Docentes Novos”, Process no. 20.1.4175.1.0 of Universidade de São Paulo.

F. Cipriano was partially supported by the Fundação para a Ciência e a Tec-
nologia (Portuguese Foundation for Science and Technology) through the project
UID/MAT/00297/2019 (Centro de Matemática e Aplicações).

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Adilson Almeida
Centro de Matemática e Aplicações (CMA), Faculdade de Ciências e Tecnologia da

Universidade Nova de Lisboa, Portugal
Email address: ama.almeida@campus.fct.unl.pt

Nikolai V. Chemetov

Department of Computing and Mathematics, University of São Paulo, 14040-901 Ribeirão
Preto - SP, Brazil

Email address: nvchemetov@gmail.com

Fernanda Cipriano
Centro de Matemática e Aplicações (CMA) FCT/UNL and Departamento de Matemática,

Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa, Portugal
Email address: cipriano@fct.unl.pt


	1. Introduction
	2. Formulation of the problem and preliminary results
	3. Uniqueness results for the control problem (2.17)
	3.1. Sufficient second-order optimality conditions
	3.2. Uniqueness of the global optimal solution
	Acknowledgments

	References