EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 4, 2019, 1595-1601
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Existence of Optimal Control for a nonlinear Partial
Differential Equation of Hyperbolic-type

Dieudonné Ampini1,∗, Vital Delmas Mabonzo1

1 Parcours Mathématiques, F.S.T, Université Marien Ngouabi, Brazzaville, Congo

Abstract. In this paper, we prove the existence of an optimal control for a nonlinear hyperbolic
problem, examined in [3]. An estimation is used which makes it possible to extract from a minimiz-
able sequence of controls and from the sequence of corresponding solutions weakly convergent sub
sequences. To prove the passage to the limit in a true equality for every element of the minimizable
sequence, Lebesgue’s theorem on the passage to the limit under the integral sign and the theorem
of immersion have been used.

2010 Mathematics Subject Classifications: 49J20, 58J45, 35L86, 81T13

Key Words and Phrases: Optimal control, hyperbolic equation, functional

1. Preliminaries notions

Before proceeding to the formulation of the problem, let us recall some fundamental
notions of [2].

1.1. Definition of Ck,λ,0(Ω̄) space: (see [4])

Let Ω be a domain of RN , k ∈ N0 and λ ∈]0, 1[. We call Ck,λ,0(Ω̄) any subset of the
functions u ∈ Ck,λ(Ω̄) for which the following condition is satisfied

∀ε > 0, ∃δ > 0 : (x, y ∈ Ω, 0 < |x− y| < δ, |α| = k) =⇒ |Dαu(x)−Dαu(y)| · |x− y|−λ < ε

where α = (α1, · · · , α2) is the multi-index.
The norm of the Ck,λ,0(Ω̄) space is deduced from Ck,λ(Ω̄), namely

‖u‖k,λ =
∑
|α|6k

sup
x∈Ω
|Dαu(x)|+

∑
|α|6k

sup
x 6=y
|Dαu(x)−Dαu(y)| · |x− y|−λ.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i4.3577

Email addresses: dieudonne.ampini@gmail.com (D. Ampini), vitalm28@gmail.com (V. D. Mabonzo)

http://www.ejpam.com 1595 c© 2019 EJPAM All rights reserved.



D. Ampini, V. D. Mabonzo / Eur. J. Pure Appl. Math, 12 (4) (2019), 1595-1601 1596

Theorem 1 ([2], P.11). Let Ω be a bounded domain of Rn, 0 < λ < 1
F : R× Ω̄ −→ R, (u, x) 7−→ F (u, x) a continuous function defined on R× Ω̄, differen-

tiable with respect to u on R for all x ∈ Ω̄ and also F ′u : Ω̄×R −→ R a continuous function
on Ω̄× R satisfied

|F ′u(u, y)− F ′u(v, z)| 6 Q1|u− v|+Q2|y − z|λ

and
|F (u, y)− F (v, z)| 6 C1|u− v|+ C2(y, z)|y − z|λ

where Q1, Q2, C1 are the constants and C2 a bounded function which verifies the condition

∀ ε > 0,∃ δ > 0 : (|y − z| < δ) =⇒ C2(y, z) < ε.

Then the ϕ(x) 7−→ F (ϕ(x), x) mapping is defined from C0,λ,0(Ω̄) to C0,λ,0(Ω̄) and is weakly
sequentially continuous.

Theorem 2 ([2], P.13). Let Ω be a bounded domain of Rn, 0 < λ < 1
K : R×R, (x, y, u) 7−→ K(x, y, u) a continuous function on R× Ω̄2, differentiable with

respect to u on R for all (x, y) ∈ Ω̄2 and also K ′u : Ω̄2×R −→ R a continuous function on
Ω̄2 × R verifying

|K ′u(t, y, u)−K ′u(s, y, u)| 6 Qr|t− s|λ, |u| 6 r

and
|K(t, y, u)−K(s, y, u)| 6 ar(t, s, y), |u| 6 r

with ar a measurable function,∫
Ω
ar(t, s, y)dy 6 br(t, s) · |t− s|λ,

and br : Q̄2
T −→ R satisfied the following conditions:

br is bounded and ∀ε > 0, ∃δ > 0 : (|t− s| < δ) =⇒ br(t, s) < ε
Then the mapping

G : [u(x)] 7−→
∫

Ω
K(x, y, u(y))dy

is defined from C0,λ,0(Ω̄) to C0,λ,0(Ω̄) and weakly sequentially continuous.

2. Main operators

We shall consider the following problem

∂2u

∂t2
−∆u+ |u|ρu = f(x, t), ρ > 0, (1)

∂u

∂~n
(x, t)|∂Ω = 0, t ∈ (0, T ) (2)



D. Ampini, V. D. Mabonzo / Eur. J. Pure Appl. Math, 12 (4) (2019), 1595-1601 1597

u(x, t)|t=0 = ϕ(x), x ∈ Ω,
∂u

∂t
(x, t)|t=0 = ψ(x), x ∈ Ω (3)

in the cylinder
QT = {x, t : x ∈ Ω ⊂ Rn, 0 < t 6 T <∞},

where Ω is a bounded domain of Rn with differentiable boundary ∂Ω, ~n designates the

outer normal to ∂Ω and ∆u =
n∑
i=1

∂2u

∂x2
i

.

Let

H1(Ω) = {v/v ∈ L2(Ω),
∂v

∂xi
∈ L2(Ω), i = 1, · · · , n}

with associated norm

‖v‖H1(Ω) =

(∫
Ω

[
|v|2 +

n∑
i=1

| ∂v
∂xi
|2
]
dx

) 1
2

.

Assume that the functions f(x, t), ϕ(x), ψ(x) are the control and then

f(x, t) ∈ Y ⊂ L2(QT ), ϕ(x) ∈ X ⊂ H1(Ω), ψ(x) ∈W ⊂ L2(Ω) (4)

where Y,X,W are respectively the convex sets, bounded and closed of L2(QT ), H1(Ω) and
L2(Ω).

Let consider the operator:

A : L2(QT )×H1(Ω)× L2(Ω) −→ C0,λ,0(Q̄T )

[A(f, ϕ, ψ)](x, t) =

∫
Ω
K1(x, t, x′, t′)f(x′, t′)dx′dt′+

∫
Ω
K2(x, x′)ϕ(x′)dx′+

∫
Ω
K3(x, x′)ψ(x)dx′

where K1,K2,K3 verify the condition of Hölder:
λ+ λ′, 0 < λ′ < λ, λ+ λ′ < 1 respectively in (x, t), x, x′ and

|K1(x, t, x′, t′)−K1(x̃, t̃, x′, t′)| 6 c3(x′, t′)|(x, t)− (x̃, t̃)|λ+λ′ ,

|K2(x, x′)−K2(x̃, x′)| 6 c4(x′)|x− x̃|λ+λ′ ,

|K3(x, x′)−K3(x̃, x′)| 6 c5(x′)|x− x̃|λ+λ′ ,

sup
(x,t)∈QT

∫
QT

K2
1 (x, t, x′, t′)dx′dt′ = c6 <∞,

sup
x∈Ω̄

∫
QT

K2
2 (x, x′)dx′ = c7 <∞,

sup
x∈Ω̄

∫
QT

K2
3 (x, x′)dx′ = c8 <∞.

with c3(x′, t′) ∈ L2(QT ), c4(x′), c5(x′) ∈ L2(Ω).



D. Ampini, V. D. Mabonzo / Eur. J. Pure Appl. Math, 12 (4) (2019), 1595-1601 1598

Note that this operator is linear, continuous and therefore it is weakly sequentially
continuous (by Theorem 1).

Let consider then the operator

[B(f, ϕ, ψ)](x, t) =

∫
QT

K(x, t, x′, t′, [A(f, ϕ, ψ)](x′, t′)dx′dt′

where

• the function K : Q̄2
T × R −→ R, K : (x, t, x′, ξ) −→ K(x, t, x′, t′, ξ) is continuous on

Q̄2
T × R, differentiable with respect to ξ on R for all (x, t, x′, t′) ∈ Q̄2

T ;

• the derived function K ′ξ : Q̄2
T × R −→ R is also continuous on Q̄2

T × R, and

|K ′ξ(x, t, x′, t′, ξ)−K ′ξ(x̃, t̃, x′, t′, ξ)| 6 QT |(x, t)− (x̃, t̃)|λ+λ′ , |ξ| 6 r

|K ′ξ(x, t, x′, t′, ξ)−K(x̃, t̃, x′, t′, ξ)| 6 ar(x, t, x̃, t̃, x
′, t′), |ξ| 6 r

here ar is a measurable function verifying∫
QT

ar(x, t, x̃, t̃, x
′, t′)dx′dt′ 6 br(x, t, x̃, t̃) · |(x, t)− (x̃, t̃)|λ+λ′

and br : Q̄2
T −→ R satisfying the following conditions:

br is bounded and

∀ε > 0,∃δ > 0 : (|(x, t)− (x̃, t̃)| < δ) =⇒ br(x, t, x̃, t̃) < ε.

This operation is a mapping defined from L2(QT )×H1(Ω)× L2(Ω) to C0,λ,0(Q̄T ) and it
is weakly sequentially continuous (by Theorem 2).

Let E ∈ (C0,λ,0(Q̄T ))′.
Remember ([2],P.5) that there exists such Borelian measures (definite positive) µ1 and

µ2 with bounded variation on Q̄T and Q̄2
T respectively for which

〈E, u〉 =

∫
QT

u(x, t)dµ1(x, t) +

∫ 2

QT

(u(x, t)− u(x̃, t̃)) · |(x, t)− (x̃, t̃)|−λdµ2(x, t, x̃, t̃)

for u ∈ C0,λ,0(Q̄T ).
In this case, the functionals of the form

F i : L2(QT )×H1(Ω)× L2(Ω) −→ R

F i(f, ϕ, ψ) = 〈Ei, Bi(f, ϕ, ψ)〉, i = 0, s1 + s2,

are also weakly sequentially continuous.



D. Ampini, V. D. Mabonzo / Eur. J. Pure Appl. Math, 12 (4) (2019), 1595-1601 1599

3. Formulation of the problem

Consider the problem (1)-(3) with the propositions (4). Then consider the functional
of the form

Ji(f, ϕ, ψ) =

∫
Q̄T

vi(x, t, u(x, t))dxdt+ F i(f, ϕ, ψ), (5)

i = 0, s1 + s2 where the functions vi(x, t, ξ) verify the following conditions:

a) the functions vi(x, t, ξ) are measurable on QT × R,

b) almost for each (x, t) ∈ QT , the functions vi(x, t, ξ) are continuous at ξ on R and

|vi(x, t, ξ)| 6 c9 + c10|ξ|2. (6)

Note that the functions Ji(f, ϕ, ψ) are weakly sequentially continuous by virtue of the
immersion theorem H1(QT ) ⊂ L2(QT ), of inequality

‖u‖H1(QT ) 6 c(T )(‖f‖L2(QT ) + ‖ϕ‖H1(Ω) + ‖ψ‖L2(Ω))

[1], and the continuity of the functional u 7−→
∫
QT

vi(x, t, u(x, t))dxdt from L2(QT ) into
R.

We thus pose the following problem:
To find out such measurable functions f0(x, t) ∈ Y, ϕ0(x) ∈ X, ψ0(x) ∈ W in such

a way that, for the solution u0(x, t) of the problem (1)-(3) corresponding to (f0, ϕ0, ψ0),
inequality-type constraints are verified,

Ji(f, ϕ, ψ) 6 0, i = 1, s1, (7)

equality-type constraints,

Ji(f, ϕ, ψ) = 0, i = s1 + 1, s1 + s2 (8)

and with that
J0(f0, ϕ0, ψ0) = inf

Y×X×W
J0(f, ϕ, ψ) (9)

4. Existence of an optimal control

Theorem 3. We suppose there is a control of the above indicated class and inf
Y×X×W

Ji(f, ϕ, ψ) >

−∞.
Then there exists an optimal control f̂0(x, t), ϕ̂0(x), ψ̂0(x).

Proof. white.
Let {fm(x, t)}m>1, {ϕm(x)}m>1, {ψm(x)}m>1 be minimizable sequences of controls and

{um(x, t)}m>1 their corresponding sequence of solution of the problem (1)-(3).
From the inequality

‖um(x, t)‖H1(QT ) + ‖um(x, t)‖Lp(QT ) 6 const



D. Ampini, V. D. Mabonzo / Eur. J. Pure Appl. Math, 12 (4) (2019), 1595-1601 1600

[3], where p = ρ + 2, it follows that the {um(x, t)}m>1 sequence is uniformly bounded
into H1(QT ); which allows to subtract a sub-sequence of solutions {umk

(x, t)}∞k=1 that
converge weakly to u(x, t) into H1(QT ) and fmk

(x, t), ϕmk
(x), ψmk

(x) converge weakly in
the spaces L2(QT ), H1(Ω), L2(Ω) to f0(x, t) ∈ Y, ϕ0(x) ∈ X,ψ0(x) ∈W .

From the weak converge in H1(QT ) of the sequence umk
(x, t) to u(x, t) and by virtue of

the complete continuity of the operator H1(QT ) into L2(QT ), result the weak convergence
into L2(QT ) of the sequence umk

(x, t) to u(x, t).

H1(QT ) ⊂ L2(QT ) ∀{um(x, t)} ⊂ H1(QT ) : ‖um(x, t)‖H1(QT ) 6 c11

∃ {umk
(x, t)} ⊂ {um(x, t)} which is fundamental in L2(QT ).

As L2(QT ) is complete then ∃ x∗(x, t) ∈ L2(QT ) : umk
(x, t) −→ u∗ converge strongly

into L2(QT ).
By virtue of the separation of L2(QT ), we have u = u∗.
We can consider that ([5],p.162)

|umk
(x, t)| 6 z(x, t) ∈ L2(QT ).

Then from the inequality (6), we obtain

|vi(x, t, umk
)| 6 c9 + c10z

2(x, t) ∈ L1(QT ).

By using the formula of the functional Ji(f, ϕ, ψ) for umk
(x, t), we have

Ji(fmk
, ϕmk

, ψmk
) =

∫
QT

vi(x, t, umk
)dxdt+ F i(fmk

, ϕmk
, ψmk

)

i = 0, s1 + s2

According to the Lebesgue theorem, we obtain

Ji(f̂
0, ϕ̂0, ψ̂0) =

∫
QT

vi(x, t, u(x, t))dxdt+ F i(f̂0, ϕ̂0, ψ̂0) (10)

i = 0, s1 + s2

As the functions fm(x, t), ϕm(x), ψm(x) are the minimizable sequences, then

J0(fm, ϕm, ψm) −→ inf
X×Y×W

J0(f, ϕ, ψ) := J∗ (11)

Under the weak sequential continuity, we have

J∗ = lim
m→∞

J0(fm, ϕm, ψm) = J0(f̂0, ϕ̂0, ψ̂0) (12)

By the same way, we have

lim
m→∞

Ji(fm, ϕm, ψm) = Ji(f̂
0, ϕ̂0, ψ̂0), (13)



REFERENCES 1601

i = 1, s1 + s2

In addition, from (7) and (8), it follows that :

Ji(fm, ϕm, ψm) 6 0, i = 1, s1

Ji(fm, ϕm, ψm) = 0, i = s1 + 1, s1 + s2

and from this, it follows that :

Ji(f̂
0, ϕ̂0, ψ̂0) 6 0, i = 1, s1 (14)

Ji(f̂
0, ϕ̂0, ψ̂0) = 0, i = s1 + 1, s1 + s2. (15)

From (12), (14), (15), it follows that f̂0, ϕ̂0, ψ̂0 is an optimal control.

Acknowledgements

The authors thank the anonymous referees of European Journal of Pure and Applied
Mathematics, for their valuable comments and suggestions which have led to an improve-
ment of the presentation.

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