Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 127, pp. 1–28.

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

INGHAM TYPE APPROACH FOR UNIFORM OBSERVABILITY

INEQUALITY OF THE SEMI-DISCRETE COUPLED

WAVE EQUATIONS

DILBERTO DA SILVA ALMEIDA JÚNIOR, ANDERSON DE JESUS ARAÚJO RAMOS,

JOÃO CARLOS PANTOJA FORTES, MAURO DE LIMA SANTOS

Abstract. This article concerns an observability inequality for a system of

coupled wave equations for the continuous models as well as for the space

semi-discrete finite difference approximations. For finite difference and stan-
dard finite elements methods on uniform numerical meshes it is known that a

numerical pathology produces a blow-up of the constant on the observability

inequality as the mesh-size h tends to zero. We identify this numerical anomaly
for coupled wave equations and we prove that there exists a uniform observ-

ability inequality in a subspace of solutions generated by low frequencies. We

use the Ingham type approach for getting a uniform boundary observability.

1. Introduction

This article concerns an observability inequality problem for 1− d coupled wave
equations for the continuous model as well as for their numerical counterpart for
the space semi-discrete finite difference approximation. The goal of the present
study is to analyze the semi-discrete counterpart in finite difference of an observ-
ability inequality. It is well recognized that observability inequalities are relevant
for control and stabilization theories as well as for inverse problems. In that direc-
tion, the understanding of these questions in finite dimensional is very important
to theoretical and numerical analysis.

In the literature, we found several contributions dealing with numerical questions
on uniform observability and its consequence to the control and stabilization of wave
equations. However, there exist only a few works dealing with analogue questions
for coupled systems of wave equations and numerical counterpart. Almeida Júnior
et al [1] first contributed in that direction for uniform boundary observability. See
also Akri and Maniar [2, 3] and Xu [17]. To fix our ideas we address the problem of
the well known observability inequality in one-dimensional setting. It is well-known
that for the wave equation

utt − uxx = 0, in (0, L)× (0, T ), (1.1)

u(0, t) = u(L, t) = 0, ∀t ≥ 0, (1.2)

2010 Mathematics Subject Classification. 35B35, 35B40, 35K57, 35Q92, 92C17.
Key words and phrases. Coupled wave equations; positivity-preserving; semi-discretization;

Ingham’s inequality.
c©2020 Texas State University.

Submitted November 14, 2019. Published December 22, 2020.

1



2 D. S. ALMEIDA JR., A. J. A. RAMOS, J. C. P. FORTES, M. L. SANTOS EJDE-2020/127

u(x, 0) = u0(x), ut(x, 0) = u1(x), ∀x ∈ (0, L), (1.3)

the total energy of the solution is estimated uniformly by the energy concentrated
near the endpoint x = L. More precisely, for any T > 2L there exists a positive
constant C(T ) satisfying

E(0) ≤ C(T )

∫ T

0

∣∣ux(L, t)
∣∣2dt, (1.4)

for every finite energy solution of (1.1)–(1.3) where the energy is

E(t) :=
1

2

∫ L

0

u2t dx+
1

2

∫ L

0

u2x dx. (1.5)

Estimate (1.4) is known as boundary observability (observability inequality/inverse
inequality) and the best constant C is the so-called observability constant. We re-
fer the readers to Lions [12] and Komornik [11] for an analysis of the equivalence
between controllability and observability through the Hilbert Uniqueness Method
(HUM). On the other hand, semi-discrete schemes generate high frequency nu-
merical spurious oscillations because standard discrete approximations of the wave
equations are, in general, non-uniformly observable. These spurious oscillations
weakly converge to zero as h → 0 and this fact is perfectly compatible with the
convergence property. But, a uniform constant C(T ) for estimate (1.4) is needed
and this is not the case of elementary numerical schemes such as finite difference
and standard finite elements. A numerical evidence to the lack of numerical ob-
servability to boundary observability problem was first observed by Glowinski et al
[6, 7, 8], in connection with the exact boundary controllability of the wave equation
and the numerical implementation of the so-called HUM method. The non-uniform
observability inequality to semi-discrete versions of (1.4) was first solved by Infante
and Zuazua [9]. They noticed that the problem to estimate the total numerical en-
ergy in terms of the numerical energy concentrated at the boundary is not uniform
as h→ 0 for the discrete dynamic equation

u′′j (t)− uj+1(t)− 2uj(t) + uj−1(t)

h2
= 0, j = 1, 2, . . . , J, 0 < t < T, (1.6)

u0(t) = uJ+1(t) = 0, 0 < t < T, (1.7)

uj(0) = u0j , u′j(0) = u1j , j = 0, 1, 2, . . . , J + 1. (1.8)

The total energy of (1.6)–(1.8) is conserved, i.e.,

Eh(t) :=
h

2

J∑
j=0

[
|u′j(t)|2 +

∣∣uj+1(t)− uj(t)
h

∣∣2] = Eh(0), ∀0 < t < T. (1.9)

In [9] there is an analysis of a discrete version of (1.4), i.e.,

Eh(0) ≤ C(T )

∫ T

0

|uJ(t)/h|2dt, (1.10)

with uJ+1(t) = 0 for 0 < t < T . The problem is to know if the positive constant
C(T ) blows-up as h→ 0. According to the result given in [9, Theorem 1.1],

sup
uj solution of (1.6)-(1.8)

Eh(0)∫ T
0
|uJ (t)

h |2dt
→∞ as h→ 0. (1.11)



EJDE-2020/127 SEMI-DISCRETE COUPLED WAVE EQUATIONS 3

Otherwise, there exists a positive counterpart to (1.11) in a subspace of solutions
generated by low frequencies. See [9, Theorem 1.2] for similar results for the stan-
dard finite element methods. In both cases, the authors used standard numerical
schemes on uniform meshes. Since the pioneering work by Infante and Zuazua [9], a
large number of papers and numerical techniques have been introduced to deal with
the uniform observability and questions related to control and stabilization theo-
ries. We refer the readers to the surveys [5, 18] for the developments in numerical
analysis dealing with uniform observability on uniform meshes. See also Ervedoza
et al [4] who opened new perspectives in the uniform observability problem by tak-
ing non-uniform numerical meshes for finite difference and standard finite element
approximations of the 1D wave equation.

This article concerns the theoretical numerical analysis of an observability in-
equality concerning a coupled system of wave equations. We use a particular scheme
in finite difference semi-discretization on a uniform mesh. Let us consider the 1D
coupled wave equations

utt − uxx + αv = 0, (0, L)× (0, T ), (1.12)

vtt − vxx + αu = 0, (0, L)× (0, T ), (1.13)

u(x, t) = v(x, t) = 0, x = 0, L, 0 < t < T, (1.14)

u(x, 0) = u0(x), ut(x, 0) = u1(x), ∀x ∈ (0, L), (1.15)

v(x, 0) = v0(x), vt(x, 0) = v1(x), ∀x ∈ (0, L), (1.16)

where the positive coupling term α is proportional to the displacements u(x, t)
and v(x, t). Here, t and x are the time and space variables, respectively. The
functions u(x, t) and v(x, t) are the displacements of two vibrating strings, measured
from their equilibrium positions and the coupling is given by coupling terms αv
and αu. This system is motivated by analogous problem in ordinary differential
equations for coupled harmonic oscillators (see [13]) and it is well-posed in the

energy space H :=
(
H1

0 (0, L)×L2(0, L)
)2

. That is to say, denoting U = (u, ut, v, vt)

we have that for any U0 ∈
(
H1

0 (0, L)×L2(0, L)
)2

there exists a unique solution U in

C0
(
[0, T ];D(A)

)
∩C1

(
[0, T ];H

)
, where D(A) :=

(
H1

0 (0, L)∩H2(0, L)×H1
0 (0, L)

)2
.

The energy of solutions to system (1.12)–(1.16) is

E(t) :=
1

2

∫ L

0

u2t dx+
1

2

∫ L

0

u2x dx+
1

2

∫ L

0

v2t dx+
1

2

∫ L

0

v2x dx+α

∫ L

0

uv dx. (1.17)

In this work, we obtain an observability inequality of system (1.12)–(1.16) and
we study such observability in the light of a semi-discrete scheme in finite difference.

The rest of this article is organized as follows. In section 2, we show a neces-
sary condition to obtain the positivity of the energy and we build an observability
inequality. In section 3, we introduce a semi-discrete numerical scheme in finite
difference and we show a uniform observability inequality no filtering. In section
4, we prove results on uniform observability for filtered solutions using Ingham
inequalities and we improve the results obtained in section 3.

2. Energy properties at continuous level

In this section, we establish some properties concerning the system (1.12)–(1.16).
We focus on the observability inequality built from multipliers. First, we develop



4 D. S. ALMEIDA JR., A. J. A. RAMOS, J. C. P. FORTES, M. L. SANTOS EJDE-2020/127

the Fourier series of the system and after that we show two important properties
concerning the energy in (1.17): its positivity and its conservation law.

2.1. Fourier expansion of the solutions. Here, we show that the solutions of
(1.12)–(1.16) admit a Fourier expansion on the specific basis of eigenvectors. To do
this, we decoupled the system into two systems, namely: for ψ := u− v we obtain
from system (1.12)–(1.16) the system given by

ψtt − ψxx − αψ = 0, in (0, L)× (0, T ), (2.1)

ψ(0, t) = ψ(L, t) = 0, 0 < t < T, (2.2)

ψ(x, 0) = ψ0(x), ψt(x, 0) = ψ1(x), ∀x ∈ (0, L), (2.3)

as well as for the system

φtt − φxx + αφ = 0, in (0, L)× (0, T ), (2.4)

φ(0, t) = φ(L, t) = 0, 0 < t < T, (2.5)

φ(x, 0) = φ0(x), φt(x, 0) = φ1(x), ∀x ∈ (0, L), (2.6)

where φ := u+ v. Both systems are conservative and their energies are

G(t) :=
1

2

∫ L

0

ψ2
t dx+

1

2

∫ L

0

ψ2
x dx−

α

2

∫ L

0

ψ2 dx, (2.7)

F (t) :=
1

2

∫ L

0

φ2t dx+
1

2

∫ L

0

φ2x dx+
α

2

∫ L

0

φ2 dx. (2.8)

The results obtained for the energy of the decoupled systems are extended to
the coupled system by taking E(t) := (F (t) +G(t))/2 for t ≥ 0.

Proposition 2.1. Assume that α ≤ π2/L2. Then, the solutions of the system
(2.1)–(2.3) admit the Fourier expansion, on the basis of eigenvectors,

ψ =
∑
k≥1

[
ak sin

(√
ν−k t

)
+ bk cos

(√
ν−k t

)]
sin
(kπx
L

)
, (2.9)

where ak, bk are the Fourier coefficients, and ν−k =
(
kπ
L

)2 − α are the eigenvalues
for k ≥ 1.

Proof. The solutions of (2.1)–(2.3) can be written as ψ(x, t) = f(x)T (t). By sub-
stituting this decomposition into (2.1) we obtain

T ′′(t)− αT (t)

T (t)
=
f ′′(x)

f(x)
= −ν2, ∀t ≥ 0. (2.10)

Thus, we obtain the eigenvalue problem

f ′′ + ν2f = 0, (2.11)

f(0) = f(L) = 0. (2.12)

To obtain non-trivial solutions, we must take ν > 0. Under this assumption, it
is immediate that the solution of (2.11)–(2.12) is

f(x) = c1 sin(νx), (2.13)



EJDE-2020/127 SEMI-DISCRETE COUPLED WAVE EQUATIONS 5

where c1 is constant. By taking the solution at x = L it results that sin(νL) = 0,
from where we obtain ν = νk = kπ/L, k ≥ 1. On the other hand, from the equation
T ′′(t) + (ν2 − α)T (t) = 0 we obtain

T (t) = aei
√
ν2−α t + be−i

√
ν2−α t. (2.14)

The solutions are valid for ν2 > α. Hence, for α ≤ π2/L2 ≤ k2π2/L2 for all
k ≥ 1 and using the linear combination of solutions, we obtain

Tk(t) = ak sin
(√

ν−k t
)

+ bk cos
(√

ν−k t
)
, ∀t ≥ 0, (2.15)

where ν−k :=
(
kπ
L

)2−α, and ak, bk are the Fourier coefficients for k ≥ 1. The proof
is complete. �

One important question is the positivity of the energy of solutions for both
systems (2.1)–(2.3) and (1.12)–(1.16). By taking the initial data

ψ0(x) =
∑
k≥1

bk sin
(kπx
L

)
, ψ1(x) =

∑
k≥1

ak

√
ν−k sin

(kπx
L

)
,

we obtain that

G(0) =
L

4

∑
k≥1

b2k

[(kπ
L

)2
− α

]
+
L

4

∑
k≥1

a2kν
−
k ≥ 0, (2.16)

if α ≤ π2/L2. Analogously, we have the following two results.

Proposition 2.2. The solutions of (2.4)–(2.6) admit the Fourier expansion, on
the basis of eigenvectors,

φ =
∑
k≥1

[
ck sin

(√
ν+k t

)
+ dk cos

(√
ν+k t

)]
sin
(kπx
L

)
, (2.17)

where ck, dk are the Fourier coefficients and ν+k =
(
kπ
L

)2
+ α are the eigenvalues

for k ≥ 1.

Proposition 2.3. Assume that α ≤ π2/L2. Then the solutions of (1.12)–(1.16)
admit the Fourier expansion, on the basis of eigenvectors,

u =
1

2

∑
k≥1

[
ak sin

(√
ν−k t

)
+ bk cos

(√
ν−k t

)
+ ck sin

(√
ν+k t

)
+ dk cos

(√
ν+k t

)]
sin
(kπx
L

)
,

(2.18)

v =
1

2

∑
k≥1

[
ck sin

(√
ν+k t

)
+ dk cos

(√
ν+k t

)
− ak sin

(√
ν−k t

)
− bk cos

(√
ν−k t

)]
sin
(kπx
L

)
,

(2.19)

where ak, bk, ck, dk are the Fourier coefficients and ν±k =
(
kπ
L

)2 ± α are the eigen-
values.

The proof of the two above results follows from Propositions 2.1 and 2.2. By
taking the initial data

u0(x) =
1

2

∑
k≥1

(bk + dk) sin
(kπx
L

)
,



6 D. S. ALMEIDA JR., A. J. A. RAMOS, J. C. P. FORTES, M. L. SANTOS EJDE-2020/127

u1(x) =
1

2

∑
k≥1

(
ak

√
ν−k + ck

√
ν+k

)
sin
(kπx
L

)
,

v0(x) =
1

2

∑
k≥1

(dk − bk) sin
(kπx
L

)
,

v1(x) =
1

2

∑
k≥1

(
ck

√
ν+k − ak

√
ν−k

)
sin
(kπx
L

)
,

and after some calculations we obtain the positivity of E(t), i.e.,

E(0) =
L

4

∑
k≥1

d2k

[(kπ
L

)2
+ α

]
+
L

4

∑
k≥1

b2k

[(kπ
L

)2
− α

]
+
L

4

∑
k≥1

(
c2kν

+
k + a2kν

−
k

)
≥ 0,

if α ≤ π2/L2.

2.2. Observability inequality for the coupled system. One of the main phys-
ical properties concerning the hyperbolic systems of wave propagations is the pos-
itivity of the energy of solutions. Before getting the observability inequality for
system (1.12)–(1.16), we show that it’s energy is positive and it obeys the energy
conservation law.

Theorem 2.4. Let U = (u, ut, v, vt) be the solution of (1.12)–(1.16). Then, for all
α ≤ π2/L2, holds

E(t) ≥ π2 − αL2

2π2

[ ∫ L

0

u2x dx+

∫ L

0

v2x dx
]
≥ 0, ∀t ∈ [0, T ], (2.20)

where E(t) is the energy defined (1.17).

Proof. We multiply formally the Eqs. (1.12) and (1.13) by ut and vt respectively,
and then we add the two resulting equations considering the boundary conditions
(1.14) to obtain

d

dt

[1

2

∫ L

0

u2t dx+
1

2

∫ L

0

u2x dx+
1

2

∫ L

0

v2t dx+
1

2

∫ L

0

v2x dx+α

∫ L

0

uv dx
]

= 0, (2.21)

for all t ∈ [0, T ]. Then, the energy conservation property is assured, i.e.,

E(t) = E(0), ∀t ∈ [0, T ], (2.22)



EJDE-2020/127 SEMI-DISCRETE COUPLED WAVE EQUATIONS 7

where E(t) is given by (1.17). On the other hand, taking into account that 2uv ≥
−u2 − v2 and using Poincaré’s inequality [16] we have

E(t) ≥ 1

2

[ ∫ L

0

u2t dx+

∫ L

0

u2x dx+

∫ L

0

v2t dx+

∫ L

0

v2x dx

− α
∫ L

0

u2 dx− α
∫ L

0

v2 dx
]

≥ 1

2

[ ∫ L

0

u2t dx+

∫ L

0

u2x dx+

∫ L

0

v2t dx+

∫ L

0

v2x dx

− αL
2

π2

∫ L

0

u2x dx− α
L2

π2

∫ L

0

v2x dx
]

=
1

2

[ ∫ L

0

u2t dx+
(

1− αL
2

π2

)∫ L

0

u2x dx+

∫ L

0

v2t dx

+
(

1− αL
2

π2

)∫ L

0

v2x dx
]
,

(2.23)

and since α ≤ π2/L2, it follows that

E(t) ≥ π2 − αL2

2π2

[ ∫ L

0

u2x dx+

∫ L

0

v2x dx
]
, (2.24)

assuring the positivity of E(t). �

The next theorem gives our result on the observability inequality.

Theorem 2.5. Let U = (u, ut, v, vt) be the solution of the system (1.12)–(1.16).
Then for all α ≤ π2/L2 there exists T0 > 0 such that for all T > T0 there exists
C(T, α) > 0 for which

E(0) ≤ C(T, α)
[
α

∫ T

0

∫ L

0

(u2 + v2) dx dt+
L

2

∫ T

0

u2x(L, t) dt

+
L

2

∫ T

0

v2x(L, t) dt
]
,

(2.25)

where E(t) is the energy given by (1.17).

Proof. Multiplying the Eq. (1.12) by xux we have∫ T

0

∫ L

0

(
utt − uxx + αv

)
xux dx dt = 0. (2.26)

From the boundary conditions in (1.14) we have∫ T

0

∫ L

0

uttxux dx dt =
[ ∫ L

0

utuxx dx
]T
0

+
1

2

∫ T

0

∫ L

0

u2t dx dt , (2.27)

−
∫ T

0

∫ L

0

uxxuxx dx dt =
1

2

∫ T

0

∫ L

0

u2x dx dt−
L

2

∫ T

0

u2x(L, t) dt. (2.28)



8 D. S. ALMEIDA JR., A. J. A. RAMOS, J. C. P. FORTES, M. L. SANTOS EJDE-2020/127

Combining (2.26), (2.27) and (2.28), one has

Xu(t)
∣∣T
0

+
1

2

∫ T

0

∫ L

0

u2t dx dt+
1

2

∫ T

0

∫ L

0

u2x dx dt+ α

∫ T

0

∫ L

0

vxux dx dt

=
L

2

∫ T

0

u2x(L, t) dt,

(2.29)

where Xu(t) =
∫ L
0
xuxut dx. Analogously, multiplying the Eq. (1.13) by xvx we

obtain

Xv(t)
∣∣T
0

+
1

2

∫ T

0

∫ L

0

v2t dx dt+
1

2

∫ T

0

∫ L

0

v2x dx dt+ α

∫ T

0

∫ L

0

uxvx dx dt

=
L

2

∫ T

0

v2x(L, t) dt,

(2.30)

where Xv(t) =
∫ L
0
xvxvt dx. Adding (2.29) and (2.30) we obtain

[
Xu(t) +Xv(t)

]∣∣T
0

+

∫ T

0

[1

2

∫ L

0

u2t dx+
1

2

∫ L

0

v2t dx+
1

2

∫ L

0

v2x dx

+
1

2

∫ L

0

v2x dx
]
dt+ α

∫ T

0

∫ L

0

(vux + uvx)x dx dt

=
L

2

∫ T

0

u2x(L, t) dt+
L

2

∫ T

0

v2x(L, t) dt.

(2.31)

Moreover,

α

∫ T

0

∫ L

0

(vux + uvx)x dx dt = α

∫ T

0

∫ L

0

x
d

dx
(uv) dx dt = −α

∫ T

0

∫ L

0

uv dx dt,

from where we arrive at

[Xu(t) +Xv(t)]
∣∣T
0

+

∫ T

0

E(t)dt

= 2α

∫ T

0

∫ L

0

uv dx dt+
L

2

∫ T

0

u2x(L, t) dt+
L

2

∫ T

0

v2x(L, t) dt.

(2.32)

Using Young’s inequality we have

|Xu(t) +Xv(t)| ≤
L

2

∫ L

0

u2t dx+
L

2

∫ L

0

u2x dx+
L

2

∫ L

0

v2t dx+
L

2

∫ L

0

v2x dx , (2.33)

and taking (2.23) into account we have

L

2

[ ∫ L

0

u2t dx+

∫ L

0

u2x dx+

∫ L

0

v2t dx+

∫ L

0

v2x dx
]

≤ αL
∫ L

0

u2 dx+ αL

∫ L

0

v2 dx+ LE(t)

≤ αL
3

π2

∫ L

0

u2x dx+ α
L3

π2

∫ L

0

v2x dx+ LE(t)

= α
L3

π2

[ ∫ L

0

u2x dx+

∫ L

0

v2x dx
]

+ LE(t) .

(2.34)



EJDE-2020/127 SEMI-DISCRETE COUPLED WAVE EQUATIONS 9

From Theorem (2.4) we conclude that

L

2

[ ∫ L

0

u2t dx+

∫ L

0

u2x dx+

∫ L

0

v2t dx+

∫ L

0

v2x dx
]

≤ αL
3

π2

[ ∫ L

0

u2x dx+

∫ L

0

v2x dx
]

+ LE(t)

≤ α L
3

2π2

2π2

π2 − αL2
E(t) + LE(t)

=
Lπ2

π2 − αL2
E(t),

(2.35)

from where we obtain

|Xu(t) +Xv(t)| ≤ L
π2

π2 − αL2
E(t). (2.36)

On the other hand, from energy conservation law (2.22), the inequality (2.32)

lead us to an observability inequality. Indeed, for T0 = 2L π2

π2−αL2 > 0 we have

E(0) ≤ C
[
2α

∫ T

0

∫ L

0

uv dx dt+
L

2

∫ T

0

u2x(L, t) dt+
L

2

∫ T

0

v2x(L, t) dt
]
, (2.37)

where C = C(T, α) := π2−αL2

T (π2−αL2)−2Lπ2 . Finally, using the Young’s inequality we

obtain the required result. �

Note that (2.25) uses observations on the boundary and observations distributed
on the whole space domain. The observability term with α which is integrated on
the whole domain is normally removable, but the literature is too brief. We refer
the reader to Tebou [15] for a study of several observability estimates for a system of
two coupled non-conservative wave equations, and for possible extensions to other
similar models with fewer controls. In any case, one can use the compactness-
uniqueness argument such performed by Zuazua [19] to prove that the observation
distributed on the whole domain can be absorbed. In that direction, it is possible
to obtain the observability inequality given by

E(0) ≤ Ĉ(T )
L

2

∫ T

0

[
u2x + v2x

]
(L, t) dt, (2.38)

where Ĉ(T ) depends also on α.

2.3. Observability inequalities for the uncoupled systems. In this section,
we obtain observability inequalities for the coupled systems and after that we re-
cover the observability inequality (2.25).

Theorem 2.6. Let (φ, φt) be the solution of system (2.4)–(2.6). Then, for all
T > 2L there exists C(T ) > 0 such that

F (0) ≤ C(T )
[
α

∫ T

0

∫ L

0

φ2 dx dt+
L

2

∫ T

0

φ2x(L, t) dt
]
, (2.39)

where F (t) is the energy given by

F (t) :=
1

2

∫ L

0

φ2x dx+
1

2

∫ L

0

φ2x dx+
α

2

∫ L

0

φ2 dx. (2.40)



10 D. S. ALMEIDA JR., A. J. A. RAMOS, J. C. P. FORTES, M. L. SANTOS EJDE-2020/127

Proof. Proceeding as in Theorem (2.5), by using multipliers of the type xφx, we
obtain

Xφ(t)
∣∣T
0

+
1

2

∫ T

0

∫ L

0

φ2t dx dt+
1

2

∫ T

0

∫ L

0

φ2x dx dt−
α

2

∫ T

0

∫ L

0

φ2 dx dt

=
L

2

∫ T

0

φ2x(L, t) dt,

(2.41)

where Xφ(t) =
∫ L
0
xφxφt dx and from where results that

Xφ(t)
∣∣T
0

+

∫ T

0

F (t)dt = α

∫ T

0

∫ L

0

φ2 dx dt+
L

2

∫ T

0

φ2x(L, t) dt. (2.42)

It is immediate that Xφ(t)
∣∣T
0
≥ −2LF (t). Moreover, having in mind the energy con-

servation law for F (t), the inequality (2.42) lead us to the observability inequality
for T > 2L, i.e.,

F (0) ≤ C(T )
[
α

∫ T

0

∫ L

0

φ2 dx dt+
L

2

∫ T

0

φ2x(L, t) dt
]
, (2.43)

where C(T ) = 1/(T − 2L). The proof is complete. �

Theorem 2.7. Let (ψ,ψt) be the solution of the system (2.1)–(2.3). Then, for all
α ≤ π2/L2 there exists T0 > 0 such that for all T > T0 there exists C(T, α) > 0
such that

G(0) ≤ C(T, α)
L

2

∫ T

0

ψ2
x(L, t) dt, (2.44)

where

C(T, α) =
π2 − αL2

T (π2 − αL2)− 2Lπ2
> 0,

and G(t) is the energy given by

G(t) :=
1

2

∫ L

0

ψ2
x dx+

1

2

∫ L

0

ψ2
x dx−

α

2

∫ L

0

ψ2 dx. (2.45)

Proof. It is immediate that

Xψ(t)
∣∣T
0

+
1

2

∫ T

0

∫ L

0

ψ2
t dx dt+

1

2

∫ T

0

∫ L

0

ψ2
x dx dt+

α

2

∫ T

0

∫ L

0

ψ2 dx dt

=
L

2

∫ T

0

ψ2
x(L, t) dt,

(2.46)

where Xψ(t) =
∫ L
0
xψxψt dx and taking into account the energy defined by (2.45),

we obtain

Xψ(t)
∣∣T
0

+

∫ T

0

G(t)dt = −α
∫ T

0

∫ L

0

ψ2 dx dt+
L

2

∫ T

0

ψ2
x(L, t) dt. (2.47)

Moreover, using Poincaré’s inequality [16], we have

G(t) ≥ 1

2

[ ∫ L

0

ψ2
t dx+

∫ L

0

ψ2
x dx− α

L2

π2

∫ L

0

ψ2
x dx

]
=

1

2

∫ L

0

[
ψ2
t +

(
1− αL

2

π2

)
ψ2
x

]
dx .

(2.48)



EJDE-2020/127 SEMI-DISCRETE COUPLED WAVE EQUATIONS 11

It follows for α ≤ π2/L2 that

G(t) ≥ π2 − αL2

2π2

∫ L

0

ψ2
x dx, (2.49)

implying the positivity of G(t). Now we estimate Xψ(t) as follows,

|Xψ(t)| ≤ L

2

∫ L

0

ψ2
t dx+

L

2

∫ L

0

ψ2
x dx

≤ LG(t) + α
L

2

∫ L

0

ψ2 dx

≤ LG(t) + α
L3

2π2

∫ L

0

ψ2
x dx .

Using (2.49), we obtain

|Xψ(t)| ≤ LG(t) + α
L3

2π2

∫ L

0

ψ2
x dx ≤ L

π2

π2 − αL2
G(t). (2.50)

Finally, having in mind the energy conservation law of G(t), inequality (2.47)

lead us to the observability inequality for T > T0 where T0 = 2L π2

π2−αL2 , i.e.,

G(0) ≤ π2 − αL2

T (π2 − αL2)− 2Lπ2

[
− α

∫ T

0

∫ L

0

ψ2 dx dt+
L

2

∫ T

0

ψ2
x(L, t) dt

]
≤ π2 − αL2

T (π2 − αL2)− 2Lπ2

L

2

∫ T

0

ψ2
x(L, t) dt,

which completes the proof. �

From Theorems (2.6) and (2.7) we have

F (0) +G(0) ≤ 1

T − 2L

[
α

∫ T

0

∫ L

0

φ2 dx dt+
L

2

∫ T

0

φ2x(L, t) dt
]

+
π2 − αL2

T (π2 − αL2)− 2Lπ2

L

2

∫ T

0

ψ2
x(L, t) dt .

Noting that

T (π2 − αL2)− 2Lπ2 = Tπ2 − TαL2 − 2Lπ2

≤ (Tπ2 + 2αL3)− TαL2 − 2Lπ2 = (T − 2L)(π2 − αL2),

we obtain
1

T − 2L
≤ π2 − αL2

T (π2 − αL2)− 2Lπ2
.

Therefore, using φ = u+ v and ψ = u− v, we obtain

F (0) +G(0)

≤ π2 − αL2

T (π2 − αL2)− 2Lπ2

[
α

∫ T

0

∫ L

0

(u+ v)2 dx dt

+
L

2

∫ T

0

(
ux + vx

)2
(L, t)dt+

L

2

∫ T

0

(
ux − vx

)2
(L, t)dt

]
=

π2 − αL2

T (π2 − αL2)− 2Lπ2

[
4α

∫ T

0

∫ L

0

uv dx dt+ L

∫ T

0

[
u2x + v2x](L, t)dt

]
,



12 D. S. ALMEIDA JR., A. J. A. RAMOS, J. C. P. FORTES, M. L. SANTOS EJDE-2020/127

and using Young’s inequality it follows that

E(0) ≤ π2 − αL2

T (π2 − αL2)− 2Lπ2

[
α

∫ T

0

∫ L

0

(u2 + v2) dx dt+
L

2

∫ T

0

[
u2x + v2x](L, t)dt

]
,

for 2E(0) = F (0) +G(0).

3. Finite difference semi-discretization

For our purposes we consider J a nonnegative integer, h = L/(J + 1) and the
partition of (0, L) given by

0 = x0 < x1 < · · · < xJ < xJ+1 = L, where xj = jh, ∀j = 0, . . . , J + 1. (3.1)

Here, we consider the following semi-discrete system for the 1D coupled wave equa-
tions (1.12)–(1.16):

u′′j (t)−∆huj(t) + αvj(t) = 0, j = 1, 2, . . . , J, 0 < t < T, (3.2)

v′′j (t)−∆hvj(t) + αuj(t) = 0, j = 1, 2, . . . , J, 0 < t < T, (3.3)

u0(t) = uJ+1(t) = 0, v0(t) = vJ+1(t) = 0, 0 < t < T, (3.4)

uj(0) = u0j , u′j(0) = u1j , vj(0) = v0j , v′j(0) = v1j ,

j = 0, 1, 2, . . . , J + 1,
(3.5)

where primes ′ denotes the derivative of u with respect to time t, and the functions
uj(t) and vj(t) are approximations to u(xj , t) and v(xj , t) respectively, being u and
v solutions of (1.12)–(1.16). We use ∆h to denote

∆huj(t) :=
uj+1(t)− 2uj(t) + uj−1(t)

h2
, ∆hvj(t) :=

vj+1(t)− 2vj(t) + vj−1(t)

h2

The energy of (3.2)–(3.5) is

Eh(t) :=
h

2

J∑
j=0

[
|u′j(t)|2 +

∣∣uj+1(t)− uj(t)
h

∣∣2 + |v′j(t)|2

+
∣∣vj+1(t)− vj(t)

h

∣∣2 + 2αuj(t)vj(t)
]
,

(3.6)

and it is a conserved quantity of the time t for the system (3.2)–(3.5). Our aim here
relies on analysis of a semi-discrete counterpart to the boundary observability (2.38)
with respect to mesh size h. To do this we use the Ingham’s type approach [10].
First of all, we show that there exists a uniform observability inequality associated
with the observability inequality (2.25) (see [14, 18]). We consider the two systems
associated to the discrete system (3.2)–(3.5). The first one is obtained by taking
φj(t) := uj(t) + vj(t) from where we have

φ′′j (t)−∆hφj(t) + αφj(t) = 0, j = 1, 2, . . . , J, 0 < t < T, (3.7)

φ0(t) = φJ+1(t) = 0, 0 < t < T, (3.8)

φj(0) = φ0j , φ′j(0) = φ1j , j = 0, 1, 2, . . . , J + 1, (3.9)

and the second one is obtained for ψj(t) := uj(t)− vj(t) from where we have

ψ′′j (t)−∆hψj(t)− αψj(t) = 0, j = 1, 2, . . . , J, 0 < t < T, (3.10)

ψ0(t) = ψJ+1(t) = 0, 0 < t < T, (3.11)

ψj(0) = ψ0
j , ψ′j(0) = ψ1

j , j = 0, 1, 2, . . . , J + 1. (3.12)



EJDE-2020/127 SEMI-DISCRETE COUPLED WAVE EQUATIONS 13

Note that for uj(t) := (φj(t) +ψj(t))/2 and vj(t) := (φj(t)−ψj(t))/2 we recover
the Eqs. (3.2)-(3.3). Their energies are given, respectively, by

Fh(t) :=
h

2

J∑
j=0

[
|φ′j(t)|2 +

∣∣φj+1(t)− φj(t)
h

∣∣2 + α|φj(t)|2
]
, (3.13)

Gh(t) :=
h

2

J∑
j=0

[
|ψ′j(t)|2 +

∣∣ψj+1(t)− ψj(t)
h

∣∣2 − αψ2
j (t)

]
. (3.14)

It is not difficult to see these energies are conservative for all time t. That is
to say, Fh(t) = Fh(0) and Gh(t) = Gh(0) for all t ∈ [0, T ]. Naturally, Eh(t) =
(Gh(t) + Fh(t))/2 for all t ∈ [0, T ].

Proposition 3.1 (Energy conserving). For any h > 0 and (u, v) solution of (3.2)–
(3.5) we have

Eh(t) = Eh(0), ∀t ∈ [0, T ], (3.15)

where Eh(t) is given in (3.6).

Proof. Multiplying the Eqs. (3.2) and (3.3) by hu′j(t) and hv′j(t) respectively, and
adding the results for indices j = 1, 2, . . . , J , we obtain

h

J∑
j=1

u′′j u
′
j − h

J∑
j=1

(∆huj)u
′
j + hα

J∑
j=1

vju
′
j = 0, (3.16)

h

J∑
j=1

v′′j v
′
j − h

J∑
j=1

(∆hvj)v
′
j + hα

J∑
j=1

ujv
′
j = 0. (3.17)

Keeping in mind the boundary conditions (3.4) and after some calculations we
arrive at

−h
J∑
j=1

(∆huj)u
′
j =

h

2

d

dt

J∑
j=0

∣∣uj+1 − uj
h

∣∣2, (3.18)

−h
J∑
j=1

(∆hvj)v
′
j =

h

2

d

dt

J∑
j=0

∣∣vj+1 − vj
h

∣∣2. (3.19)

On the other hand,

αh

J∑
j=1

vju
′
j + αh

J∑
j=1

ujv
′
j = αh

d

dt

J∑
j=0

ujvj . (3.20)

Substituting (3.18), (3.19) and (3.20) into (3.16) and (3.17) we obtain

d

dt

J∑
j=0

[h
2
|u′j |2+

h

2

J∑
j=0

∣∣uj+1 − uj
h

∣∣2+
h

2
|v′j |2+

h

2

∣∣vj+1 − vj
h

∣∣2+αhujvj

]
= 0, (3.21)

or
d

dt
Eh(t) = 0 ⇒ Eh(t) = Eh(0), ∀t ∈ [0, T ]. (3.22)

The proof is complete. �

Next, we show the positivity of Eh(t).



14 D. S. ALMEIDA JR., A. J. A. RAMOS, J. C. P. FORTES, M. L. SANTOS EJDE-2020/127

Proposition 3.2 (Energy positivity). For any h > 0 and (u, v) solution of (3.2)–
(3.5) we have

Eh(t) ≥ π2 − αL2

π2

h

2

J∑
j=0

[∣∣uj+1(t)− uj(t)
h

∣∣2 +
∣∣vj+1(t)− vj(t)

h

∣∣2], (3.23)

since α ≤ π2/L2, where E(t) is given by (3.6).

Proof. Proceeding as in Theorem (2.4), we have

2Eh(t) = h

J∑
j=0

[
|u′j(t)|2 +

∣∣uj+1(t)− uj(t)
h

∣∣2 + |v′j(t)|2

+
∣∣vj+1(t)− vj(t)

h

∣∣2 + 2αuj(t)vj(t)
]

≥ h
J∑
j=0

[
|u′j(t)|2 +

∣∣uj+1(t)− uj(t)
h

∣∣2 + |v′j(t)|2 +
∣∣vj+1(t)− vj(t)

h

∣∣2
− α|vj(t)|2 − α|uj(t)|2

]
.

Now, using the embedding Theorem (see [9])

h

J∑
j=0

|uj(t)|2 ≤
L2

π2
h

J∑
j=0

∣∣uj+1(t)− uj(t)
h

∣∣2 , (3.24)

and for h sufficiently small, we obtain

2Eh(t) ≥ h
J∑
j=0

[
|u′j(t)|2 +

(
1− αL

2

π2

)∣∣uj+1(t)− uj(t)
h

∣∣2 + |v′j(t)|2

+
(

1− αL
2

π2

)∣∣vj+1(t)− vj(t)
h

∣∣2]
= h

J∑
j=0

[
|u′j(t)|2 +

π2 − αL2

π2

∣∣uj+1(t)− uj(t)
h

∣∣2 + |v′j(t)|2

+
π2 − αL2

π2

∣∣vj+1(t)− vj(t)
h

∣∣2],
from where we obtain the positivity of Eh(t) for α ≤ π2/L2. �

3.1. Uniform observability inequality: no filtering. In this section, we prove
that there exists a uniform observability inequality which is the discrete counterpart
of the observability inequality (2.25). The proof requires a set of results obtained
using discrete multipliers. The first result concerns to the system (3.10)–(3.12).

Lemma 3.3. For any h > 0 and ψ solution of (3.10)–(3.12) we have

TGh(0) + χh(t)
∣∣T
0

=
h3

4

J∑
j=0

∫ T

0

(ψ′j+1 − ψ′j
h

)2
dt+

L

2

∫ T

0

∣∣ψJ
h

∣∣2dt, (3.25)

where

χh(t) = h

J∑
j=1

jψ′j

(ψj+1 − ψj−1
2

)
. (3.26)



EJDE-2020/127 SEMI-DISCRETE COUPLED WAVE EQUATIONS 15

Proof. Multiplying the Eq. (3.10) by j(ψj+1 − ψj−1)/2 we obtain

h

J∑
j=1

∫ T

0

jψ′′j

(ψj+1 − ψj−1
2

)
dt

− h
J∑
j=1

∫ T

0

j
ψj+1 − 2ψj + ψj−1

h2

(ψj+1 − ψj−1
2

)
dt

− αh
J∑
j=1

∫ T

0

jψj

(ψj+1 − ψj−1
2

)
dt = 0.

(3.27)

It is immediate that

h

J∑
j=1

∫ T

0

jψ′′j

(ψj+1 − ψj−1
2

)
dt

= h

J∑
j=1

jψ′j

(ψj+1 − ψj−1
2

)∣∣T
0
− h

J∑
j=1

∫ T

0

jψ′j

(ψ′j+1 − ψ′j−1
2

)
dt

= χh(t)
∣∣T
0

+
h

2

J∑
j=0

∫ T

0

ψ′jψ
′
j+1dt

= χh(t)
∣∣T
0
− h3

4

J∑
j=0

∫ T

0

(ψ′j+1 − ψ′j
h

)2
dt+

h

2

J∑
j=0

∫ T

0

∣∣ψ′j∣∣2dt.

(3.28)

On the other hand, it is not difficult to see that

h

J∑
j=1

∫ T

0

j
ψj+1 − 2ψj + ψj−1

h2
ψj+1 − ψj−1

2
dt

=
h

2h2

J∑
j=0

∫ T

0

j|ψj+1|2dt−
h

2h2

J∑
j=0

∫ T

0

j|ψj |2dt−
h

2h2

J∑
j=0

∫ T

0

|ψj |2dt

+
(J + 1)h

2h2

∫ T

0

|ψJ |2dt+
h

h2

J∑
j=0

∫ T

0

ψjψj+1dt .

Keeping in mind the Dirichlet boundary conditions, we obtain

− h
J∑
j=1

∫ T

0

j
ψj+1 − 2ψj + ψj−1

h2
ψj+1 − ψj−1

2
dt

=
h

2

J∑
j=0

∫ T

0

∣∣ψj+1 − ψj
h

∣∣2dt− L

2

∫ T

0

∣∣ψJ
h

∣∣2dt. (3.29)

Moreover,

αh

J∑
j=1

∫ T

0

jψj

(ψj+1 − ψj−1
2

)
dt = −α

2
h

J∑
j=0

∫ T

0

ψj+1ψjdt. (3.30)



16 D. S. ALMEIDA JR., A. J. A. RAMOS, J. C. P. FORTES, M. L. SANTOS EJDE-2020/127

Substituting (3.28), (3.29) and (3.30) into (3.27), we arrive at

h

2

J∑
j=0

∫ T

0

[∣∣ψ′j∣∣2 +
∣∣ψj+1 − ψj

h

∣∣2 − α|ψj |2]dt
+
α

2
h

J∑
j=0

∫ T

0

[|ψj |2 + ψjψj+1]dt+ χh(t)
∣∣T
0

=
h3

4

J∑
j=0

∫ T

0

(ψ′j+1 − ψ′j
h

)2
dt+

L

2

∫ T

0

∣∣ψJ
h

∣∣2dt.
To finish the proof, we observe that

α

2
h

J∑
j=0

∫ T

0

[|ψj |2 + ψjψj+1]dt =
α

4
h

J∑
j=0

∫ T

0

(
ψj+1 + ψj

)2
dt,

from where we obtain the required result

TGh(0) + χh(t)
∣∣T
0

=
h3

4

J∑
j=0

∫ T

0

(ψ′j+1 − ψ′j
h

)2
dt+

L

2

∫ T

0

∣∣ψJ
h

∣∣2dt.
�

Lemma 3.4. The energy Gh(t) of the system (3.10)–(3.12) preserves the positivity
since that α ≤ π2/L2, i.e.,

Gh(0) ≥ π2 − αL2

2π2
h

J∑
j=0

∣∣ψj+1 − ψj
h

∣∣2 ≥ 0. (3.31)

Proof. The proof is immediate. Indeed, from the energy defined in (3.14) and using
the discrete Poincaré’s inequality (3.24), we have

Gh(t) ≥ h

2

J∑
j=0

[
|ψ′j |2 +

∣∣ψj+1 − ψj
h

∣∣2 − αL2

π2

∣∣ψj+1 − ψj
h

∣∣2]

≥ h

2

J∑
j=0

[
|ψ′j |2 +

(
1− αL

2

π2

)
h

J∑
j=0

∣∣ψj+1 − ψj
h

∣∣2],
from where the result follows for α ≤ π2/L2. �

Lemma 3.5. For any h > 0, 0 ≤ t ≤ T , α < π2/L2 and ψ solution of (3.10)–(3.12)
we have

|χh(t)| ≤ L π2

π2 − αL2
Gh(0), (3.32)

where χh(t) is given in (3.26).

Proof. Based on [9, Lemma 2.5], we have

|χh(t)| ≤ L
[ J∑
j=0

|ψ′j |2
]1/2[

h

J∑
j=0

∣∣ψj+1 − ψj
h

∣∣2]1/2.



EJDE-2020/127 SEMI-DISCRETE COUPLED WAVE EQUATIONS 17

Now we use Young’s inequality and the energy defined in (3.14) to obtain

|χh(t)| ≤ L
[h

2

J∑
j=0

|ψ′j |2 +
h

2

J∑
j=0

∣∣ψj+1 − ψj
h

∣∣2] ≤ L[Gh(t) + α
h

2

J∑
j=0

|ψj |2
]
,

and using inequality (3.24) we obtain

|χh(t)| ≤ L
[
Gh(t) + α

h

2

(L
π

)2 J∑
j=0

∣∣ψj+1 − ψj
h

∣∣2].
To complete the proof, we use the inequality (3.31) from Lemma (3.4) to obtain

|χh(t)| ≤ L
[
Gh(t) +

α

2

(L
π

)2 2π2

π2 − αL2
Gh(0)

]
,

from where we conclude the proof by taking the energy conservation law for energy
Gh(t). �

Now we prove the uniform observability inequality concerning to system (3.10)–
(3.12).

Theorem 3.6. Let ψ be the solution of the system (3.10)–(3.12). Then for all
α ≤ π2/L2 there exists T0 > 0 such that for each T > T0 there exists C(T, α) > 0
which is independent of h such that, for all h > 0,

Gh(0) ≤ C(T, α)
[h3

4

J∑
j=0

∫ T

0

(ψ′j+1 − ψ′j
h

)2
dt+

L

2

∫ T

0

∣∣ψJ
h

∣∣2dt], (3.33)

where Gh(·) is the energy in (3.14) and C(T, α) = π2−αL2

T (π2−αL2)−2Lπ2 .

Proof. From Lemmas 3.3 and 3.5, we have

TGh(0)− 2L
π2

π2 − αL2
Gh(0) ≤ h3

4

J∑
j=0

∫ T

0

(ψ′j+1 − ψ′j
h

)2
dt+

L

2

∫ T

0

∣∣ψJ
h

∣∣2dt,
(3.34)

and for T > T0 where the time T0 is the same as in Theorem 2.7, we obtain

Gh(0) ≤ C(T, α)
[h3

4

J∑
j=0

∫ T

0

(ψ′j+1 − ψ′j
h

)2
dt+

L

2

∫ T

0

∣∣ψJ
h

∣∣2dt], (3.35)

where C(T, α) = π2−αL2

T (π2−αL2)−2Lπ2 . �

In a similar way, any solution φ of the system (3.7)–(3.9) obeys a uniform ob-
servability inequality. Indeed, we have the following Theorem.

Theorem 3.7. Let φ be the solution of the system (3.7)–(3.9). Then for all T > 2L
there exists C(T ) > 0 which is independent of h such that, for all h > 0,

Fh(0) ≤ C(T )
[h3

4

J∑
j=0

∫ T

0

(φ′j+1 − φ′j
h

)2
dt+ αh

J∑
j=0

∫ T

0

∣∣φj∣∣2dt+
L

2

∫ T

0

∣∣φJ
h

∣∣2dt],
where Fh(·) is the energy in (3.13) and C(T ) = 1/(T − 2L).



18 D. S. ALMEIDA JR., A. J. A. RAMOS, J. C. P. FORTES, M. L. SANTOS EJDE-2020/127

The insertion, at semi-discrete level, of the extra terms

h3

4

J∑
j=0

∫ T

0

(ψ′j+1 − ψ′j
h

)2
dt and

h3

4

J∑
j=0

∫ T

0

(φ′j+1 − φ′j
h

)2
dt,

is a suitable one to reestablish the uniform observability inequality. For example,
in light of the arguments discussed in [18], it can be shown that this extra term has
order of h2ν−j Gh(0) (see ν−j from Proposition 3.9) and it is important to overcome

the high-frequency when ν−j = O(h−2). Moreover, the terms involving observation
on the boundary and observation distributed on the whole space domain are nec-
essary for estimate (3.33). See [18] for details of several remedies to overcome the
lack of uniform observability.

Now we are in a position to prove the uniform observability inequality to the
coupled system (3.2)–(3.5).

Theorem 3.8. Let (u, v) be the solution of system (3.2)–(3.5). Then for all α ≤
π2/L2 there exists T0 > 0 such that for all T > T0 there exists C(T, α) > 0 such
that, for all h > 0,

Eh(0) ≤ C(T, α)
[h3

4

J∑
j=0

∫ T

0

(u′j+1 − u′j
h

)2
dt+

h3

4

J∑
j=0

∫ T

0

(v′j+1 − v′j
h

)2
dt

+ αh

J∑
j=0

∫ T

0

(
u2j + v2j

)
dt+

L

2

∫ T

0

(∣∣uJ
h

∣∣2 +
∣∣vJ
h

∣∣2)dt]. (3.36)

Proof. From uniform observability inequalities built in Theorems 3.6 and 3.7 and
taking into account the decompositions of the solutions φj and ψj , we have

Fh(0) +Gh(0)

≤ C(T )
[h3

4

J∑
j=0

∫ T

0

(φ′j+1 − φ′j
h

)2
dt+ αh

J∑
j=0

∫ T

0

∣∣φj∣∣2dt+
L

2

∫ T

0

∣∣φJ
h

∣∣2dt]

+ C(T, α)
[h3

4

J∑
j=0

∫ T

0

(ψ′j+1 − ψ′j
h

)2
dt+

L

2

∫ T

0

∣∣ψJ
h

∣∣2dt]

≤ C(T, α)
[h3

2

J∑
j=0

∫ T

0

(u′j+1 − u′j
h

)2
dt+

h3

2

J∑
j=0

∫ T

0

(v′j+1 − v′j
h

)2
dt

+ 2αh

J∑
j=0

∫ T

0

(
u2j + v2j

)
dt+ L

∫ T

0

∣∣uJ
h

∣∣2dt+ L

∫ T

0

∣∣vJ
h

∣∣2dt],
and therefore we conclude the proof by taking E(0) = (Fh(0) +Gh(0))/2. �

3.2. Spectral analysis. In this section, we study a uniform semi-discrete version
to the boundary observability (2.38). We solved the semi-discrete versions to the
Propositions 2.1, 2.2 and 2.3.

Proposition 3.9. Assume that α ≤ π2/L2. Then, the Fourier series expansion of
system (3.10)–(3.12) is

Ψh(t) =

J∑
k=1

[
ak sin

(√
ν−k (h) t

)
+ bk cos

(√
ν−k (h) t

)]
ϕk,



EJDE-2020/127 SEMI-DISCRETE COUPLED WAVE EQUATIONS 19

where ak, bk are the Fourier coefficients, ν−k (h) = 4
h2 sin2

(
kπh
2L

)
− α are the eigen-

values, and ϕk = (ϕk,1, . . . , ϕk,J) are the eigenvectors ϕk,j = sin
(kπxj

L

)
, j, k =

1, . . . , J .

Proof. Making ψj(t) := ϕjT (t) and substituting into (3.10) we obtain

T ′′(t)

T (t)
=
[ϕj+1 − 2ϕj + ϕj−1

h2
+ αϕj

] 1

ϕj
= ν, (3.37)

for all t ≥ 0 and j = 0, 1, 2, . . . , J, J + 1. Then, we have the eigenvalue problem

ϕj+1 − 2ϕj + ϕj−1
h2

− (ν − α)ϕj = 0, j = 1, . . . , J (3.38)

ϕ0 = ϕJ+1 = 0. (3.39)

Proceeding as in Proposition 2.1, we arrive at

ν−k (h) =
4

h2
sin2

(kπh
2L

)
− α, ∀k = 1, . . . , J, (3.40)

for α ≤ 4
h2 sin2

(
kπh
2L

)
, i.e., α ≤ π2/L2 for h sufficiently small. Now, from identity

(3.37) we obtain the equation T ′′(t) − νT (t) = 0 for all t > 0 and solving this
equation we obtain

Tk(t) = ak sin
(√

ν−k (h) t
)

+ bk cos
(√

ν−k (h) t
)
, k = 1, . . . , J, ∀t > 0, (3.41)

where ak, bk are the Fourier coefficients. The proof is complete. �

In a similar way, the following holds.

Proposition 3.10. The Fourier series expansion of system (3.7)–(3.9) is

Φh(t) =

J∑
k=1

[
ck sin

(√
ν+k (h) t

)
+ dk cos

(√
ν+k (h) t

)]
ϕk

where ck, dk are the Fourier coefficients, ν+k (h) = 4
h2 sin2

(
kπh
2L

)
+ α are the eigen-

values, and ϕk = (ϕk,1, . . . , ϕk,J) are the eigenvectors, where ϕk,j = sin
(kπxj

L

)
for

j, k = 1, . . . , J .

The next result follows from Propositions (3.9) and (3.10).

Proposition 3.11. Assume that α ≤ π2/L2. Then the Fourier series expansions
of the system (3.2)–(3.5) are

Uh(t) =
1

2

J∑
k=1

[
ak sin(

√
ν−k (h) t) + bk cos(

√
ν−k (h) t) + ck sin(

√
ν+k (h) t)

+ dk cos(
√
ν+k (h) t)

]
ϕk,

(3.42)

Vh(t) =
1

2

J∑
k=1

[
ck sin(

√
ν+k (h) t) + dk cos(

√
ν+k (h) t)− ak sin(

√
ν−k (h) t)

− bk cos(
√
ν−k (h) t)

]
ϕk,

(3.43)

where ak, bk, ck, dk are the Fourier coefficients, ν±k (h) = 4
h2 sin2

(
kπh
2L

)
± α are the

eigenvalues, and ϕk = (ϕk,1, . . . , ϕk,J) are the eigenvectors associated where each

component ϕk,j is ϕk,j = sin
(kπxj

L

)
, j, k = 1, . . . , J .



20 D. S. ALMEIDA JR., A. J. A. RAMOS, J. C. P. FORTES, M. L. SANTOS EJDE-2020/127

The next two lemmas play an important role for getting the uniform boundary
observability of system (3.2)–(3.5). These lemmas concern the boundary observ-
ability of the spectral problem associated to the uncoupled systems (3.10)-(3.12)
and (3.7)–(3.9). The proofs are trivial and we omit them.

Lemma 3.12. For any eigenvector ϕ = (ϕ1, . . . , ϕJ) of (3.10)-(3.12) we have

h

J∑
j=0

∣∣ϕj+1 − ϕj
h

∣∣2 =
2L

4− (ν−k (h) + α)h2

∣∣ϕJ
h

∣∣2, (3.44)

where ν−k (h) = 4
h2 sin2

(
kπh
2L

)
− α for k = 1, . . . , J .

Lemma 3.13. For each eigenvector ϕ = (ϕ1, . . . , ϕJ) of (3.7)–(3.9) we have

h
J∑
j=0

∣∣ϕj+1 − ϕj
h

∣∣2 =
2L

4− (ν+k (h)− α)h2

∣∣ϕJ
h

∣∣2, (3.45)

where ν+k (h) = 4
h2 sin2

(
kπh
2L

)
+ α for k = 1, . . . , J .

Equalities (3.44) and (3.45) provide an explicit relation between the total energy
of the eigenvectors and the energy concentrated on x = L according to the quantity
measured by |ϕJ/h|2. In both cases, the observability inequalities are written as

h

J∑
j=0

∣∣ϕj+1 − ϕj
h

∣∣2 =
2L

4− 4 sin2
(
kπh
2L

)∣∣ϕJ
h

∣∣2, (3.46)

and

4 sin2
(Jπh

2L

)
= 4 cos2

(πh
2L

)
→ 4, as h→ 0,

from where the blow-up happens on the right hand side of (3.44) and (3.45). There-
fore, the lack of numerical boundary observability follows in these cases.

4. Uniform observability: non-harmonic Fourier series

In this section, we use the Ingham’s theorem [10] to get a uniform boundary
observability (2.38).

Theorem 4.1 (Ingham’s Inequality). Let {µk}k∈I be a sequence of real numbers
such that

µk+1 − µk ≥ γ > 0, ∀k ∈ I. (4.1)

Then, for any T > 2π
γ there exist positive constants Ci(T, γ) > 0, i = 1, 2 such that

C1(T, γ)
∑
k∈I

|ak|2 ≤
∫ T

0

∣∣∣∑
k∈I

ake
iµkt
∣∣∣2dt ≤ C2(T, γ)

∑
k∈I

|ak|2, (4.2)

for all sequences of complex numbers {ak} ∈ l2.

To apply Ingham’s inequality, we need an estimate between the roots of consecu-
tive eigenvalues entering in the Fourier expansions to the solutions of the decoupled
systems (3.7)–(3.9) and (3.10)-(3.12) into subspaces of filtered solutions. We have
in mind that, these numerical solutions, are the filtered solutions generate by spec-
tral problem such that λh2 ≤ γ. In that direction and taking into account the



EJDE-2020/127 SEMI-DISCRETE COUPLED WAVE EQUATIONS 21

Fourier series expansions, given any 0 < γ < 4 we introduce the following class of
filtered solutions to the system (3.10)-(3.12),

Ph(γ) :=
{
ψ =

∑
λk(h)≤γh−2

[
ak sin

(√
ν−k (h) t

)
+ bk cos

(√
ν−k (h) t

)]
ϕk
}
, (4.3)

where λk(h) := 4
h2 sin2

(
kπh
2L

)
and ak, bk ∈ R. Moreover, we introduce the class of

filtered solutions to (3.7)–(3.9) given by

Qh(γ) :=
{
φ =

∑
λk(h)≤γh−2

[
ck sin

(√
ν+k (h) t

)
+ dk cos

(√
ν+k (h) t

)]
ϕk
}
, (4.4)

where ck, dk ∈ R. In that direction, we consider the gap between the consecutive
eigenvalues ν+j (h) and ν−j (h).

Lemma 4.2. Assume that

iγ = 4 sin2
(πε

2

)
and α <

γ(ε)

h2
, (4.5)

for some 0 ≤ ε < 1. Then,√
ν+j+1(h)−

√
ν+j (h) ≥ π

L
cos
(πε

2

)
> 0, (4.6)

for all eigenvalues ν+j (h) = λj(h) + α such λh2 ≤ γ where λj(h) = 4
h2 sin2

(
jπh
2L

)
,

j = 1, . . . , J .

Proof. Note that√
λj(h) + α =

√
λj(h)

√
1 +

α

λj(h)
≈
√
λj(h)

(
1 +

α

2λj(h)

)
, (4.7)

for j = 1, . . . , J ; from where we obtain√
λj+1(h) + α−

√
λj(h) + α ≈

(√
λj+1(h)−

√
λj(h)

)(
1− α

2
√
λj+1(h)λj(h)

)
,

for j = 1, . . . , J . On the other hand, using λj+1(h) > λj(h), j = 1, . . . , J , we obtain√
λj+1(h) + α−

√
λj(h) + α ≥

(√
λj+1(h)−

√
λj(h)

)(
1− α

2
√
λj(h)λj(h)

)
≥
(√

λj+1(h)−
√
λj(h)

)(
1− α

2

h2

4 sin2(jπh/2L)

)
≥
(√

λj+1(h)−
√
λj(h)

)(
1− α h2

4 sin2(πε/2)

)
≥
(√

λj+1(h)−
√
λj(h)

)(
1− αh2

γ(ε)

)
,

for γ = 4 sin2
(
πε
2

)
. Now, by taking α = γ(ε)/2h2 ≤ γ(ε)/h2 we arrive at√

λj+1(h) + α−
√
λj(h) + α ≥ 1

2

(√
λj+1(h)−

√
λj(h)

)
. (4.8)

The estimate for the right hand side of (4.8) can be found in [1, 9] which completes
the proof of this lemma. �



22 D. S. ALMEIDA JR., A. J. A. RAMOS, J. C. P. FORTES, M. L. SANTOS EJDE-2020/127

Lemma 4.3. Assume that α ≤ π2/L2 and γ = 4 sin2
(
πε
2

)
for some 0 ≤ ε < 1.

Then, √
ν−j+1(h)−

√
ν−j (h) ≥ π

L
cos
(πε

2

)
> 0, (4.9)

for all eigenvalues ν−j (h) = λj(h) − α such λh2 ≤ γ where λj(h) = 4
h2 sin2

(
jπh
2L

)
,

j = 1, . . . , J .

Proof. The proof is immediate. Indeed, for α ≤ π2/L2, one has√
λj(h)− α =

√
λj(h)

√
1− α

λj(h)
≈
√
λj(h)

(
1− α

2λj(h)

)
, j = 1, . . . , J, (4.10)

and then√
λj+1(h)− α−

√
λj(h)− α ≈

(√
λj+1(h)−

√
λj(h)

)(
1 +

α

2
√
λj+1(h)λj(h)

)
,

≥
(√

λj+1(h)−
√
λj(h)

)
, ∀j = 1, . . . , J ;

The remainder of the proof can be found in [1, 9]. �

4.1. Proof of the uniform boundary observability to uncoupled systems.
In this section, we obtain the proof of two theorems concerning the uniform bound-
ary observability to uncoupled systems (3.7)–(3.9) and (3.10)-(3.12).

Theorem 4.4. For any 0 < γ < 4 there exists T (γ) > 0 such that for any T > T (γ)
there exists a positive constant C(T, α, γ(ε)) such that

Fh(0) ≤ C(T, α, γ(ε))
L

2

∫ T

0

∣∣φJ
h

∣∣2dt, (4.11)

for any solution of system (3.7)–(3.9) in the class Qh(γ(ε)) in (4.4).

Proof. According to Ingham’s inequality and in view of Lemma 4.2, for any 0 ≤ ε <
1 and T > 2L

cos(πε/2) there exist positive constants Ci(T, ε) > 0, i = 1, 2 satisfying

C1(T, ε)
∑

|µk|h≤
√
γ(ε)

|ak|2 ≤
∫ T

0

∣∣∣ ∑
|µk|h≤

√
γ(ε)

ake
iµkt
∣∣∣2dt

≤ C2(T, ε)
∑

|µk|h≤
√
γ(ε)

|ak|2,
(4.12)

where γ(ε) = 4 sin2
(
πε
2

)
. By Lemma 3.13, we have

h

J∑
j=0

∣∣ϕj+1 − ϕj
h

∣∣2 =
2L

4− (ν+(h)− α)h2
∣∣ϕJ
h

∣∣2
=

2L

4− λ(h)h2
∣∣ϕJ
h

∣∣2
≤ L

2 cos2
(
πε/2

) ∣∣ϕJ
h

∣∣2,
(4.13)

for any eigenvector ϕ associated with an eigenvalue λ satisfying λh2 ≤ γ(ε).



EJDE-2020/127 SEMI-DISCRETE COUPLED WAVE EQUATIONS 23

Now, let us consider the solution φ of system (3.7)–(3.9) into the class Qh(γ(ε)),
which is written as

φ =
∑

|µk(h)|h≤
√
γ(ε)

ake
iµk(h)tϕk, (4.14)

where µk(h) =
√
ν+j (h). Then, we can deduce that for T > 2L

cos(πε/2) it holds

L

2

∫ T

0

∣∣φJ
h

∣∣2dt =
L

2h2

∫ T

0

∣∣∣ ∑
|µk|h≤

√
γ(ε)

ake
iµk(h)tϕk,J

∣∣∣2dt
≥ C1(T, ε) cos2

(
πε/2

) ∑
|µk|h≤

√
γ(ε)

|ak|2h
J∑
j=0

∣∣ϕk,j+1 − ϕk,j
h

∣∣2.
(4.15)

Moreover, it is not difficult to see that∑
|µk|h≤

√
γ(ε)

[
|ak|2h

J∑
j=0

∣∣ϕk,j+1 − ϕk,j
h

∣∣2] =
ν+J (h)− α
ν+J (h)

Fh(0) =
λJ(h)

λJ(h) + α
Fh(0).

Therefore, substituting the above expression into (4.15), for T > 2L
cos(πε/2) and

φ ∈ Qh(γ(ε)) we have

Fh(0) ≤ λJ(h) + α

λJ(h)

1

cos2
(
πε/2

)
C1(T, ε)

L

2

∫ T

0

∣∣φJ
h

∣∣2dt, (4.16)

and we conclude the proof by taking

C(T, α, γ(ε)) =
λJ(h) + α

λJ(h)

1

C1(T, ε) cos2
(
πε/2

) .
�

We observe that for T > 2L
cos(πε/2) , since γ = γ(ε), Theorem 4.4 holds with

T (γ) =
2L√

1− γ/4
, C(T, α, γ(ε)) =

λJ(h) + α

λJ(h)

1

C1(T, ε)(1− γ/4)
. (4.17)

The time observability is such that: T (γ) ↗ ∞ as γ ↗ 4 and T (γ) ↘ 2L as
γ ↘ 0. Clearly, the Ingham’s type approach improves the estimates on observability
inequalities (see Theorem 3.7). Indeed, the uniform observability inequality from
Theorem 3.7 holds for T > 2L and it is given by

Fh(0) ≤ 1

T − 2L

[h3
4

J∑
j=0

∫ T

0

(φ′j+1 − φ′j
h

)2
dt+αh

J∑
j=0

∫ T

0

∣∣φj∣∣2dt+L

2

∫ T

0

∣∣φJ
h

∣∣2dt],
which represents the discrete counterpart (uniformly) of

F (0) ≤ 1

T − 2L

[
α

∫ T

0

∫ L

0

φ2 dx dt+
L

2

∫ T

0

φ2x(L, t) dt
]
. (4.18)

Here, in contrast to the above result, we obtain a uniform observability inequality
given by

Fh(0) ≤ C(T, α, γ(ε))
L

2

∫ T

0

∣∣φJ
h

∣∣2dt, (4.19)



24 D. S. ALMEIDA JR., A. J. A. RAMOS, J. C. P. FORTES, M. L. SANTOS EJDE-2020/127

which represents the discrete counterpart (uniformly) of the boundary observability
inequality given by

F (0) ≤ C(T, α)
L

2

∫ T

0

φ2x(L, t) dt. (4.20)

In that direction, the above constant C(T, α) is the limit of C(T, α, γ(ε)) when
h→ 0 and γ → 0. Indeed, when h and γ approaches to zero, one has

C(T, α, γ(ε)) =
(

1 +
α

λJ(h)

) 1

C1(T, ε)(1− γ/4)

→ C(T, α) =
[
1 + α

( L
Jπ

)2] 1

C1(T, ε)
> 0,

where λJ(h) = 4
h2 sin2

(
Jπh
2L

)
→
(
Jπ
L

)2
when h approaches to zero. Analogously

to the previous case, we have the following theorem:

Theorem 4.5. Assume that 0 < γ < 4. Then, for each α < λJ(h) there exists
T (γ) > 0 such that for any T > T (γ) there exists a positive constant C(T, α, γ(ε)) >
0 such that

Gh(0) ≤ C(T, α, γ(ε))
L

2

∫ T

0

∣∣ψJ
h

∣∣2dt, (4.21)

for any solution of system (3.10)-(3.12) in the class Ph(γ(ε)) in (4.3).

Proof. According to Ingham’s inequality and in view of Lemma 4.3, for any 0 ≤
ε < 1 and T > 2L/ cos(πε/2) there exist positive constants Di(T, ε) > 0, i = 1, 2,
satisfying

D1(T, ε)
∑

|µk|h≤
√
γ(ε)

|ak|2 ≤
∫ T

0

∣∣∣ ∑
|µk|h≤

√
γ(ε)

ake
iµkt
∣∣∣2dt

≤ D2(T, ε)
∑

|µk|h≤
√
γ(ε)

|ak|2,
(4.22)

where γ(ε) = 4 sin2
(
πε
2

)
. Now, from Lemma 3.12, we have

h

J∑
j=0

∣∣ϕj+1 − ϕj
h

∣∣2 =
2L

4− (ν−(h) + α)h2
∣∣ϕJ
h

∣∣2 =
2L

4− λ(h)h2
∣∣ϕJ
h

∣∣2
≤ L

2 cos2
(
πε/2

) ∣∣ϕJ
h

∣∣2, (4.23)

for any eigenvector ϕ associated with an eigenvalue λ satisfying λh2 ≤ γ(ε). Let
us consider the solution ψ of the system (3.10)-(3.12) in the class Ph(γ(ε)), which
is written as

ψ =
∑

|µk(h)|h≤
√
γ(ε)

ake
iµk(h)tϕk, (4.24)

where we are denoting µk(h) =
√
ν−j (h). Then, combining (4.23) and (4.24) and

taking T > 2L
cos(πε/2) , we obtain

L

2

∫ T

0

∣∣ψJ
h

∣∣2dt =
L

2h2

∫ T

0

∣∣∣ ∑
|µk|h≤

√
γ(ε)

ake
iµk(h)tϕk,J

∣∣∣2dt



EJDE-2020/127 SEMI-DISCRETE COUPLED WAVE EQUATIONS 25

≥ D1(T, ε) cos2
(
πε/2

) ∑
|µk|h≤

√
γ(ε)

|ak|2h
J∑
j=0

∣∣ϕk,j+1 − ϕk,j
h

∣∣2
= D1(T, ε) cos2

(
πε/2

) λJ(h)

λJ(h)− α
Gh(0),

and for any ψ ∈ Ph(γ(ε)), we have

Gh(0) ≤ C(T, γ)
L

2

∫ T

0

∣∣ψJ
h

∣∣2dt, (4.25)

where

C(T, α, γ(ε)) =
λJ(h)− α
λJ(h)

1

D1(T, ε) cos2(πε/2)
.

This observability constant is positive for α < λJ(h) = 4
h2 sin2

(
Jπh
2L

)
. Therefore,

the proof is complete. �

Observe that for T > 2L
cos(πε/2) and since γ = γ(ε) the previous theorem holds

since

T (γ) =
2L√

1− γ/4
, C(T, α, γ(ε)) =

λJ(h)− α
λJ(h)

1

D1(T, ε) cos2
(
πε/2

) . (4.26)

It is clear that T (γ)↗∞ as γ ↗ 4 and T (γ)↘ 2L as γ ↘ 0. The improvement in
comparison with Theorem 3.6 is basically on uniform observability inequality and
also on time observability. In Theorem 3.6, the uniform observability inequality

Gh(0) ≤ C(T, α)
[h3

4

J∑
j=0

∫ T

0

(ψ′j+1 − ψ′j
h

)2
dt+

L

2

∫ T

0

∣∣ψJ
h

∣∣2dt],
holds for T > T0 = 2L π2

π2−αL2 and for

C(T, α) =
π2 − αL2

T (π2 − αL2)− 2Lπ2
. (4.27)

Moreover, the uniform boundary observability

Gh(0) ≤ C(T, α, γ(ε))
L

2

∫ T

0

∣∣ψJ
h

∣∣2dt, (4.28)

represents the discrete counterpart (uniformly) of

G(0) ≤ C(T, α)
L

2

∫ T

0

ψ2
x(L, t) dt, (4.29)

and the constant C(T, α) is the limit of C(T, α, γ(ε)) when h → 0 and γ → 0.
Indeed, when h and γ approach to zero, one has

C(T, α, γ(ε)) =
λJ(h)− α
λJ(h)

1

D1(T, ε) cos2
(
πε/2

)
→ C(T ) =

[
1− α

( L
Jπ

)2] 1

D1(T, ε)
> 0,

where λJ(h) = 4
h2 sin2

(
Jπh
2L

)
→
(
Jπ
L

)2
as h→ 0.



26 D. S. ALMEIDA JR., A. J. A. RAMOS, J. C. P. FORTES, M. L. SANTOS EJDE-2020/127

4.2. Proof of the uniform boundary observability for coupled systems.
In this section, we use the Theorems 4.4 and 4.5 to show a uniform boundary
observability to the system (3.2)-(3.5). In order to do this we consider the class of
filtered solutions Rh(γ) which the solutions are given by

Uh =
1

2

∑
λk(h)≤γh−2

[
ak sin(

√
ν−k (h) t) + bk cos(

√
ν−k (h) t)

+ ck sin(
√
ν+k (h) t) + dk cos(

√
ν+k (h) t)

]
ϕk,

Vh =
1

2

∑
λk(h)≤γh−2

[
ck sin(

√
ν+k (h) t) + dk cos(

√
ν+k (h) t)

− ak sin(
√
ν−k (h) t)− bk cos(

√
ν−k (h) t)

]
ϕk,

where ak, bk, ck, dk ∈ R.

Theorem 4.6. Assume that 0 < γ < 4. Then, for each α < λJ(h) there exists

T (γ) > 0 such that for any T > T (γ) there exists a positive constant C̃(T, α, γ(ε)) >
0, such that

Eh(0) ≤ C̃(T, α, γ(ε))
L

2

[ ∫ T

0

∣∣uJ
h

∣∣2dt+

∫ T

0

∣∣vJ
h

∣∣2dt], (4.30)

for any solution of system (3.2)-(3.5) in the class Rh(γ(ε)).

Proof. The proof follows by a combination of the results obtained in Theorems 4.5
and 4.4 and from decomposition of the energy Eh(t). Indeed, we have

2Eh(0) = Fh(0) +Gh(0)

≤ max
{
C(T, α, γ(ε)), C(T, α, γ(ε))

}L
2

∫ T

0

[∣∣φJ
h

∣∣2 +
∣∣ψJ
h

∣∣2]dt,
and taking into account the decompositions φj = uj + vj and ψj = uj − vj , we
arrive at

Eh(0) ≤ C̃(T, α, γ(ε))
L

2

[ ∫ T

0

∣∣uJ
h

∣∣2dt+

∫ T

0

∣∣vJ
h

∣∣2dt], (4.31)

where

C̃(T, α, γ(ε)) = 2 max
[ 1

cos2
(
πε/2

)
D(T, ε)

,
λJ(h) + 2α

λJ(h)

1

cos2
(
πε/2

)
C1(T, ε)

]
,

(4.32)
and the conclusions are similar to the previous cases. �

Acknowledgments. D. S. Almeida Júnior wants to thank CAPES/INCTMAT/
LNCC for their financial support, Grant 88887.351763/2019-00. A. J. A. Ramos
wants to thank CNPq for financial support through the projects Asymptotic stabi-
lization and numerical treatment for carbon nanotubes (CNPq Grant 310729/2019-
0). M. L. Santos wants to thank CNPq for financial support through the projects:
CNPq Grant 303026/2018-9 and CNPq Grant Pós-doc Sênior /PDS114563/ 2018-7.

The authors are grateful to the referees for their constructive remarks, that
enhanced the presentation of this article.



EJDE-2020/127 SEMI-DISCRETE COUPLED WAVE EQUATIONS 27

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Dilberto da Silva Almeida Júnior
PhD Program in Mathematics, Federal University of Pará, Augusto Corrêa Street, 01,
66075-110, Belém - Pará - Brazil

Email address: dilberto@ufpa.br, dilxingu@gmail.com

Anderson de Jesus Araújo Ramos
Faculty of Sciences, Federal University of Pará, Raimundo Santana Street, s/n, 68721-

000, Salinópolis - Pará - Brazil
Email address: ramos@ufpa.br



28 D. S. ALMEIDA JR., A. J. A. RAMOS, J. C. P. FORTES, M. L. SANTOS EJDE-2020/127

João Carlos Pantoja Fortes

PhD Program in Mathematics, Federal University of Pará, Augusto Corrêa Street, 01,

66075-110, Belém - Pará - Brazil
Email address: joaocarlos.fortes@yahoo.com.br

Mauro de Lima Santos
PhD Program in Mathematics, Federal University of Pará, Augusto Corrêa Street, 01,

66075-110, Belém - Pará - Brazil

Email address: ls@ufpa.br


	1. Introduction
	2. Energy properties at continuous level
	2.1. Fourier expansion of the solutions
	2.2. Observability inequality for the coupled system
	2.3. Observability inequalities for the uncoupled systems

	3. Finite difference semi-discretization
	3.1. Uniform observability inequality: no filtering
	3.2. Spectral analysis

	4. Uniform observability: non-harmonic Fourier series
	4.1. Proof of the uniform boundary observability to uncoupled systems
	4.2. Proof of the uniform boundary observability for coupled systems
	Acknowledgments

	References