2 The Fourier series expansion of the solution

Partial observability of a wave-Petrovsky system with memory

Paola Loreti and Daniela Sforza

Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Università di Roma, via A. Scarpa n. 16, Roma, I–00161, Italy

Received 15 May 2015, appeared 8 December 2015 Communicated by Vilmos Komornik

Abstract. Our goal is to show partial observability results for coupled systems with memory terms. To this end, by means of non-harmonic analysis techniques we prove Theorem3.2and Theorem3.7below.

Keywords: coupled systems, convolution kernels, Fourier series, Ingham estimates.

2010 Mathematics Subject Classification: 42A38, 45K05, 93C20.

1 Introduction

In this paper we consider a coupled system obtained by combining a wave equation with an integral relaxation term and a Petrovsky type equation, that is



















u_1tt−u_1xx+β Z _t

0 e^−η(t−s)u_1xx(s,x)ds+Au2=0 in (0,∞)×(0,π), u2tt+u2xxxx+Bu₁ =0 in (0,∞)×(0,π),

u₁(·, 0) =u₁(·,π) =u₂(·, 0) =u₂(·,π) =u_2xx(·, 0) =u_2xx(·,π) =_{0 in}(_0,∞)_, u₁(0,·) =u₁₀, u_1t(0,·) =u₁₁, u₂(0,·) =u₂₀, u_2t(0,·) =u₂₁ in(0,π),

(1.1)

where 0< β<ηandA, Bare real constants.

In [14] we proved that the observation of the solution at a point of the boundary allows us to recognize the unknown initial data. In the following theorem we recall that result.

Theorem 1.1. Letη>3β/2and T >2π. For any(u₁₀,u₁₁)∈ H₀¹(0,π)×L²(0,π)and(u₂₀,u₂₁)∈ H₀¹(0,π)×H⁻¹(0,π),if(u₁,u₂)is a solution of problem(1.1)we have

Z _T

0

|u_1x(t,π)|²+|u_2x(t,π)|² dt ku₁₀k²

H¹₀(0,π)+ku₁₁k²+ku₂₀k²

H¹₀(0,π)+ku₂₁k²_H−1(0,π). (1.2)

BCorresponding author. Email: daniela.sforza@sbai.uniroma1.it

Throughout the paper, we will use the notation k · k = k · k_L2(0,π). Moreover, we will adopt the convention to write f g if there exist two positive constants c₁ andc2 such that c₁f ≤ g≤c₂f.

Our goal is to establish a partial observability result where we only observeu₁ oru₂at the boundary. Indeed, we will show sufficient conditions guaranteeing the validity of estimates

Z _T

0

|u_1x(t,π)|² dt ku₁₀k²_H1

0(0,π)+ku₁₁k²+ku20k²_H1

0(0,π)+ku₂₁k²_H−1(_0,π), (1.3) or

Z _T

0

|u2x(t,π)|² dt ku₁₀k²_H1

0(0,π)+ku₁₁k²+ku₂₀k²_H1

0(0,π)+ku₂₁k²_H−1(0,π). (1.4) It is evident that the direct inequality in (1.3) and in (1.4) follows from (1.2), and hence the key point is to prove the inverse inequalities. In fact, by writing the solution of system (1.1) as a Fourier series and using typical techniques of non-harmonic analysis, we are able to establish Theorem3.2and Theorem 3.7below. It is noteworthy to observe that in Theorem3.2we have to assume that the initial data u₂₀ and u₂₁ are null, while the same condition on the initial datau₁₀andu₁₁is not required in Theorem3.7.

For references related to integral equations and viscoelasticity theory see e.g [1–3,15,16]. It is worthwhile to mention a partial observability problem for a wave-Petrovsky system (with- out memory) analyzed in [7]. For a classical overview about exact controllability problems see [8–11,17].

2 The Fourier series expansion of the solution

Let T > 0. Fix two real numbers A, B different from zero. For any (u₁₀,u₁₁) ∈ H₀¹(0,π)× L²(0,π)and(u₂₀,u₂₁)∈ H₀¹(0,π)×H⁻¹(0,π)there exists a uniqueweaksolution(u₁,u₂)with u₁ ∈C(_R+;H₀¹(0,π))∩C¹(_R+;L²(0,π))andu₂ ∈C(_R+;H₀¹(0,π))∩C¹(_R+;H⁻¹(0,π))of the following coupled system



















u_1tt−u_1xx+β Z _t

0 e^−η(t−s)u_1xx(s,x)ds+Au₂=0 in (0,∞)×(0,π), u_2tt+u_2xxxx+Bu₁ =0 in (0,∞)×(0,π),

u₁(·, 0) =u₁(·,π) =u₂(·, 0) =u₂(·,π) =u_2xx(·, 0) =u_2xx(·,π) =0 in(0,∞), u₁(0,·) =u₁₀, u_1t(0,·) =u₁₁, u₂(0,·) =u₂₀, u_2t(0,·) =u₂₁ in(0,π).

(2.1)

If we expand the initial data according to the eigenfunctions of the Laplacian sin(nx), n∈N, then we obtain the expressions

u₁₀(x) =

∑

α_1nsin(nx), ku₁₀k²_H1

0(0,π)= ^π 2

∑

α²_1nn², u₁₁(x) =

∑

χ_1nsin(nx), ku₁₁k²= ^π 2

∑

χ²_1n,

(2.2)

u₂₀(x) =

∑

α_2nsin(nx), ku₂₀k²

H₀¹(0,π)= ^π 2

∑

α²_2nn², u₂₁(x) =

∑

χ2nsin(nx), ku₂₁k²_H−1(_0,π)= ^π 2

∑

χ²_2n n² .

(2.3)

By applying the spectral analysis developed in Hilbert spaces, see [14, Section 4], we are able to write the solution (u₁,u2)of problem (2.1) as Fourier series.

Theorem 2.1. For any (u₁₀,u₁₁) ∈ H¹₀(0,π)×L²(0,π)and (u20,u₂₁) ∈ H₀¹(0,π)×H⁻¹(0,π), the weak solution(u₁,u₂)of problem(2.1)is given by

u₁(t,x) =

∑

R_ne^rnt+C_ne^iωnt+C_ne^−iωnt+D_ne^ipnt+D_ne^−ipnt

sin(nx) u₂(t,x) =

∑

d_nD_ne^ipnt+d_nD_ne^−ipnt−^2β

An²< ^Dn η+ipn

e^−ηt

sin(nx)_,

(2.4)

for t ≥0and x∈ (0,π), where r_n,R_n∈_Randω_n,C_n,p_n,D_n,d_n∈_Care defined by r_n =β−η+O

1 n²

, Rn = ^β

n²(α_1n(β−η) +χ_1n) + (α_1n+χ_1n)O 1

n⁴

, ω_n =n+ ^β

2 3

4β−η 1

n+iβ 2 +O

1 n²

, C_n = ^α1n

2 − ⁱ

4n(βα_1n+2χ_1n) + (α_1n+χ_1n)O 1

n²

, p_n =n²+O

1 n⁶

, (2.5)

Dn = ^Aα2n

2n⁴ + (α2n−iχ2n) ^A

2n⁶ + (α2n+χ2n)O 1

n⁷

, (2.6)

d_n = ¹ A

p²_n−n²+ ^βn

2

η+ipn

. Moreover, for any n∈_None has

|dn| |pn|², (2.7)

n²|C_n|² α²_1nn²+χ²_1n, (2.8) n²|p_n|⁴|D_n|² α²_2nn²+^χ

22n

n² . (2.9)

3 Partial observability results

To establish the result concerning the observation of the first component of the solution of problem (2.1), we need an inverse estimate of Ingham type (see [5]), involving only the terms Rne^rnt andCne^iωnt, see (2.4). For the reader’s convenience, we recall a known theorem.

Theorem 3.1([12,13]). Letω_n∈ _Cand r_n∈ _Rbe sequences of pairwise distinct numbers such that r_n6=iω_m for any n, m∈_N. Assume

lim inf

n→∞ (<ω_n+1− <ω_n) =γ>0 ,

nlim→_∞=ω_n=α, r_n≤ −=ω_n ∀ n≥n⁰,

|R_n| ≤ ^µ

n^ν|C_n| ∀n≥n⁰, |R_n| ≤µ|C_n| ∀n≤n⁰,

for some n⁰ ∈_N,α∈_R,µ>0andν>1/2.

Then, for any T> ^2π_γ there exists c(T)>0such that Z _T

0

∑

R_ne^rnt+C_ne^iωnt+C_ne^−iωnt

2dt≥c(T)

∑

(1+e^{−2(=ωn−α)T})|C_n|². (3.1) Now, we are in condition to show our first result.

Theorem 3.2. Letη >3β/2and T>2π. If(u₁,u₂)is a solution of problem(2.1)with(u₁₀,u₁₁)∈ H₀¹(0,π)×L²(0,π)and

u₂₀=u₂₁=0 , (3.2)

then we have

Z _T

0

|u_1x(t,π)|² dt≥c(T) ku₁₀k²

H₀¹(0,π)+ku₁₁k², (3.3) where c(T)is a positive constant.

Proof. If we bear in mind formulas (2.3), from (3.2) it follows α2n=χ2n=0 for any n∈_N, whence, in virtue of (2.6) we get

D_n =0 for any n∈_N. Therefore, from (2.4) it follows

u_1x(t,π) =

∑

(−1)ⁿn R_ne^rnt+C_ne^iωnt+C_ne^−iωnt .

Now, we can employ Theorem3.1 (γ = ν = 1, α = β/2) for dealing with the previous sum.

Indeed, applying formula (3.1) to u_1x(t,π)we obtain Z _T

0

u_1x(t,π)

2dt≥ c(T)

∑

n²|C_n|², whence, in virtue of (2.8) and (2.2) our statement follows.

We note that, in the above result, we have to assume the condition (3.2) just as in the non-integral case, see [7, Theorem 1.2].

Before studying the observation of the second component, we have to show an inverse estimate regarding only the second component of the solution of problem (2.1), see (2.4).

Proposition 3.3. Let{p_n}_n∈Nbe a sequence of pairwise distinct nonzero complex numbers, satisfying

nlim→_∞(<p_n+1− <pn) = +_∞, lim

n→_∞=pn =0 . Then, for any T>0there exists a positive constant c(T)such that

Z _T

0

∑

d_nD_ne^ipnt+d_nD_ne^−ipnt

− ^2β Ae^−ηt

∑

n²< ^Dn η+ip_n

2

dt

≥c(T)





∑

|p_n|⁴|D_n|²+

∑

n²< ^Dn η+ip_n

2

 . (3.4)

To prove that inverse estimate, we need some preliminary results. The first step is to state inverse and direct inequalities for Fourier series without a finite number of terms. The following result follows from [14, Propositions 5.8–5.9].

Lemma 3.4. There exist n₀ ∈ _N such that for any sequence {E_n} of complex numbers, with

∑^∞n=1 |En|² <+_∞and En=0for any n<n₀, we have Z _T

0

∑

E_ne^ipnt+E_ne^−ipnt

2

dt

∑

n=n₀

|E_n|². (3.5)

The second step is to recover the finite number of terms in the series. To this end, we need to establish a so-called Haraux type estimate.

For the sake of completeness, we introduce a family of integral operators which annihilate a finite number of terms in the Fourier series. We begin by recalling the definition of operators, which was given in [12] and is slightly different from that introduced in [4] and [6].

Given δ > 0 and z ∈ _C arbitrarily, the symbol I_δ,z denotes the linear operator defined as follows: for every continuous function u : R →_C the function I_δ,zu :R → _C is given by the formula

I_δ,zu(t):=u(t)−¹ δ

Z _δ

0 e^−izsu(t+s)ds, t∈_R. (3.6) For the reader’s convenience, we list some known properties verified by the operators I_δ,z, see e.g. [4,6,12].

Lemma 3.5. For anyδ >_{0and z}∈_Cthe following statements hold true.

(i) I_δ,z(_e^izt) =_0.

(ii) For any z⁰ ∈ _{C, z}⁰ 6=z, we have

I_δ,z(e^iz0t) =

1− ^e

i(z⁰−z)δ−1 i(z⁰−z)δ

e^iz0t. (iii) The linear operators I_δ,zcommute: for anyδ⁰ >₀and z⁰ ∈_Cwe have

I_δ,z◦I_δ⁰_,z⁰ = I_δ⁰_,z⁰◦I_δ,z,

where the symbol◦denotes the standard composition among operators.

(iv) For any T>0and continuous function u:R→_Cwe have Z _T

0

|I_δ,zu(t)|² dt≤2(1+e^2|=z|δ)

Z _T+δ

0

|u(t)|²dt. (3.7) The following result is similar to [6, Prop. 1.9], but due to the presence of another term (see inequality (3.11) below), we prefer to prove it, to make also the paper as self-contained as possible.

Proposition 3.6. Let{p_n}_n∈Nbe a sequence of pairwise distinct nonzero complex numbers such that pn6=iη, for any n∈_N,

nlim→_∞|p_n|= +_∞, the sequence {=p_n}is bounded. (3.8)

Assume that there exists n₀ ∈_Nsuch that for any sequence{E_n}the estimates Z _T

0

∑

E_ne^ipnt+E_ne^−ipnt

2

dt≥c₁

∑

|E_n|², (3.9)

Z _T

0

∑

E_ne^ipnt+E_ne^−ipnt

2

dt≤c₂

∑

|E_n|², (3.10)

hold for some constants c₁,c₂>0.

Then, there exists C₁>0such that for any sequence{E_n}andD ∈_Rthe estimate Z _T

0

∑

E_ne^ipnt+E_ne^−ipnt

+De^−ηt

2

dt≥C₁

∑

|E_n|²+|D|²

!

(3.11) is true.

Proof. To begin with, we will transform u(t) =

∑

E_ne^ipnt+E_ne^−ipnt

+De^−ηt

into a function without those terms corresponding to indices n = 1, . . . ,n₀−1 and without the termDe^−ηt, so we can apply the assumptions (3.9) and (3.10).

To this end, we fix ε > 0 and choose δ ∈ 0,_2n^ε

0

, where n₀ is the integer for which the estimates (3.9) and (3.10) hold. Let us denote by I the composition of I_δ,iη and all linear operatorsI_δ,pj◦I_δ,−pj,j=1, . . . ,n₀−1. We note that by Lemma3.5(iii) the definition ofIdoes not depend on the order of the operators.

By using Lemma3.5, we get Iu(t) =

∑

E⁰_ne^ipnt+E_n⁰e^−ipnt where

E⁰_n:= 1−^e

i(pn−iη)δ−1 i(p_n−iη)δ

!n0−1

∏

∏

z∈{p_j,−p_j}

1− ^e

i(pn−z)δ−1 i(p_n−z)δ

! En. Therefore, estimate (3.9) holds for functionIu(t), that is

Z _T

0

|_Iu(t)|² dt≥c₁

∑

|E⁰_n|². (3.12)

Next, we observe that we can chooseδ ∈ _0,2n^ε

0

such that for anyn≥ n₀none of the products

1− ^e

i(pn−iη)δ−1 i(p_n−iη)δ

!n0−1

∏

∏

z∈{pj,−pj}

1− ^e

i(pn−z)δ−1 i(p_n−z)δ

!

(3.13)

vanishes. Indeed, that is possible because the analytic function w7−→1− ^e

w−1 w

does not vanish identically, and hence, keeping in mind that every number p_n−z with z ∈ {iη,p_j,−p_j : j=1, . . . ,n0−1}is different from zero, we have to exclude only a countable set of values ofδ.

Then, we note that there exists a constantc⁰ >0 such that for anyn≥n₀

1− ^e

i(pn−iη)δ−1 i(p_n−iη)δ

!_n

0−1

∏

∏

z∈{p_j,−p_j}

1− ^e

i(pn−z)δ−1 i(p_n−z)δ

!

2

≥c⁰. (3.14)

Actually, it is sufficient to observe that for z∈ {iη,p_j,−p_j}, we have

e^i(pn−z)δ−1 i(p_n−z)δ

≤ ^e

−=(pn−z)δ+1

|p_n−z|δ →0 asn→_∞,

thanks to (3.8). As a result, the product in (3.13) tends to 1 asn→_∞and hence, for example, we can take it greater than 1/2 fornlarge enough. Therefore, (3.12) and (3.14) yield

Z _T

0

|_Iu(t)|² dt≥c⁰c₁

∑

|E_n|². (3.15)

On the other hand, applying (3.7) repeatedly withz=iη,z = p_jandz=−p_j,j=1, . . . ,n₀−1, we have

Z _T

0

|_Iu(t)|² dt≤2²ⁿ⁰⁻¹(1+e^2|η|δ)

n0−1

∏

(1+e^2|=pj|δ)²

Z _T+(2n₀−1)δ 0

|u(t)|²dt. From the above inequality, by using (3.15) and 2n₀δ <ε, it follows

∑

|En|²≤ ²

2n0−1

c⁰c₁ (1+e^ηε/n0)

n0−1

∏

(1+e^|=pj|ε/n0)²

Z _T+ε 0

|u(t)|²dt, whence, passing to the limit asε→0⁺, we have

∑

|E_n|² ≤ ²

4n0−2

c⁰c₁ Z _T

0

|u(t)|² dt. (3.16)

Moreover, thanks to the triangle inequality, we get Z _T

0

n0−1 n

∑

E_ne^ipnt+E_ne^−ipnt

+De^−ηt

2

dt=

Z _T

0

u(t)−

∑

n=n0

E_ne^ipnt+E_ne^−ipnt

2

dt

≤2 Z _T

0

|u(_t)|²dt+₂

Z _T

0

∑

Ene^ipnt+_Ene^−ipnt

2

dt. By using (3.10) and (3.16) in the previous inequality, we have

Z _T

0

n0−1 n

∑

E_ne^ipnt+E_ne^−ipnt

+De^−ηt

2

dt≤2 Z _T

0

|u(t)|² dt+2c₂

∑

|E_n|²

≤2

1+c₂2⁴ⁿ⁰⁻² c⁰c₁

_{Z T}

0

|u(t)|²dt. (3.17)

Let us note that the expression Z _T

0

n0−1 n

∑

E_ne^ipnt+E_ne^−ipnt

+De^−ηt

2

dt

is a positive semidefinite quadratic form of the variable {En}_n<n₀,D∈_Cⁿ⁰⁻¹×_R. Moreover, it is positivedefinite, because the functionse^−ηt,e^ipnt,n<n₀, are linearly independent. Hence, there exists a constantc⁰⁰ >0 such that

Z _T

0

n₀−1 n

∑

Ene^ipnt+Ene^−ipnt

+De^−ηt

2

dt≥ c⁰⁰

n₀−1 n

∑

|En|²+|D|²

! . So, from (3.17) and the above inequality we deduce that

n0−1 n

∑

|En|²+|D|²≤ ² c⁰⁰

1+c₂2⁴ⁿ⁰⁻² c⁰c₁

_{Z T}

0

|u(t)|² dt. Finally, the above estimate and (3.16) yield the required inequality (3.11).

Finally, we are able to prove Proposition3.3

Proof of Proposition3.3. If we consider the sequence{dnDn}, thanks to (2.7) and (2.9) we have

∑^∞n=1 |d_nD_n|²< +∞. Therefore, we can apply Lemma3.4toE_n =d_nD_n: for a suitable integer n₀ one has

Z _T

0

∑

dnDne^ipnt+dnDne^−ipnt

2

dt

∑

|dnDn|², (3.18) that is, the estimates (3.9) and (3.10) hold for the sequence{d_nD_n}. Moreover, we note that the sum

∑

n²< ^Dn η+ipn

is a real number. Indeed, in view of the inequality

∑

n²

< ^Dn η+ip_n

≤

∑

n=1

n² |Dn|

|η+ip_n|^, we get

∑

n²

< ^Dn η+ipn

2

≤

∑

n²|Dn|

|η+ipm|

m²|Dm|

|η+ipn|

≤ ¹ 2

∑

n⁴|D_n|²

∑

m=1

1

(η− =p_m)²+<p²_m +¹

2

∑

m⁴|D_m|²

∑

n=1

1

(η− =p_n)²+<p²_n

=

∑

1

(η− =p_n)²+<p²_n

∑

n⁴|D_n|² <+_∞, thanks to (2.5) and (2.9).

At last, we are in condition to apply Proposition3.6: the estimate (3.11) holds when En= dnDn andD =−^2β_{A ∑}^∞_n=1n²<_η+^D_ipⁿ

n; in consequence, thanks also to (2.7), it follows (3.4).

Finally, we are able to show a partial observability result for the second component. We note that, unlike Theorem3.2, we do not need to assume that the initial data u₁₀ andu₁₁ are null.

Theorem 3.7. Let T > 0. For any (u₁₀,u₁₁) ∈ H₀¹(0,π)×L²(0,π)and (u₂₀,u₂₁) ∈ H₀¹(0,π)× H⁻¹(0,π), if(u₁,u₂)is a solution of problem(2.1), then we have

Z _T

0

|u_2x(t,π)|² dt≥c(T) ku₂₀k²

H¹₀(0,π)+ku₂₁k²_H−1(0,π)

, (3.19)

where c(T)is a positive constant.

Proof. From (2.4) it follows u2x(t,π) =

∑

(−1)ⁿn

dnDne^ipnt+dnDne^−ipnt−^2β

An²< ^Dn η+ip_ne^−ηt

.

We can employ Proposition 3.3 to treat the previous sum. Indeed, applying formula (3.4) to u_2x(t,π)we obtain

Z _T

0

u_2x(t,π)

2 dt≥c(T)

∑

n²|p_n|⁴|D_n|², whence, in virtue of (2.9) and (2.3), our statement follows.

In conclusion, the partial observability of the first component has been established in Theorem 3.2, while by Theorem 3.7 and assuming u₁₀ = u₁₁ = 0 the partial observability of the second component follows.

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