We consider a linear wave equation

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

CONTROLLABILITY FOR THE WAVE EQUATION WITH MOVING BOUNDARY

ISA´IAS P. DE JESUS, EUGENIO CABANILLAS LAPA, JUAN LIMACO

Abstract. In this article, we study the boundary controllability for a one- dimensional string equation on a domain with time-dependent boundary. This equation models small vibrations of a string with one of its endpoint fixed and other moving with speedk(t). We use an inverse inequality to obtain a controllability result. We consider a linear wave equation.

1. Introduction

1.1. Statement of the problem and main result. The controllability of linear and non-linear PDEs has been the subject of much work in the last decades. Theo- retical aspects and their connection to applications have been considered by many authors and a lot of advances can be fortunately mentioned. Initially, the concept ofhierarchic control was introduced by J.-L. Lions (see [17, 18]), where some tech- niques are presented. Also we mention the papers [15, 16, 28, 29, 9, 7, 8, 6] where the authors combine the concepts of multi-criteria optimization and controllability.

As in [3], givenT >0, we consider the non-cylindrical domain defined by Qb={(x, t)∈R²; 0< x < αk(t), t∈(0, T)},

where α_k(t) = 1 +kt, 0< k <1. Its lateral boundary is defined byΣ =b Σb₀∪Σb^∗₀, with

Σb0={(0, t); t∈(0, T)}, Σb^∗₀=Σ\b Σb0={(αk(t), t); t∈(0, T)}.

We also represent by Ωtand Ω0 the intervals (0, αk(t)) and (0,1), respectively.

Motivated by the arguments contained in the work of J.-L. Lions [19], we consider the following wave equation in the non-cylindrical domainQ:b

u⁰⁰−uxx= 0 in Q,b u(x, t) =

(

w(t)e onΣb₀, 0 onΣb^∗₀,

u(x,0) =u0(x), u⁰(x,0) =u1(x) in Ω0,

(1.1)

whereuis the state variable,weis the control variable and (u0(x), u1(x))∈L²(0,1)×

H⁻¹(0,1). Byu⁰ =u⁰(x, t) we represent the derivative ^∂u_∂t and by u_xx =u_xx(x, t)

2010Mathematics Subject Classification. 93B05, 93C05, 93C20.

Key words and phrases. Wave equation; Stackelberg-Nash strategies; controllability;

inverse inequality.

c

2021. This work is licensed under a CC BY 4.0 license.

Submitted November 1, 2020. Published July 5, 2021.

1

the second order partial derivative ^∂_∂x^2u2. Equation (1.1) models the motion of a string with a fixed endpoint and a moving one. The constantk is called the speed of the moving endpoint.

The novelty of this paper is the consideration of a domain with moving boundary.

Indeed, instead of transforming the problem (1.1) from a non-cylindrical domain into a cylindrical domain, we study the controllability problem directly in a non- cylindrical domain, when the control is put on the fixed point. For this, we use an inverse inequality.

Now, letα(t) =t+α_k(t),β(t) =t−α_k(t) andγ=α◦β⁻¹. We assume that

T > γ(0), (1.2)

0< k <1. (1.3)

The main result of this paper reads as follows.

Theorem 1.1. Assume that (1.2) and (1.3) hold. Let us consider we1 ∈ L²(bΣ1) andwe2 a Nash equilibrium in the sense (1.10). Then the pair

(u(T), u⁰(T)) = (u(., T,we1,we2), u⁰(., T,we1,we2)),

whereusolves the system (2.8), generates a dense subset of L²(ΩT)×H⁻¹(ΩT).

As in [19], we divideΣb0 into two parts

Σb0=Σb1∪Σb2, (1.4)

and consider

we={we1,we2}, wei= control function inL²(bΣi), i= 1,2. (1.5) We can also write

we=we₁+we₂, withΣb₀=Σb₁=Σb₂. (1.6) Thus, we observe that system (1.1) can be rewritten as follows:

u⁰⁰−u_xx= 0 in Q,b u(x, t) =







we₁(t) onΣb₁, we2(t) onΣb2, 0 onΣ\b Σb₀, u(x,0) =u0(x), u⁰(x,0) =u1(x) in Ω0.

(1.7)

In decomposition (1.4), (1.5) we establish a hierarchy. We think of we1 as being the “main” control, the leader, and we think of we2 as the follower, in Stackelberg terminology.

Associated with the solutionu=u(x, t) of (1.7), we will consider the (secondary) functional

Je2(we1,we2) = 1 2

Z Z

Qb

(u(we1,we2)−ue2)²dx dt+σe 2

Z

Σb2

we₂²dbΣ, (1.8) and the (main) functional

J(ewe₁) =1 2

Z

Σb1

we₁²dbΣ, (1.9)

whereσ >e 0 is a constant andue₂ is a given function inL²(Q).b

Now, let us describe the Stackelberg-Nash strategy. Thus, for each choice of the leaderwe1, we try to find a Nash equilibrium for the cost functionalJe2, that is, we look for a controlwe2=F(we1), depending onwe1, satisfying

Je₂(we₁,we₂) = inf

wb₂∈L²(bΣ₂)

Je₂(we₁,wb₂). (1.10) After this, we consider the stateu(we1,F(we1)) given by the solution of

u⁰⁰−uxx= 0 in Q,b u(x, t) =







we1 onΣb1, F(we1) onΣb2, 0 onΣ\b Σb₀, u(x,0) =u0(x), u⁰(x,0) =u1(x) in Ω0.

(1.11)

We will look for any optimal controlwf1 such that J(ewe1,F(we1)) = inf

w₁∈L²(bΣ₁)

Je(w1,F(we1)), (1.12) subject to the following restriction of the approximate controllability type

(u(x, T;we1,F(we1)), u⁰(x, T;we1,F(we1)))

∈B_L2(ΩT)(u⁰, ρ0)×B_H⁻¹_(ΩT)(u¹, ρ1), (1.13) whereBX(C, r) denotes the ball in X with centerCand radiusr.

The control problems to be studied in this paper are described as follows:

Problem 1Fixed any leader controlwe₁, find the follower controlwe₂=F(we₁) (de- pending onwe₁) and the associated stateu, solution of (1.7) satisfying the condition (1.10) (Nash equilibrium) related toJe2, defined in (1.8).

Problem 2 Assuming that the existence and uniqueness of the Nash equilib- rium we₂ was proved, then when we₁ varies in L²(bΣ₁), prove that the solutions (u(x, t;we₁,we₂), u⁰(x, t;we₁,we₂)) of the state equation (1.7), evaluated att=T, that is, (u(x, T;we₁,we₂), u⁰(x, T;we₁,we₂)), generate a dense subset ofL²(Ω_T)×H⁻¹(Ω_T).

Remark 1.2. By the linearity of system (1.11), without loss of generality we may assume thatu0= 0 =u1.

1.2. Related problems. Controllability of system (1.1) has been extensively stud- ied in the recent past years; most of the papers in this direction dealt with the case of one moving endpoint with boundary conditions of the form

u(0, t) = 0, u(1 +kt, t) =w(t),e k∈(0,1), t∈(0,∞).

In [24], it has been shown that exact controllability holds at any time T > e^2k(1+k)1−k −1

2 .

The same authors came back in [25] and improved the latter result to T > e

2k(1+k) (1−k)3 −1

2 .

Later, in [11], the controllability time has been improved to be T > _1−k² . In these papers, only a sufficient condition is provided for the exact controllability.

Concerning the two moving endpoints case, the boundary functions considered in [1] are of the form

α(t) =−kt, β(t) = 1 +rt, t∈(0,∞), k, r∈[0,1) with r+k >0.

It has been shown that exact controllability holds if, and only if T ≥ _{(1−k)(1−r)}² . More general boundary functions are considered in [3] with boundary conditions

u(0, t) = 0, u(s(t), t) =w(t),e t∈(0,∞),

where s : [0,∞) → (0,∞) is assumed to be C¹ function with ks⁰kL^∞(0,∞) < 1.

Furthermore, it has been assumed thatsmust be in some admissible class of curves (see [3] for more details). Under these assumptions, the authors proved that exact controllability holds if, and only ifT ≥s⁺◦(s⁻)⁻¹(0), wheres^±(t) =t±s(t). Also, they provided a controllability result when the control is located on the non-moving part of the boundary. By considering the boundary conditions

u(0, t) =w(t),e u(s(t), t) = 0, t∈(0,∞),

they proved that exact controllability holds if, and only if T ≥(s⁻)⁻¹(1). In all the cited works, the proofs rely on the multipliers technique or the non-harmonic Fourier analysis.

Recently, in [2], a new Carleman estimate has been established for the wave equation in time-dependent domain in a more general setting. The existence of solutions of the initial boundary value problem for the nonlinear wave equation in non-cylindrical domains has been studied in [5, 20]. The controllability prob- lem for a multi-dimensional wave equation in a non-cylindrical domain has been investigated in [4, 26, 27]. Also, about the one-dimension cases, there have been extensive study of the controllability problem in a non-cylindrical domain. We re- fer the reader to [12, 14, 21, 22, 23]. Finally, we can mention also the paper by Yang and Feng [31], where the authors present the approximate controllability of Euler-Bernoulli viscoelastic systems.

The content of this article is organized as follows. Section 2 is devoted to establish the optimality system for the follower control. In Section 3, we investigate the approximate controllability proving the density Theorem 1.1. Finally, we present the optimality system for the leader control in Section 4.

2. Optimal system for the follower control

In this section, fixed any leader controlwe1∈L²(bΣ1) we determine the existence and uniqueness of solutions to the problem

inf

we₂∈L²(bΣ₂)

Je₂(we₁,we₂), (2.1) and a characterization of this solution in terms of an adjoint system.

For this, we consider U_ad = {(u,we₂) ∈ L²(Q)×L²(bΣ₂);usolution of (1.7)}, and Je2 : Uad → Rdefined by (1.8). Note that Uad is a nonempty closed convex subset of L²(Q)×L²(Σ2), and Je2 is weakly coercive, weakly sequentially lower semicontinuous and strictly convex. Therefore, there exists a unique solutionwe2of (2.1), i.e.,

Je2(we1,we2) = inf

wb₂∈L²(bΣ₂)

Je2(we1,wb2).

The Euler-Lagrange equation for problem (2.1) is Z T

0

Z

Ωt

(u−ue2)u dx dtb +σe Z

Σb2

we2wb2dbΣ = 0, ∀wb2∈L²(bΣ2), (2.2) whereubis solution of the system

ub⁰⁰−buxx= 0 in Q,b bu=







0 onΣb1, wb₂ onΣb₂,

0 onΣ\(bb Σ1∪Σb2), bu(x,0) = 0, ub⁰(x,0) = 0, xin Ω0.

(2.3)

To express (2.2) in a convenient form, we introduce the adjoint state defined by p⁰⁰−pxx=u−ue2 in Q,b

p(T) =p⁰(T) = 0, xin ΩT, p= 0 onΣ.b

(2.4)

Multiplying (2.4) byuband integrating by parts, we find that Z T

0

Z

Ω_t

(u−ue2)u dx dtb + Z

Σb₂

pxwb2dbΣ = 0, (2.5) so that (2.2) becomes

p_x=σewe₂ onΣb₂. (2.6) We summarize these results in the following theorem.

Theorem 2.1. For each we1 ∈ L²(Σ1) there exists a unique Nash equilibrium we2

in the sense of (1.10). Moreover, the followerwe2 is given by we2=F(we1) = 1

σepx onΣb2, (2.7)

where{v, p}is the unique solution of (the optimality system) u⁰⁰−u_xx= 0 inQ,b

p⁰⁰−pxx=u−eu2 inQ,b u=







we1 onΣb1,

1

eσpx onΣb2, 0 onΣ\b Σb0, p= 0 onΣ,b u(0) =u⁰(0) = 0, xin Ω₀, p(T) =p⁰(T) = 0, xin Ω_T.

(2.8)

Of course,{u, p}depends on we₁:

{u, p}={u(we₁), p(we₁)}. (2.9)

3. Proof of Theorem 1.1

Since we have proved the existence, uniqueness and characterization of the fol- lowerwe₂, the leaderwe₁now wants that solutionsuandu⁰, evaluated at timet=T, to be as close as possible to (u⁰, u¹). This will be possible if the system (2.8) is approximately controllable. We are looking for

inf 1 2

Z

Σb₁we²₁dbΣ, (3.1)

wherewe₁ is subject to

(u(T;we1), u⁰(T;we1))∈BL²(Ω_T)(u⁰, ρ0)×BH⁻¹(Ω_T)(u¹, ρ1), (3.2) assuming that we1 exists, ρ0 and ρ1 being positive numbers arbitrarily small and {u⁰, u¹} ∈L²(ΩT)×H⁻¹(ΩT).

Lemma 3.1(Inverse Inequality). If (g⁰, g¹)∈H₀¹(Ω₀)×L²(Ω₀), then there exists γ(0)>0such that for T > γ(0), the weak solution of problem

z⁰⁰−z_xx= 0 inQ,b z= 0 onΣ,b

z(0) =g⁰, z⁰(0) =g¹ inΩ0

(3.3)

satisfies

Z γ(0) 0

|zx(0, t)|²dt≥C(|g⁰|²+|g¹|²), (3.4) whereC is given in [3].

Proof. We construct a solution of the form z(x, t) =X

n∈Z

An(e^2ξinϕ(t+x)−e^{2ξinϕ(t−x)}), (3.5) whereϕ∈C² is a solution to the functional equation

ϕ(t+αk(t))−ϕ(t−αk(t)) = 1.

We letF(x) =Rx

0 g¹(s)dsand h(x) =

(₁

2 g⁰(x) +F(x)

for 0≤x≤1,

1

2 −g⁰(−x) +F(−x)

for −1≤x≤0.

We note that

A_n= Z 1

−1

h(x)e^2ξinϕ(x)ϕ⁰(x)dx.

Then z∈C² and their derivatives can be calculated term by term. Let us define some often appearing values:

m(t) = min{ϕ⁰(x) :x∈[t−α_k(t), t+α_k(t)]}, M(t) = max{ϕ⁰(x) :x∈[t−α_k(t), t+α_k(t)]}.

Now, we adapt the proof of [3, Theorem 2.1], with 0< k <1. Indeed, we consider β(t) = t−αk(t). Then β⁰(t) = 1−k > 0. Therefore, β(t) is strictly increasing and since β(0) = −1 < 0, there exists a unique t₀ such that β(t₀) = 0. Let τ₀=t₀+α_k(t₀) =γ(0).

Differentiatingzterm by term in (3.5), and evaluating atx= 0, for allτ >0 we obtain

|zx(0, t)|²_L2(0,τ0)≥m(t₀)|zx(0, t)|²_L2(0,τ0,_ϕ0¹_(t))

=m(t0)16ξ²X

n∈Z

n²|An|²

≥C(|g⁰|²+|g¹|²),

(3.6)

where the last inequality in (3.6) is obtained from [3, Proposition 1.4].

Remark 3.2. The proof of Lemma 3.1 also holds forαk(t) = 1 +kT −kt.

Now as in the case (1.6), we conclude this section with the proof of Theorem 1.1.

Proof of Theorem 1.1. We decompose the solution (u, p) of (2.8) by setting u=ϑ0+g,

p=p0+q, (3.7)

whereϑ0,p0 is given by

ϑ⁰⁰₀−(ϑ0)xx= 0 inQ,b ϑ0=







0 onΣb1,

1

σe(p₀)_x onΣb₂,

0 onΣ\b Σb0, ϑ₀(0) =ϑ⁰₀(0) = 0, xin Ω₀,

(3.8)

and

p⁰⁰₀−(p₀)_xx=u₀−eu₂ inQ,b p₀= 0 onΣ,b

p₀(T) =p⁰₀(T) = 0, xin Ω_T;

(3.9) And{g, q}is given by

g⁰⁰−gxx= 0 inQ,b g=







we1 onΣb1,

1

eσqx onΣb2, 0 onΣ\b Σb₀, g(0) =g⁰(0) = 0, xin Ω0,

(3.10)

and

q⁰⁰−qxx=g inQ,b q= 0 onΣ,b

q(T) =q⁰(T) = 0, xin Ω_T.

(3.11)

We next setA:L²(bΣ₁)→H⁻¹(Ω_T)×L²(Ω_T) as Awe1=

g⁰(T;we1), −g(T;we1) , (3.12) which defines

A∈ L L²(bΣ1); H⁻¹(ΩT)×L²(ΩT) . Using (3.7) and (3.12), we can rewrite (3.2) as

Awe₁∈ {−ϑ⁰₀(T) +B_H−1(Ω_T)(u¹, ρ₁), −ϑ0(T) +B_L2(Ω_T)(u⁰, ρ₀)}. (3.13)

We will show that Awe1 generates a dense subspace of H⁻¹(ΩT)×L²(ΩT). For this, let{f⁰, f¹} ∈H₀¹(ΩT)×L²(ΩT) and consider the following systems (“adjoint states”):

ϕ⁰⁰−ϕ_xx=ψ inQ,b ϕ= 0 onΣ,b

ϕ(T) =f⁰, ϕ⁰(T) =f¹, xin ΩT,

(3.14)

and

ψ⁰⁰−ψxx= 0 inQ,b ψ=







0 onΣb1,

1

σeϕx onΣb2, 0 on Σ\b Σb0, ψ(0) =ψ⁰(0) = 0, xin Ω₀.

(3.15)

Multiplying (3.15)₁byq, (3.14)₁byg, whereq,gsolve (3.11) and (3.10), respec- tively, and integrating inQbwe obtain

Z T 0

Z

Ω_t

g ψ dx dt=−1 σe

Z

Σb₂

q_xϕ_xdbΣ, (3.16) hg⁰(T), f⁰i_H−1(Ω_T)×H¹₀(Ω_T)− g(T), f¹

=− Z

Σb1

ϕxwe1dbΣ. (3.17) Considering the left-hand side of this equation as the inner product of{g⁰(T),−g(T)}

and{f⁰, f¹} inH⁻¹(ΩT)×L²(ΩT) andH₀¹(ΩT)×L²(ΩT), we obtain hhAwe1, fii=−

Z

Σb₁

ϕxwe1dbΣ,

wherehh·,·iirepresent the duality pairing betweenH⁻¹(Ω_T)×L²(Ω_T) andH₀¹(Ω_T)×

L²(Ω_T). Therefore, if

hhAwe₁, fii= 0, for allwe1∈L²(bΣ1), then

ϕx= 0 onΣb1. (3.18)

Hence, in case (1.6),

ψ= 0 onΣ,b so thatψ≡0. (3.19)

Therefore,

ϕ⁰⁰−ϕ_xx= 0, ϕ= 0 onΣ,b (3.20) and satisfies (3.18). Therefore, by Lemma 3.1 and by Remark 3.2, withz(x, t) = ϕ(x, T−t),g⁰=f⁰, andg¹=f¹we have thatf⁰= 0,f¹= 0. This completes the

proof.

4. Optimal system for the leader control

In the previous sections, we have seen that no matter what strategy, the leader assumes that the follower make their choice we2 satisfying the Nash equilibrium.

The goal of this section is to obtain a optimal system for the leader control. More precisely, we obtain the following result.

Theorem 4.1. Assume that the hypotheses (1.2), (1.3) and (1.6) are satisfied.

Then for {f⁰, f¹} inH₀¹(ΩT)×L²(ΩT)we uniquely define{ϕ, ψ, u, p} by ϕ⁰⁰−ϕxx=ψ in Q,b

ψ⁰⁰−ψ_xx= 0 in Q,b u⁰⁰−uxx= 0 inQ,b p⁰⁰−pxx=u−eu2 inQ,b

ϕ= 0 onΣ,b ψ=







0 onΣb1,

1

eσ ϕx onΣb2, 0 onΣ\b Σb0, u=







−ϕx on Σb1,

1

eσpx on Σb2, 0 on Σ\bb Σ0, p= 0 onΣ,b

ϕ(·, T) =f⁰, ϕ⁰(·, T) =f¹ inΩT, u(0) =u⁰(0) = 0 inΩ₀, p(T) =p⁰(T) = 0 inΩ_T.

(4.1)

We uniquely define{f⁰, f¹} as the solution of the variational inequality u⁰(T, f)−u¹,fb⁰−f⁰

H⁻¹(ΩT)×H₀¹(ΩT)− u(T, f)−u⁰,fb¹−f¹ +ρ₁ kfb⁰k − kf⁰k

+ρ₀ |fb¹| − |f¹|

≥0, ∀fb∈H₀¹(Ω_T)×L²(Ω_T).

(4.2) Then the optimal leader is

we1=−ϕx on Σb1, whereϕcorresponds to the solution of (4.1).

Proof. LetAbe the continuous linear operator defined by (3.12) and we introduce the following two convex proper functions: F1:L²(bΣ1)→R∪ {∞}by

F₁(we₁) =1 2

Z

Σb₁we²₁dbΣ (4.3)

andF₂:H⁻¹(Ω_T)×L²(Ω_T)→R∪ {∞}by

F2(ξ, µ) =











0, if (ξ, µ)∈u¹−ϑ⁰₀(T) +ρ1BH⁻¹(Ω_T)

−u⁰+ϑ0(T)−ρ0B_L2(ΩT), +∞, otherwise.

(4.4)

With these notation, problems (3.1)–(3.2) become equivalent to inf

we1∈L²(bΣ1)

F1(we1) +F2(Awe1)

(4.5) provided that we prove that the range ofAis dense in H⁻¹(Ω_T)×L²(Ω_T), under conditions (1.2) and (1.3).

By the Duality Theorem of Fenchel and Rockafellar [30] (see also [10, 13]), we have

inf

we₁∈L²(bΣ₁)

[F₁(we₁) +F₂(Awe₁)]

=− inf

(fb⁰,fb¹)∈H₀¹(ΩT)×L²(ΩT)

[F₁^∗ A^∗{fb⁰,fb¹}

+F₂^∗{−fb⁰,−fb¹}], (4.6) whereF_i^∗ is the conjugate function ofFi (i= 1,2) and A^∗ the adjoint ofA.

We haveA^∗:H₀¹(ΩT)×L²(ΩT)→L²(bΣ1) as

(f⁰, f¹)7→A^∗f =−ϕ_x, (4.7) whereϕis given in (3.14).

We see easily that

F₁^∗(we1) =F1(we1) (4.8) and

F₂^∗({fb⁰,fb¹}) =hu¹−ϑ⁰₀(T),fb⁰i_H−1(ΩT)×H₀¹(ΩT)

+ ϑ₀(T)−u⁰,fb¹

+ρ₁kfb⁰k+ρ₀|fb¹|.

(4.9) Therefore the (opposite of) right-hand side of (4.6) is given by

− inf

f∈Hb ¹₀(Ω_T)×L²(Ω_T)

n1 2

Z

Σb1

ϕ²_xdbΣ + u⁰−ϑ0(T),fb¹

− hu¹−ϑ⁰₀(T),fb⁰i_H−1(Ω_T)×H₀¹(Ω_T)+ρ1kfb⁰k+ρ0|fb¹|o .

This is the dual problem of (3.1) and (3.2). Hence, we can use the primal or the dual problem to derive the optimality system for the leader control.

Acknowledgments. Isa´ıas P. de Jesus was supported by grant 307488/2019-5 from CNPq/Brazil. The authors want to express their gratitude to the anonymous reviewers for their questions and comments; they were very helpful in improving this article.

References

[1] A. Sengouga; Exact boundary observability and controllability of the wave equation in a interval with two moving endpoints, Mathematical Control and Related Fields, 9, (2020), 1-25.

[2] A. Shao; On Carleman and observability estimates for wave equations on time-dependent domains,Proc. Lond. Math. Soc.,119, (2019), 998-1064.

[3] B. Haak, D. Hoang;Exact observability of a 1D wave on a non-cylindrical domain, SIAM J.

Control Opitim.57, (2019), 570-589.

[4] C. Bardos, G. Chen;Control and stabilization for the wave equation, part III: domain with moving boundary, SIAM J. Control Optim.,19, (1981) 123-138.

[5] C. Bardos, J. Cooper; A nonlinear wave equation in a time dependent domain, J. Math.

Anal. Appl.,42, (1973) 29-60.

[6] F. Araruna., E. Fern´andez-Cara, L. Silva;Hierarchic control for the wave equation, Journal of Optimization Theory and Applications,178, (1), (2018) 264-288.

[7] F. Araruna, E. Fern´andez-Cara, M. Santos;Stackelberg-Nash exact controllability for linear and semilinear parabolic equations, ESAIM : COCV21, (3), (2015) 835-856.

[8] F. Araruna, E. Fern´andez-Cara, S. Guerrero, M. Santos; New results on the Stackelberg - Nash exact control of linear parabolic equations, Systems & Control Letters, 104, (2017) 78-85.

[9] G. Gonz´alez, F. Lopes, M. Rojas-Medar;On the approximate controllability of Stackelberg- Nash strategies for Stokes equations Proc. Amer. Math. Soc.141(5), (2013) 1759-1773.

[10] H. Brezis; Functional analysis, Sobolev spaces and partial differential equations, Springer- Verlag, 2010.

[11] H. Sun, H. Li, L. Lu; Exact controllability for a string equation in domains with moving boundary in one dimension,Eletronic Journal of Differential Equations,2015, (2015) no. 98, 1-7.

[12] H. Wang, Y. He, S. Li;Exact controllability problem of a wave equation in non-cylindrical domains, Eletronic Journal of Differential Equations,2015, (2015) no. 31, 1-13.

[13] I. Ekeland, R. Temam;Analyse convexe et probl`emes variationnels, Dunod, Gauthier-Villars, Paris, 1974.

[14] I. Jesus;Controllability for a one-dimensional wave equation in a non-cylindrical domain, Mediterranean Journal of Mathematics,16(2019), 111.

[15] J. D´ıaz, J.-L. Lions;On the approximate controllability of Stackelberg-Nash strategies. in: J.I.

D´ıaz (Ed.), Ocean Circulation and Pollution Control Mathematical and Numerical Investi- gations, 17-27, Springer, Berlin, (2005).

[16] J. D´ıaz;On the von Neumann problem and the approximate controllability of Stackelberg - Nash strategies for some environmental problems, Rev. R. Acad. Cien., Serie A. Math.,96 (3), (2002), 343-356.

[17] J.-L. Lions;Contrˆole de Pareto de Syst`emes Distribu´es. Le cas d’ ´evolution, C.R. Acad. Sc.

Paris, s´erie I302(11) (1986), 413-417.

[18] J.-L. Lions;Some remarks on Stackelberg’s optimization, Mathematical Models and Methods in Apllied Sciences,4, (1994) 477-487.

[19] J.-L. Lions;Hierarchic control, Mathematical Science, Proc. Indian Academic Science,104, (1994) 295-304.

[20] L. A. Medeiros; Nonlinear wave equations in domains with variable boundary, Arch. Rat.

Mech. Anal.,47(1972), 47-58.

[21] L. Cui , H. Gao;Exact controllability for a wave equation with mixed boundary conditions in a non-cylindrical domain,Eletronic Journal of Differential Equations,101, (2014), no. 101, 1-12.

[22] L. Cui, L. Song;Exact controllability for a wave equation with fixed boundary control. Bound- ary Value Problems, (2014). doi: 10.1186/1687-2770-2014-47.

[23] L. Cui, L. Song;Controllability for a wave equation with moving boundary, Journal of Applied Mathematics, (2014). doi: 10.1155/2014/827698.

[24] L. Cui, X. Liu, H. Gao,Exact controllability for a one-dimensional wave equation in non- cylindrical domains.J. Math. Anal. Appl.402, (2013) 612-625.

[25] L. Cui, Y. Jiang, Y. Wang;Exact controllability for a one-dimensional wave equation with the fixed endpoint control. Boundary Value Problems, (2015). doi: 10.1186/s13661-015-0476-4.

[26] M. Milla Miranda;Exact controllability for the wave equation in domains with variable bound- ary, Rev. Mat. Univ.,9(1996), 435-457.

[27] M. Milla Miranda;HUM and the wave equation with variable coeficients, Asymptotic Analysis 11, (1995), 317-341.

[28] R. Glowinski, A. Ramos, J. Periaux;Nash equilibria for the multi-objective control of linear differential equations, Journal of Optimization Theory and Applications112(3) (2002), 457- 498.

[29] R. Glowinski, A. Ramos, J. Periaux;Pointwise Control of the Burgers Equation and Related Nash Equilibrium Problems: Computational Approach, Journal of Optimization Theory and Applications112(3) (2002), 499-516.

[30] R. Rockafellar;Convex Analysis, Princeton University Press, Princeton, N. J., 1969.

[31] Z. Yang, Z. Feng; Approximate controllability of Euler-Bernoulli viscoelastic systems, Eletronic Journal of Differential Equations,1012019 (2019), no. 19, 1-16.

Addendum posted on July 21, 2020

In response to a reader’s comments, the first author wants to indicate that the statements from page 3 line 25 to page 4 line 18 are quoted from the article

Mokhtari Yacine;Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical do- mains, Evolution Equations & Control Theory, doi: 10.3934/eect.2021004.

End of addendum.

Isa´ıas P. de Jesus

Universidade Federal do Piau´ı, DM, Teresina, PI, Brazil Email address:[email protected]

Eugenio Cabanillas Lapa

Universidad Nacional Mayor de San Marcos, Facultad de Ciencias Matem´aticas Insti- tuto de Investigaci´on, Lima, Per´u

Email address:[email protected]

Juan Limaco

Universidade Federal Fluminense, IME, Niter´oi, RJ, Brazil Email address:[email protected]