Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 60, pp. 1–12. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu CONTROLLABILITY FOR THE WAVE EQUATION WITH MOVING BOUNDARY ISAÍAS 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 speed k(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 of hierarchic 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], given T > 0, we consider the non-cylindrical domain defined by Q̂ = {(x, t) ∈ R2; 0 < x < αk(t), t ∈ (0, T )}, where αk(t) = 1 + kt, 0 < k < 1. Its lateral boundary is defined by Σ̂ = Σ̂0 ∪ Σ̂∗0, with Σ̂0 = {(0, t); t ∈ (0, T )}, Σ̂∗0 = Σ̂\Σ̂0 = {(αk(t), t); t ∈ (0, T )}. We also represent by Ωt and 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 domain Q̂: u′′ − uxx = 0 in Q̂, u(x, t) = { w̃(t) on Σ̂0, 0 on Σ̂∗0, u(x, 0) = u0(x), u′(x, 0) = u1(x) in Ω0, (1.1) where u is the state variable, w̃ is the control variable and (u0(x), u1(x)) ∈ L2(0, 1)× H−1(0, 1). By u′ = u′(x, t) we represent the derivative ∂u ∂t and by uxx = uxx(x, t) 2010 Mathematics 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 2 I. P. DE JESUS, E. CABANILLAS LAPA, J. LIMACO EJDE-2021/60 the second order partial derivative ∂2u ∂x2 . Equation (1.1) models the motion of a string with a fixed endpoint and a moving one. The constant k 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, let α(t) = t+ αk(t), β(t) = t− αk(t) and γ = 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 w̃1 ∈ L2(Σ̂1) and w̃2 a Nash equilibrium in the sense (1.10). Then the pair (u(T ), u′(T )) = (u(., T, w̃1, w̃2), u′(., T, w̃1, w̃2)) , where u solves the system (2.8), generates a dense subset of L2(ΩT )×H−1(ΩT ). As in [19], we divide Σ̂0 into two parts Σ̂0 = Σ̂1 ∪ Σ̂2, (1.4) and consider w̃ = {w̃1, w̃2}, w̃i = control function in L2(Σ̂i), i = 1, 2. (1.5) We can also write w̃ = w̃1 + w̃2, with Σ̂0 = Σ̂1 = Σ̂2. (1.6) Thus, we observe that system (1.1) can be rewritten as follows: u′′ − uxx = 0 in Q̂, u(x, t) =  w̃1(t) on Σ̂1, w̃2(t) on Σ̂2, 0 on Σ̂\Σ̂0, u(x, 0) = u0(x), u′(x, 0) = u1(x) in Ω0. (1.7) In decomposition (1.4), (1.5) we establish a hierarchy. We think of w̃1 as being the “main” control, the leader, and we think of w̃2 as the follower, in Stackelberg terminology. Associated with the solution u = u(x, t) of (1.7), we will consider the (secondary) functional J̃2(w̃1, w̃2) = 1 2 ∫ ∫ Q̂ (u(w̃1, w̃2)− ũ2) 2 dx dt+ σ̃ 2 ∫ Σ̂2 w̃2 2 dΣ̂, (1.8) and the (main) functional J̃(w̃1) = 1 2 ∫ Σ̂1 w̃2 1 dΣ̂, (1.9) where σ̃ > 0 is a constant and ũ2 is a given function in L2(Q̂). EJDE-2021/60 CONTROLLABILITY FOR THE WAVE EQUATION 3 Now, let us describe the Stackelberg-Nash strategy. Thus, for each choice of the leader w̃1, we try to find a Nash equilibrium for the cost functional J̃2, that is, we look for a control w̃2 = F(w̃1), depending on w̃1, satisfying J̃2(w̃1, w̃2) = inf ŵ2∈L2(Σ̂2) J̃2(w̃1, ŵ2). (1.10) After this, we consider the state u (w̃1,F(w̃1)) given by the solution of u′′ − uxx = 0 in Q̂, u(x, t) =  w̃1 on Σ̂1, F(w̃1) on Σ̂2, 0 on Σ̂\Σ̂0, u(x, 0) = u0(x), u′(x, 0) = u1(x) in Ω0. (1.11) We will look for any optimal control w̃1 such that J̃(w̃1,F(w̃1)) = inf w1∈L2(Σ̂1) J̃(w1,F(w̃1)), (1.12) subject to the following restriction of the approximate controllability type (u(x, T ; w̃1,F(w̃1)), u′(x, T ; w̃1,F(w̃1))) ∈ BL2(ΩT )(u 0, ρ0)×BH−1(ΩT )(u 1, ρ1), (1.13) where BX(C, r) denotes the ball in X with center C and radius r. The control problems to be studied in this paper are described as follows: Problem 1 Fixed any leader control w̃1, find the follower control w̃2 = F(w̃1) (de- pending on w̃1) and the associated state u, solution of (1.7) satisfying the condition (1.10) (Nash equilibrium) related to J̃2, defined in (1.8). Problem 2 Assuming that the existence and uniqueness of the Nash equilib- rium w̃2 was proved, then when w̃1 varies in L2(Σ̂1), prove that the solutions (u(x, t; w̃1, w̃2), u′(x, t; w̃1, w̃2)) of the state equation (1.7), evaluated at t = T , that is, (u(x, T ; w̃1, w̃2), u′(x, T ; w̃1, w̃2)), generate a dense subset of L2(ΩT )×H−1(ΩT ). Remark 1.2. By the linearity of system (1.11), without loss of generality we may assume that u0 = 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) = w̃(t), 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 > 2 1−k . In these papers, only a sufficient condition is provided for the exact controllability. 4 I. P. DE JESUS, E. CABANILLAS LAPA, J. LIMACO EJDE-2021/60 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 ≥ 2 (1−k)(1−r) . More general boundary functions are considered in [3] with boundary conditions u(0, t) = 0, u(s(t), t) = w̃(t), t ∈ (0,∞), where s : [0,∞) → (0,∞) is assumed to be C1 function with ‖s′‖L∞(0,∞) < 1. Furthermore, it has been assumed that s must 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 if T ≥ s+ ◦(s−)−1(0), where s±(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) = w̃(t), 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 control w̃1 ∈ L2(Σ̂1) we determine the existence and uniqueness of solutions to the problem inf w̃2∈L2(Σ̂2) J̃2(w̃1, w̃2), (2.1) and a characterization of this solution in terms of an adjoint system. For this, we consider Uad = {(u, w̃2) ∈ L2(Q) × L2(Σ̂2);u solution of (1.7)}, and J̃2 : Uad → R defined by (1.8). Note that Uad is a nonempty closed convex subset of L2(Q) × L2(Σ2), and J̃2 is weakly coercive, weakly sequentially lower semicontinuous and strictly convex. Therefore, there exists a unique solution w̃2 of (2.1), i.e., J̃2(w̃1, w̃2) = inf ŵ2∈L2(Σ̂2) J̃2(w̃1, ŵ2). EJDE-2021/60 CONTROLLABILITY FOR THE WAVE EQUATION 5 The Euler-Lagrange equation for problem (2.1) is∫ T 0 ∫ Ωt (u− ũ2)û dx dt+ σ̃ ∫ Σ̂2 w̃2ŵ2dΣ̂ = 0, ∀ŵ2 ∈ L2(Σ̂2), (2.2) where û is solution of the system û′′ − ûxx = 0 in Q̂, û =  0 on Σ̂1, ŵ2 on Σ̂2, 0 on Σ̂\(Σ̂1 ∪ Σ̂2), û(x, 0) = 0, û′(x, 0) = 0, x in Ω0. (2.3) To express (2.2) in a convenient form, we introduce the adjoint state defined by p′′ − pxx = u− ũ2 in Q̂, p(T ) = p′(T ) = 0, x in ΩT , p = 0 on Σ̂. (2.4) Multiplying (2.4) by û and integrating by parts, we find that∫ T 0 ∫ Ωt (u− ũ2)û dx dt+ ∫ Σ̂2 px ŵ2 dΣ̂ = 0, (2.5) so that (2.2) becomes px = σ̃ w̃2 on Σ̂2. (2.6) We summarize these results in the following theorem. Theorem 2.1. For each w̃1 ∈ L2(Σ1) there exists a unique Nash equilibrium w̃2 in the sense of (1.10). Moreover, the follower w̃2 is given by w̃2 = F(w̃1) = 1 σ̃ px on Σ̂2, (2.7) where {v, p} is the unique solution of (the optimality system) u′′ − uxx = 0 in Q̂, p′′ − pxx = u− ũ2 in Q̂, u =  w̃1 on Σ̂1, 1 σ̃px on Σ̂2, 0 on Σ̂\Σ̂0, p = 0 on Σ̂, u(0) = u′(0) = 0, x in Ω0, p(T ) = p′(T ) = 0, x in ΩT . (2.8) Of course, {u, p} depends on w̃1: {u, p} = {u(w̃1), p(w̃1)}. (2.9) 6 I. P. DE JESUS, E. CABANILLAS LAPA, J. LIMACO EJDE-2021/60 3. Proof of Theorem 1.1 Since we have proved the existence, uniqueness and characterization of the fol- lower w̃2, the leader w̃1 now wants that solutions u and u′, evaluated at time t = 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 ∫ Σ̂1 w̃2 1 dΣ̂, (3.1) where w̃1 is subject to (u(T ; w̃1), u′(T ; w̃1)) ∈ BL2(ΩT )(u 0, ρ0)×BH−1(ΩT )(u 1, ρ1), (3.2) assuming that w̃1 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) ∈ H1 0 (Ω0)× L2(Ω0), then there exists γ(0) > 0 such that for T > γ(0), the weak solution of problem z′′ − zxx = 0 in Q̂, z = 0 on Σ̂, z(0) = g0, z′(0) = g1 in Ω0 (3.3) satisfies ∫ γ(0) 0 |zx(0, t)|2 dt ≥ C(|g0|2 + |g1|2), (3.4) where C is given in [3]. Proof. We construct a solution of the form z(x, t) = ∑ n∈Z An(e2ξinϕ(t+x) − e2ξinϕ(t−x)), (3.5) where ϕ ∈ C2 is a solution to the functional equation ϕ(t+ αk(t))− ϕ(t− αk(t)) = 1. We let F (x) = ∫ x 0 g1(s)ds and 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 An = ∫ 1 −1 h(x)e2ξinϕ(x)ϕ′(x) dx. Then z ∈ C2 and their derivatives can be calculated term by term. Let us define some often appearing values: m(t) = min{ϕ′(x) : x ∈ [t− αk(t), t+ αk(t)]}, M(t) = max{ϕ′(x) : x ∈ [t− αk(t), t+ αk(t)]}. Now, we adapt the proof of [3, Theorem 2.1], with 0 < k < 1. Indeed, we consider β(t) = t − αk(t). Then β′(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 + αk(t0) = γ(0). EJDE-2021/60 CONTROLLABILITY FOR THE WAVE EQUATION 7 Differentiating z term by term in (3.5), and evaluating at x = 0, for all τ > 0 we obtain |zx(0, t)|2L2(0,τ0) ≥ m(t0)|zx(0, t)|2L2(0,τ0, 1 ϕ′(t) ) = m(t0)16ξ2 ∑ 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 for αk(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) where ϑ0, p0 is given by ϑ′′0 − (ϑ0)xx = 0 in Q̂, ϑ0 =  0 on Σ̂1, 1 σ̃ (p0)x on Σ̂2, 0 on Σ̂\Σ̂0, ϑ0(0) = ϑ′0(0) = 0, x in Ω0, (3.8) and p′′0 − (p0)xx = u0 − ũ2 in Q̂, p0 = 0 on Σ̂, p0(T ) = p′0(T ) = 0, x in ΩT ; (3.9) And {g, q} is given by g′′ − gxx = 0 in Q̂, g =  w̃1 on Σ̂1, 1 σ̃ qx on Σ̂2, 0 on Σ̂\Σ̂0, g(0) = g′(0) = 0, x in Ω0, (3.10) and q′′ − qxx = g in Q̂, q = 0 on Σ̂, q(T ) = q′(T ) = 0, x in ΩT . (3.11) We next set A : L2(Σ̂1)→ H−1(ΩT )× L2(ΩT ) as Aw̃1 = { g′(T ; w̃1), −g(T ; w̃1) } , (3.12) which defines A ∈ L ( L2(Σ̂1); H−1(ΩT )× L2(ΩT ) ) . Using (3.7) and (3.12), we can rewrite (3.2) as Aw̃1 ∈ {−ϑ′0(T ) +BH−1(ΩT )(u 1, ρ1), −ϑ0(T ) +BL2(ΩT )(u 0, ρ0)}. (3.13) 8 I. P. DE JESUS, E. CABANILLAS LAPA, J. LIMACO EJDE-2021/60 We will show that Aw̃1 generates a dense subspace of H−1(ΩT ) × L2(ΩT ). For this, let {f0, f1} ∈ H1 0 (ΩT )×L2(ΩT ) and consider the following systems (“adjoint states”): ϕ′′ − ϕxx = ψ in Q̂, ϕ = 0 on Σ̂, ϕ(T ) = f0, ϕ′(T ) = f1, x in ΩT , (3.14) and ψ′′ − ψxx = 0 in Q̂, ψ =  0 on Σ̂1, 1 σ̃ϕx on Σ̂2, 0 on Σ̂\Σ̂0, ψ(0) = ψ′(0) = 0, x in Ω0. (3.15) Multiplying (3.15)1 by q, (3.14)1 by g, where q, g solve (3.11) and (3.10), respec- tively, and integrating in Q̂ we obtain∫ T 0 ∫ Ωt g ψ dx dt = − 1 σ̃ ∫ Σ̂2 qxϕxdΣ̂, (3.16) 〈g′(T ), f0〉H−1(ΩT )×H1 0 (ΩT ) − ( g(T ), f1 ) = − ∫ Σ̂1 ϕx w̃1 dΣ̂. (3.17) Considering the left-hand side of this equation as the inner product of {g′(T ),−g(T )} and {f0, f1} in H−1(ΩT )× L2(ΩT ) and H1 0 (ΩT )× L2(ΩT ), we obtain 〈〈Aw̃1, f〉〉 = − ∫ Σ̂1 ϕx w̃1 dΣ̂, where 〈〈·, ·〉〉 represent the duality pairing betweenH−1(ΩT )×L2(ΩT ) andH1 0 (ΩT )× L2(ΩT ). Therefore, if 〈〈A w̃1, f〉〉 = 0, for all w̃1 ∈ L2(Σ̂1), then ϕx = 0 on Σ̂1. (3.18) Hence, in case (1.6), ψ = 0 on Σ̂, so that ψ ≡ 0. (3.19) Therefore, ϕ′′ − ϕxx = 0, ϕ = 0 on Σ̂, (3.20) and satisfies (3.18). Therefore, by Lemma 3.1 and by Remark 3.2, with z(x, t) = ϕ(x, T − t), g0 = f0, and g1 = f1 we have that f0 = 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 w̃2 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. EJDE-2021/60 CONTROLLABILITY FOR THE WAVE EQUATION 9 Theorem 4.1. Assume that the hypotheses (1.2), (1.3) and (1.6) are satisfied. Then for {f0, f1} in H1 0 (ΩT )× L2(ΩT ) we uniquely define {ϕ,ψ, u, p} by ϕ′′ − ϕxx = ψ in Q̂, ψ′′ − ψxx = 0 in Q̂, u′′ − uxx = 0 in Q̂, p′′ − pxx = u− ũ2 in Q̂, ϕ = 0 on Σ̂, ψ =  0 on Σ̂1, 1 σ̃ ϕx on Σ̂2, 0 on Σ̂\Σ̂0, u =  −ϕx on Σ̂1, 1 σ̃px on Σ̂2, 0 on Σ̂\Σ̂0, p = 0 on Σ̂, ϕ(·, T ) = f0, ϕ′(·, T ) = f1 in ΩT , u(0) = u′(0) = 0 in Ω0, p(T ) = p′(T ) = 0 in ΩT . (4.1) We uniquely define {f0, f1} as the solution of the variational inequality〈 u′(T, f)− u1, f̂0 − f0 〉 H−1(ΩT )×H1 0 (ΩT ) − ( u(T, f)− u0, f̂1 − f1 ) + ρ1 ( ‖f̂0‖ − ‖f0‖ ) + ρ0 ( |f̂1| − |f1| ) ≥ 0, ∀f̂ ∈ H1 0 (ΩT )× L2(ΩT ). (4.2) Then the optimal leader is w̃1 = −ϕx on Σ̂1, where ϕ corresponds to the solution of (4.1). Proof. Let A be the continuous linear operator defined by (3.12) and we introduce the following two convex proper functions: F1 : L2(Σ̂1)→ R ∪ {∞} by F1(w̃1) = 1 2 ∫ Σ̂1 w̃2 1 dΣ̂ (4.3) and F2 : H−1(ΩT )× L2(ΩT )→ R ∪ {∞} by F2(ξ, µ) =  0, if (ξ, µ) ∈ u1 − ϑ′0(T ) + ρ1BH−1(ΩT ) −u0 + ϑ0(T )− ρ0BL2(ΩT ), +∞, otherwise. (4.4) With these notation, problems (3.1)–(3.2) become equivalent to inf w̃1∈L2(Σ̂1) [ F1(w̃1) + F2(Aw̃1) ] (4.5) provided that we prove that the range of A is dense in H−1(ΩT )× L2(ΩT ), under conditions (1.2) and (1.3). 10 I. P. DE JESUS, E. CABANILLAS LAPA, J. LIMACO EJDE-2021/60 By the Duality Theorem of Fenchel and Rockafellar [30] (see also [10, 13]), we have inf w̃1∈L2(Σ̂1) [F1(w̃1) + F2(Aw̃1)] = − inf (f̂0,f̂1)∈H1 0 (ΩT )×L2(ΩT ) [F ∗1 ( A∗{f̂0, f̂1} ) + F ∗2 {−f̂0,−f̂1}], (4.6) where F ∗i is the conjugate function of Fi (i = 1, 2) and A∗ the adjoint of A. We have A∗ : H1 0 (ΩT )× L2(ΩT )→ L2(Σ̂1) as (f0, f1) 7→ A∗f = −ϕx, (4.7) where ϕ is given in (3.14). We see easily that F ∗1 (w̃1) = F1(w̃1) (4.8) and F ∗2 ({f̂0, f̂1}) = 〈u1 − ϑ′0(T ), f̂0〉H−1(ΩT )×H1 0 (ΩT ) + ( ϑ0(T )− u0, f̂1 ) + ρ1‖f̂0‖+ ρ0|f̂1|. (4.9) Therefore the (opposite of) right-hand side of (4.6) is given by − inf f̂∈H1 0 (ΩT )×L2(ΩT ) {1 2 ∫ Σ̂1 ϕ2 xdΣ̂ + ( u0 − ϑ0(T ), f̂1 ) − 〈u1 − ϑ′0(T ), f̂0〉H−1(ΩT )×H1 0 (ΩT ) + ρ1‖f̂0‖+ ρ0|f̂1| } . 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. Isáıas 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. 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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. 12 I. P. DE JESUS, E. CABANILLAS LAPA, J. LIMACO EJDE-2021/60 End of addendum. Isáıas P. de Jesus Universidade Federal do Piaúı, DM, Teresina, PI, Brazil Email address: isaias@ufpi.edu.br Eugenio Cabanillas Lapa Universidad Nacional Mayor de San Marcos, Facultad de Ciencias Matemáticas Insti- tuto de Investigación, Lima, Perú Email address: cleugenio@yahoo.com Juan Limaco Universidade Federal Fluminense, IME, Niterói, RJ, Brazil Email address: jlimaco@id.uff.br 1. Introduction 1.1. Statement of the problem and main result 1.2. Related problems 2. Optimal system for the follower control 3. Proof of Theorem 1.1 4. Optimal system for the leader control Acknowledgments References Addendum posted on July 21, 2020