Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 49, pp. 1–12. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXACT BOUNDARY CONTROLLABILITY FOR THE WAVE EQUATION WITH MOVING BOUNDARY DOMAINS IN A STAR-SHAPED HOLE RUIKSON S. O. NUNES Abstract. We consider an exact boundary control problem for the wave equa- tion in a moving bounded domain which has a star-shaped hole. The boundary domain is composed by two disjoint parts, one is the boundary of the hole, which is fixed, and the other one is the external boundary which is moving. The initial data has finite energy and the control obtained is square integrable and is obtained by means of the conormal derivative. We use the method of controllability presented by Russell in [20], and assume that the control acts only in the moving part of the boundary. 1. Introduction There is a large number of works available for the exact boundary controllabil- ity problems of wave equations, because such problems have a great importance both from a practical and theoretical point of view. Since the 1970s, many works have appeared giving seminal contributions to the advancement of this branch of mathematics, among them we have [3, 12, 18] which are canonical references for the subject. Many works have appeared dealing with control problems for wave equations on Euclidean domains with diverse types of geometry, for example, do- mains with fixed and moving boundary and domains with perforated interior; see [2, 4, 6, 7, 11, 13, 17]. In this work we study an exact boundary control problem for the standard wave equation on a domain with moving boundary which has a single fixed hole. The boundary of such domains is composed by two disjoint parts: one it is the boundary on hole which is fixed, and the other one is the external boundary which is moving. We shall consider the control acting only on the moving boundary part. In practical situations, many processes involve domains with a geometry as described above. For example, a flexible body that is crossed by a cylindrical pillar and is fixed to it. Without any variation in the temperature of the environment the body has no dilation and thus its external boundary remains static. However, if there is a variation in the temperature, the body would have a dilation or a contraction, causing the mobility of the its external boundary. In this work when we deal with 2010 Mathematics Subject Classification. 35L05, 35L20, 35L53, 35B40, 93B05, 49J20. Key words and phrases. Wave equation; energy decay; exact boundary controllability; non-cylindrical domains; holed domains. c©2021 Texas State University. Submitted August 16, 2020. Published June 1, 2021. 1 2 R. S. O. NUNES EJDE-2021/49 a domain with a hole, and we refer by external boundary as being the part of the boundary of the domain that does not coincide with that one of the hole. To establish these concepts in more detail, we consider B ⊂ Rn, n ≥ 2, a convex compact set with the origin in its interior with smooth boundary Γ0. We set Ω∞ = Rn − B. Let Ξ ⊂ Rn be a simply connected bounded domain with piecewise smooth boundary Γ1, with no cusps, such that B ⊂ Ξ. We assume that dist(Γ0,Γ1) ≥ ε > 0 and set Ω = Ξ−B. Hence, the boundary of Ω is ∂Ω = Γ0∪Γ1. Note that Ω is a holed domain whose hole has the shape of B. We also consider the moving boundary domain Ξt ⊂ Rn where Ξt = {x ∈ Rn : x = α(t)y, y ∈ Ξ}, t ∈ [0,+∞) whose boundary is denoted by Γt and Ξ0 = Ξ. Here α : R+∪{0} → R is a piecewise bounded smooth function, where Ξt × R ⊂ ∪x∈Ξ{(x, t) ∈ RN × R : |x− x|2 ≤ t2}, (1.1) with B ⊂ Ξt and dist(Γ0 ∩Γt) ≥ ε > 0, for all t > 0. The boundedness of α implies in the existence of r > 0 such that Ξt ⊂ B(0, r) for all t ∈ [0,+∞). Defining Ω̃ = B(0, r)−B and Ωt = Ξt −B we can see that Ωt ⊂ Ω̃ for all t ∈ [0,+∞). The boundary of Ωt is ∂Ωt = Γ0∪Γt. Now, for T > 0, let us consider the non-cylindrical domain of Rn+1, QT = ∪0 0 sufficiently large and a control function h(·, t) ∈ L2(ΣT ) such that the solution u ∈ H1(QT ) of the problem utt −∆u = 0 in QT u(·, 0) = f, ut(·, 0) = g, in Ω u(·, t) = 0, on Γ0 × [0, T ] νtut −∇u · νx = h(·, t), on ΣT . (1.2) satisfy the final condition u(·, T ) = 0 = ut(·, T ) in ΩT . (1.3) EJDE-2021/49 EXACT BOUNDARY CONTROLLABILITY FOR THE WAVE EQUATION 3 In the literature some papers deal with exact boundary controllability problems for wave equation on domains with holes. However, as far as we know, there are no papers dealing with such problems in holed moving domains, as it is proposed in the present paper. In this paper [12] the author seeking to show the wide applicability of the HUM method, considers control problems on domains with different types of geometry. In particular, he has shown how the HUM can be applied to solve some exact boundary control problems on a fixed domain which has a hole. In [6] also using the HUM method the authors have considered exact boundary control problems for wave equation in domains which have one or a family of small holes. In the literature there are also many works dealing with exact boundary control problems on domains that has a mobile boundary but with no hole; see [2, 4, 13] and citing references. At this point we highlight the contribution of the present paper by making available in the literature a work that deals with exact control problems on non-cylindrical holed domains. Russell [20] developed a technique, based in [19], for studding an exact boundary control problem for wave equation with control acting only on a part of boundary of the domain. Here we shall utilize such technique to obtain the desirable exact boundary control problem proposed in Theorem 1.1. The applicability of Russell’s method requires some properties of the system to be considered, the principals are: linearity, time reversibility, local energy decay in a exterior domain (obtained via [10, 14]) and suitable traces theorems that are obtained in [21]. As seen above the Russell’s method requires many properties of the system to be controlled but it has the advantage of requiring very little on the geometry of the domain. From this fact we can consider the limiting function α, defined above, to be only piecewise smooth. Here, it is proposed that the boundary of the domain be comprise two disjoint parts: the internal part (the boundary of the hole) fixed, and the other part which is moving. An interesting point is to consider an exact boundary control problem for wave equation on a holed domain where both external and internal boundary moving. In the literature there are papers [8, 9] that obtain local energy decay estimates for the wave equation in the exterior of a domain with moving boundary. Another interesting point it is to study on exact boundary control problems, in holed domains, for systems of coupled waves equations as proposed in [5, 16]. We intend to return on this questions in posterior works. The rest of this article is organized as follows. In Section 2 we presents a brief summary with respect to trace and extension properties. In Section 3 we obtain local energy decay estimates for the wave equation in exterior domain. In Section 4 we explore exact boundary controllability results in a holed domain with fixed boundary. In Section 5 we prove Theorem 1.1. 2. Extension and traces In this section, we make a brief presentation about the trace and extension theorems, such properties are essential in the proof of Theorem 1.1. Remembering that we are considering Ω as a holed domain whose its boundary is performed by two disjoint parts which is denoted by Γ0 and Γ1 respectively, with dist(Γ0,Γ1) > 0 and that Γ0 is smooth surface and Γ1 a surface smooth by parts with no cusps. In this work we shall use the following extension lemma. 4 R. S. O. NUNES EJDE-2021/49 Lemma 2.1. Let Ω ⊂ Ω∞ as defined above and V ⊂ Ω∞ an open set such that Ω ⊂ V . So, there exist a bounded linear operator E : H1(Ω)×L2(Ω)→ H1(Ω∞)× L2(Ω∞) such that for u ∈ H1(Ω)× L2(Ω):Please define A b B (1) Eu = u in Ω; (2) suppEu b V ; (3) ‖Eu‖H1(Ω∞)×L2(Ω∞) ≤ C‖u‖H1(Ω)×L2(Ω), where C is a positive real con- stant independent on u. Proof. To sketch the proof let us firstly to define the projection operators Pi−1 : H1(Ω)× L2(Ω)→ Hi−1(Ω∞), i = 1, 2. As we are considering dist(Γ0,Γ1) > 0 and Γ1 is a piecewise smooth surface, we obtain a finite cover (Ui) k i=1 of Γ1 where Ui ⊂ Ω∞, i = 1, · · · , k. So, we can adapt the proof of the [15, Theorems 3.9 and 3.10] to obtain the extension operators E1 : H1(Ω)→ H1(Ω∞), where E1u = u in Ω with ‖E1u‖H1(Ω∞) ≤ C‖u‖H1(Ω). On the other hand, using the classical extension by zero out Ω, we obtain the extension operator E0 : L2(Ω) → L2(Ω∞) where ‖E0u‖L2(Ω∞) ≤ ‖u‖L2(Ω). Thus, we obtain the desirable extension operator by defines E = (E1P1, E0P0). � Next we mention a result on the regularity of the traces of the solution of the wave equation which it is essential in the proof of Theorem 1.1. Let us begin with some notation and definitions. Let P (ξ,D) be a linear second order hyperbolic partial differential equation with C∞ coefficients depending on ξ in some open bounded domain Ξ ⊂ RN . Being Σ ⊂ Ξ an oriented smooth hypersurface which is time-like and non-characteristic with respect to P (ξ,D). Let η = (η1, · · · , ηN ) be a unit normal to Σ. If ∑ aij ∂2 ∂ξi∂ξj is the principal part of P (ξ,D), then the expression ∂u ∂η = ∑ aij ∂u∂ξi ηj defines the conormal derivative of u relative to the P (ξ,D) along Σ. An important fact it is to know what the regularity of the traces of the conormal derivative on surfaces, for this purpose we turn to [21]. Considering Ξ ⊂ RN , with N ≥ 2, [21, Theorem 2] proves that if u ∈ H1 loc(Ξ) is such that P (ξ,D)u ∈ L2 loc(Ξ) then ∂u ∂η ∈ L 2 loc(Σ). Particularly, if we consider P (ξ,D) as being the standard wave operator, its principal part will be ∂2 ∂t2 − ∑N i=1 ∂2 ∂x2 i . Now, if γ is a smooth hypersurface in RN we consider the surface γ × R whose unit normal vector is ν = (νx, νt), where νx = (ν1, · · · , νN ). In this case the conormal derivative of u along γ × R is ∂u ∂ν = νtut −∇u · νx. Particularly, if we apply the trace result mentioned in the previous paragraph for the wave operator we obtain the following result. Lemma 2.2. Let u be the solution of the initial-boundary value problem utt −∆u = 0 in Ω∞ × R u(·, 0) = u0, ut(·, 0) = u1, in Ω∞ u(·, t) = 0, on Γ0 × R (2.1) with initial data (u0, u1) ∈ H1(Ω∞) × L2(Ω∞), where supp(u0), supp(u1) b Ω∞. Let γ be a smooth hypersurface in Ω∞, with no self intersection and considers the surface γ × R which the unit normal vector is ν = (νx, νt). Then the conormal derivative of u along γ × R has trace νtut −∇u · νx ∈ L2 loc(γ × R). EJDE-2021/49 EXACT BOUNDARY CONTROLLABILITY FOR THE WAVE EQUATION 5 3. Local energy decay There are many publications dedicated to obtain local energy decay estimates for hyperbolic equations in exterior domains; see [10, 14, 22] and references there in. In this paper such estimates play a fundamental role in the proof of Theorem 1.1. So, we dedicate this section to such subject. Let us consider the initial initial-boundary value problem utt −∆u = 0 in Ω∞ × (0,+∞) u(·, 0) = f, ut(·, 0) = g, in Ω∞ u(·, t) = 0, in Γ0 × (0,+∞). (3.1) Let O ⊂ Ω∞ a bounded domain, the energy of the solution u of the (3.1) confined in O is defined by E(t,O, u) = 1 2 ∫ O [|∇u|2 + u2 t ](x, t)dx. (3.2) If there exist a positive constant C and a function p(t) such that E(t,O, u) ≤ Cp(t)E(0,O, u), (3.3) with p(t) → 0, as t → +∞, we say the energy of (3.1) decays locally. Adapting process in [14, 10] we show the validity of the following local energy decay estimate. Lemma 3.1. Let (f, g) ∈ H1(Ω∞) × L2(Ω∞) with supp(f), supp(g) ⊂ O ⊂ Ω∞, then there exist a positive real constant K, independent of f and g, such that the solution u of (3.1) satisfies ‖u(·, t)‖2H1(O) + ‖ut(·, t)‖2L2(O) ≤ K t−R { ‖u(·, 0)‖2H1(O) + ‖ut(·, 0)‖2L2(O) } , (3.4) for t > R sufficiently large, where R is such that O ⊂ ΩR being B(0, R) the ball of center 0 and radius R and ΩR = B(0, R) ∩ Ω∞. The proof of Lemma 3.1 follows the ideas presented by Morawetz [14], so it is essential to prove some preliminaries results. The compactness of the initial data f and g implies finite propagation speed property for solution of the system (3.1), that is the solution u of (3.1), in the instant t, has support contained in some bounded region of the space Ω∞. So if we take R > 0 and the ball B(0, R) of center 0 and radius R, such that supp(f)∪supp(g) ⊂ ΩR = B(0, R)∩Ω∞, then from some t > R the function u(x, t) as well as its derivatives are null for x ∈ Ω∞ with |x| ≥ t. Other important and classical property of the solution u of the system (3.1) is that his total energy E(t,Ω∞, u) = 1 2 ∫ Ω∞ [|∇u|2 + u2 t ](x, t)dx, (3.5) is conserved with respect to t, that is, E(t,Ω∞, u) = E(0,Ω∞, u), for all t > 0. (3.6) In the proof of estimate (3.4) we will used the following result. 6 R. S. O. NUNES EJDE-2021/49 Lemma 3.2. Let u be the solution of the boundary-value problem (3.1), if w is the solution of the problem wtt −∆w = 0 in Ω∞ × (0,+∞) w(·, 0) = h, wt(·, 0) = f, in Ω∞ w(·, t) = 0, in Γ0 × (0,+∞) (3.7) with ∆h = g and h = 0 in B. Then wt = u. Proof. Note that v = wt satisfies vtt −∆v = 0 in Ω∞ × (0,+∞) v(·, 0) = f, vt(·, 0) = ∆h, in Ω∞ v(·, t) = 0, in Γ0 × (0,+∞) (3.8) with ∆h = g and h = 0 on B. Then v = wt satisfies (3.1) and by the uniqueness of solution we conclude that v = wt = u � From Lemma 3.2 we obtain the estimate∫ Ω∞ |u(x, t)|2dx = ∫ Ω∞ |wt(x, t)|2dx ≤ E(t,Ω∞, wt) = E(0,Ω∞, wt) ≤ CE(0,Ω∞, u). (3.9) Now, for T > 0, let D = Ω∞ × [0, T ] be an exterior region which boundary is ∂D = ∂D1 ∪ ∂D2 ∪ ∂D3, where ∂D1 = Ω∞ × {0}, ∂D2 = Ω∞ × {T}, ∂D3 = Γ0 × [0, T ]. Multiplying the equality utt −∆u = 0 by (x · ∇u) + tut + N−1 2 u and integrating it on D, from the Gauss’ divergence formula, as in [10], we obtain tE(t,Ω∞, u) = ∫ ΩR (x · ∇u(·, 0))ut(·, 0)dx+ (N − 1) 2 ∫ ΩR u(·, 0)ut(·, 0)dx − ∫ Ω∞ (x · ∇u(·, t))ut(·, t)dx− (N − 1) 2 ∫ Ω∞ u(·, t)ut(·, t)dx + 1 2 ∫ D3 {x · ν(·)} ∣∣∂u ∂ν (., s) ∣∣2 dx ds. (3.10) Since B = Ωc∞ is star-shaped with respect to origin, that is {x · ν(·)} ≤ 0, it follow that 1 2 ∫ D3 {x · ν(·)} ∣∣∂u ∂ν (., s) ∣∣2 dx ds ≤ 0. So, from (3.10) we obtain the estimate tE(t,Ω∞, u) ≤ ∫ ΩR (x · ∇u(·, 0))ut(·, 0)dx+ (N − 1) 2 ∫ B(0,R) u(·, 0)ut(·, 0)dx − ∫ Ω∞ (x · ∇u(·, t))ut(·, t)dx− (N − 1) 2 ∫ Ω∞ u(·, t)ut(·, t)dx. (3.11) EJDE-2021/49 EXACT BOUNDARY CONTROLLABILITY FOR THE WAVE EQUATION 7 Now observe that∣∣ ∫ Ω∞ (x · ∇u(·, t))ut(·, t)dx ∣∣ ≤ R ∫ ΩR |∇u(·, t)| |ut(·, t)|dx+ ∫ |x|≥R |x||∇u(·, t)||ut(·, t)|dx. Because of the finite propagation speed property, we can take t > R such that u(x, t) and its derivatives are null, so∫ |x|≥R |x||∇u(·, t)||ut(·, t)|dx = ∫ t≥|x|≥R |x||∇u(·, t)||ut(·, t)|dx ≤ t 2 ∫ |x|≥R {|∇u(·, t)|2 + |ut(·, t)|2}dx. Therefore,∣∣ ∫ Ω∞ (x · ∇u(·, t))ut(·, t)dx ∣∣ ≤ R 2 ∫ ΩR {|∇u(·, t)|2 + |ut(·, t)|2}dx+ t 2 ∫ |x|≥R {|∇u(·, t)|2 + |ut(·, t)|2}dx. (3.12) On the other hand,∣∣ ∫ Ω∞ u(·, t)ut(·, t)dx ∣∣ ≤ 1 2 ∫ Ω∞ [ |u(·, t)|2 + |ut(·, t)|2 ] dx. (3.13) Joining (3.9) and (3.13) we obtain∣∣ ∫ Ω∞ u(·, t)ut(·, t)dx ∣∣ ≤ CE(0,Ω∞, u), (3.14) where C is a positive constant independent of the initial data f and g. The following inequalities are also valid∣∣ ∫ ΩR (x · ∇u(·, 0))ut(·, 0)dx ∣∣ ≤ C(R)E(0,ΩR, u), (3.15) ∣∣ ∫ ΩR u(·, 0)ut(·, 0)dx ∣∣ ≤ C(R)E(0,ΩR, u), (3.16) where C(R) is a positive constant which vary from line to line and depend on R but not on of the initial data f and g. Now, joining and manipulating the the inequalities (3.11)-(3.16) we obtain tE(t,ΩR, u) + tE(t, |x| ≥ R, u) ≤ CE(0,ΩR, u) +RE(t,ΩR, u) +tE(t, |x| ≥ R, u), which implies E(t,ΩR, u) ≤ K t−R E(0,ΩR, u), (3.17) where K is a positive constant independent of the initial data. 8 R. S. O. NUNES EJDE-2021/49 Applying (3.17) to the solution w of (3.7), and by Lemma 3.2 we obtain∫ ΩR |u|2dx = ∫ B(0,R) |wt|2dx ≤ E(t,ΩR, wt) ≤ E(t,ΩR, u) ≤ K t−R E(0,ΩR, u). (3.18) Joining (3.17) and (3.18) and considering O ⊂ ΩR, we obtain (3.4), completes the proof. Lemma 3.1. 4. Control in a holed domain with fixed boundary In this section we prove an exact boundary control problem for the wave equation on a holed domain with fixed boundary. This section plays a important rule in the proof Theorem 1.1. Let Ω be as define in the first section and T > 0. The boundary of Ω is ∂Ω = Γ0 ∪ Γ1. Lemma 4.1. Given (v0, v1) ∈ H1(Ω) × L2(Ω) and T > 0, there exists a control function h(·, t) ∈ L2(Γ1 × [0, T ]) such that the solution v ∈ H1(Ω × [0, T ]) of the problem vtt −∆v = 0 in Ω× [0, T ] v(·, T ) = v0, vt(·, T ) = v1, in Ω v(·, t) = 0, on Γ0 × [0, T ] νtvt −∇v · νx = h(·, t), on Γ1 × [0, T ]. (4.1) satisfies the condition v(·, 0) = 0 = vt(·, 0) in Ω. (4.2) Proof. Let δ be a positive number, and Ωδ = {y ∈ Ω∞ : ∃x ∈ Ω; |x− y| < δ} be an open neighborhood of Ω. Given an arbitrary (w0, w1) ∈ H1(Ω)× L2(Ω), according Lemma 2.1 we can extend (w0, w1), for all Ω∞. Let (w̃0, w̃1) be the extension of (w0, w1), that is, (w̃0, w̃1) = E(w0, w1). Let w the solution of the backward initial boundary value problem wtt −∆w = 0 in Ω∞ × (0,+∞) w(·, T ) = w̃0, wt(·, T ) = w̃1, in Ω∞ w(·, t) = 0, in Γ0 × (0,+∞). (4.3) Now, for T > 0 we define the bounded linear operator ST : H1 0(Ωδ)× L2(Ωδ)→ H1(Ω∞)× L2(Ω∞) such that ST (w(·, T ), wt(·, T )) = (w(·, 0), wt(·, 0)), where w is the solution of (4.3). From the decay estimate (3.4), with O = Ωδ, applied to w we obtain the estimate ‖(w(·, 0), wt(·, 0))‖2H1(Ωδ)×L2(Ωδ) ≤ K T −R ‖(w̃0, w̃1)‖2H1(Ω∞)×L2(Ω∞) , (4.4) for T > R sufficiently large and K is a constant independent on data (w̃0, w̃1). EJDE-2021/49 EXACT BOUNDARY CONTROLLABILITY FOR THE WAVE EQUATION 9 Now we consider the cut off function φ ∈ C∞0 (Ω∞) such that φ ≡ 1 in Ωδ/2, and φ ≡ 0 out side of Ωδ. Then we solve the forward initial boundary value problem ztt −∆z = 0 in Ω∞ × (0,+∞) z(·, 0) = φw(·, 0), zt(·, 0) = φwt(·, 0), in Ω∞ z(·, t) = 0, in Γ0 × (0,+∞). (4.5) We define the linear operator ST : H1 0(Ωδ) × L2(Ωδ) → H1(Ω∞) × L2(Ω∞) by ST (z(·, 0), zt(·, 0)) = (z(·, T ), zt(·, T )). Applying again the decay estimate (3.4), with O = Ωδ, for function z, we obtain ‖(z(·, T ), zt(·, T ))‖2H1(Ωδ)×L2(Ωδ) ≤ K T −R ‖(z(·, 0), zt(·, 0)))‖2H1(Ωδ)×L2(Ωδ) , (4.6) for T > R sufficiently large and K is a constant independent on data (z0, z1). We define ṽ(·, t) = w(·, t)− z(·, t) and see that ṽ satisfies ṽtt −∆ṽ = 0 in Ω∞ × (0,+∞) ṽ(·, T ) = w(·, T )− z(·, T ), ṽt(·, T ) = wt(·, T )− zt(·, T ) in Ω∞ ṽ(·, t) = 0, in Γ0 × (0,+∞) (4.7) and ṽ(·, 0) = w(·, 0)− φw(·, 0) = 0 in Ω, ṽt(·, 0) = wt(·, 0)− φwt(·, 0) = 0 in Ω, since φ = 1 in Ω. Note that the function ṽ solves the homogeneous wave equation and has the desirable final state (ṽ(·, 0), ṽt(·, 0)) = (0, 0) in Ω. Now an important step it is to know if we may obtain T > 0 such that (ṽ(·, T ), ṽt(·, T )) extend the initial data (v0, v1). That is, we wish establish solution for the equations w(·, T )− z(·, T ) = v0, wt(·, T )− zt(·, T ) = v1 in Ω. The last equations can be rewriting as E(w0, w1)− (z(·, T ), zt(·, T )) = (v0, v1) in Ω. (4.8) We want to solve (4.8) for unknown (w0, w1) ∈ H1(Ω) × L2(Ω). For this purpose we rewrite equation (4.8) in terms of the operators ST and ST . Note that (z(·, T ), zt(·, T )) = ST (φw(·, 0), φwt(·, 0)) = STMφ(w(·, 0), wt(·, 0)) = STMφST (w(·, T ), wt(·, T )) = [STMφSTE](w0, w1), where Mφ is the operator multiplication by φ. Thus, (4.8) becomes (w0, w1)−RSTMφSTE(w0, w1) = (v0, v1) in Ω, (4.9) where R denotes the restriction to Ω. Denoting RSTMφSTE by KT , equation (4.9) can be rewritten as (I −KT ) (w0, w1) = (v0, v1) in Ω, (4.10) where I is the identity operator in H1(Ω)× L2(Ω). 10 R. S. O. NUNES EJDE-2021/49 Now, for solving equation (4.9) it is sufficient to show that KT is a contraction in H1(Ω)×L2(Ω). It is in this point where the energy decay takes place, by considering inequalities (4.4) and (4.6) note that ‖KT (w0, w1)‖H1(Ω)×L2(Ω) = ‖(z(·, T ), zt(·, T ))‖H1(Ω)×L2(Ω) ≤ ‖(z(·, T ), zt(·, T ))‖2H1(Ωδ)×L2(Ωδ) ≤ K T −R ‖(z(·, 0), zt(·, 0))‖2H1(Ωδ)×L2(Ωδ) = K T −R ‖(φw(·, 0), φwt(·, 0))‖2H1(Ωδ)×L2(Ωδ) ≤ K̃ T −R ‖(w(·, 0), wt(·, 0))‖2H1(Ωδ)×L2(Ωδ) ≤ KK̃ (T −R)2 ‖E(w0, w1)‖2H1(Ω)×L2(Ω) ≤ KK̃ (T −R)2 ‖(w0, w1)‖2H1(Ω)×L2(Ω). From the inequalities above we obtain ‖KT (w0, w1)‖H1(Ω)×L2(Ω) ≤ √ KK̃ T −R ‖(w0, w1)‖H1(Ω)×L2(Ω), (4.11) for T > R sufficiently large, where K̃ is a positive constant depending only K and Ω. So, we choose a T > R such that √ KK̃ T−R ≤ c < 1 and for such T , KT is a contraction. Thus, we take the solution (w0, w1) for (4.10) and take it to the begin of the proof in order to obtain w, z and ṽ = w− z, where ṽ solves (4.7) and has the desirable final condition (ṽ(·, 0), ṽ(·, 0)) = (0, 0). To complete the proof we define v = ṽ|Ω and apply Lemma 2.2 from the previous section to read off the trace of νtut −∇u · νx as a function of ∈ L2(Γ1 × [0, T ]) completing the proof. � 5. Proof of Theorem 1.1 Let Ω, Ω∞ and Ω̃ as defined in the initial section. See that the Ω̃ is a holed domain with fixed boundary ∂Ω̃ = Γ̃ ∪ Γ0, where Γ0 is the boundary of the hole and Γ̃ is the external boundary. Let (f̃ , g̃) ∈ H1(Ω∞) × L2(Ω∞) such that f̃ and g̃ are extensions of f and g respectively, with supp(f̃) ⊂ B(0, r) ∩ Ω∞ and supp(g̃) ⊂ B(0, r) ∩ Ω∞. Here the number r is that one defined in the Section 1. Let ũ be the solution of the initial-boundary value problem ũtt −∆ũ = 0 in Ω∞ × (0,+∞) ũ(·, 0) = f̃ , ũt(·, 0) = g̃, in Ω∞ ũ(·, t) = 0, in Γ0 × (0,+∞). (5.1) Now, for a T > r, we take the state (ũ(·, T ), ũt(·, T )) ∈ H1(Ω̃) × L2(Ω̃) and according to Lemma 4.1, changing Ω by Ω̃, we solve the exact boundary control EJDE-2021/49 EXACT BOUNDARY CONTROLLABILITY FOR THE WAVE EQUATION 11 problem vtt −∆v = 0 in Ω̃× [0, T ] v(·, T ) = ũ(·, T ), vt(·, T ) = ũt(·, T ), in Ω̃ v(·, t) = 0, on Γ0 × [0, T ] νtvt −∇v · νx = h(·, t), on Γ̃× [0, T ], (5.2) which satisfies, at the instant t = 0, the condition v(·, 0) = 0 = vt(·, 0) in Ω̃. (5.3) Considering Ωt as defined in the Section 1, note that Ωt ⊂ Ω̃ for all t > 0, so follows that for each T > 0 we have QT = ∪0