Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 25, pp. 1–14. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.25 EXACT BOUNDARY CONTROLLABILITY FOR WAVE EQUATIONS WITH FIXED AND MOVING BOUNDARIES IN TWO-DIMENSIONAL CONVEX-COMPLEMENTED DOMAINS RUIKSON S. O. NUNES, MIGUEL R. NUÑEZ-CHÁVEZ Abstract. The purpose of this article is to study exact boundary problems for the standard wave equation in domains that are the exterior of a convex compact set of R2, where both have a common boundary part. We consider two cases: first where the boundary domain is fixed, and where a part of the boundary is moving. In both cases we consider control problems with controls acting only one part of the boundary. For the fixed boundary case the control is of Neumann type, and for the moving boundary case the control is a conormal derivative type. The controllability method used here was developed by Russell [17]. 1. Introduction Let A ⊂ R2 be a bounded domain whose boundary ∂A is smooth by parts. Let γ ⊊ ∂A be a part of the boundary. The pair (A, γ) is called convex-complemented when there is a convex compact region A∗ ⊆ R2 with A ⊆ R2 −A∗ and γ ⊆ ∂A∗. Let Ω ⊂ R2 be a convex bounded domain whose boundary ∂Ω is smooth by parts with no cusps, with ∂Ω = Γ0 ∪ Γ1 where Γ0 ⊊ ∂Ω, and Γ0 ∩ Γ1 = {P0, P1} where P0, P1 ∈ R2. We also requires that Ω be in only one side of its boundary ∂Ω and the pair (Ω,Γ0) be a convex-complemented with respect to the convex compact domain Ω∗. Here, we requires that the boundary ∂Ω∗ be smooth, connected, and compact curve as illustrated in the Figure 1. Now, let us consider the moving boundary domain Ωt ⊂ R2 whose boundary ∂Ωt = Γ0 ∪ Γ1t for all t ∈ R. We require that Γ0 ∩ Γ1t = {P0, P1}, for all t ∈ R. The moving part Γ1t of the boundary ∂Ωt is such that Γ10 = Γ1 and it may has two configurations. Firstly, we can have Γ1t = { x ∈ R2 : x = α(t)y, y ∈ Γ1 } , for all t ∈ R. 2020 Mathematics Subject Classification. 35L05, 35L20, 35L53, 35B40, 93B05, 49J20. Key words and phrases. Wave equation; energy decay; exact boundary controllability; non-cylindrical domains; moving boundary domains. ©2025. This work is licensed under a CC BY 4.0 license. Submitted November 21, 2024. Published March 4, 2025. 1 2 R. S. O. NUNES, M. R. NUÑEZ-CHÁVEZ EJDE-2025/25 Γ1Ω Ω∗ Γ0 Figure 1. The pair (Ω,Γ0) is convex-complemented. Ω∗ is the convex complement. In this case, Ωt is illustrated in Figure 2. Γ1t Γ0 Ω∗ Ωt Γ1Ω Figure 2. The pairs (Ω,Γ0) and (Ωt,Γ0) are convex- complemented. In both cases, Ω∗ is the convex complement. Another configuration for Ωt is Γ1t = { x ∈ R2 : x = α(t)y, y ∈ Γ1 } ∪ γ1 ∪ γ0 , for all t ∈ R, where γ0 and γ1 are the straight segments joining the points P0 to α(t)P0 and P1 to α(t)P1 respectively. Here α : R → R+ ∪ {0} is a piecewise continuous bounded function, where α such that, for each t ∈ R, the pair (Ωt,Γ0) is a convex-complemented with respect to Ω∗. An illustration for the domain Ωt is shown in Figure 3. Γ1t γ0 γ1 Γ0 Ω∗ Ωt Γ1Ω Figure 3. The pairs (Ω,Γ0) and (Ωt,Γ0) are convex- complemented. In both cases, Ω∗ is the convex complement. For the well posedness of the initial boundary value problem we require Ωt × R ⊂ ∪x∈Ω { (x, t) ∈ R2 × R : |x− x|2 ≤ t2 } . (1.1) The boundedness of movement function α implies in the existence of a bounded domain Ω̃ ⊂ R2 such that Ωt ⊆ Ω̃, for all t ∈ R. The boundary of Ω̃ is denoted by ∂Ω̃ and it is such that Γ0 ⊂ ∂Ω̃. Furthermore, the pair (Ω̃,Γ0) has the convex- completed property with respect to the domain Ω∗. EJDE-2025/25 EXACT BOUNDARY CONTROLLABILITY FOR WAVE EQUATIONS 3 Now, for T > 0, let us consider the non-cylindrical domain of R2+1, QT = ∪0 0 sufficiently large and a control function h(·, t) ∈ L2(∂Ω × [0, T ]) such that the solution u ∈ H1 loc(Ω× [0, T ]) of the problem utt −∆u = 0 in Ω× [0, T ] u(·, 0) = f, ut(·, 0) = g in Ω ∂νu(·, t) = 0 on Σ0, ∂νu(·, t) = h(·, t) on Σ1, (1.3) 4 R. S. O. NUNES, M. R. NUÑEZ-CHÁVEZ EJDE-2025/25 satisfy the final condition u(·, T ) = 0 = ut(·, T ) in Ω. (1.4) Theorem 1.3. Let Ω and Ωt be as defined above. Given (f, g) ∈ H1(Ω)× L2(Ω), there exist T > 0 sufficiently large and a control function h(·, t) ∈ L2(Γ1t × [0, T ]) such that the solution u ∈ H1 loc(QT ) of the problem utt −∆u = 0 in QT u(·, 0) = f, ut(·, 0) = g in Ω ∂νu(·, t) = 0 on Σ0, νtut −∇u · νx = h(·, t) on ΣT , (1.5) satisfy the final condition u(·, T ) = 0 = ut(·, T ) in ΩT . (1.6) Note that in both problems (1.3) and (1.5) the controls acts only one part of the boundary. In the problem (1.3) the control is of Neumann type and acts on Σ1 while in (1.5) the control acts on the moving boundary part ΣT and the control is established via conormal derivative of the solution u. Russell [17] developed a technique, based in [16], for studding an exact boundary control problem for wave equation with control acting only on a part of boundary of the domain. However, in [17] it is considered domains only fixed boundaries and with Dirichlet null condition on one part of the boundary. Here, it is used the Russell’s technique but we go a step more by studying both exact boundary control problem with moving and fixed boundaries with Neumann null condition on the part of the boundary where the controls do not act. In the literature there are also many works dealing with exact boundary control problems on non-cylindrical domains, using as HUM method (see [9]) as Russell’s method, to cite a few see [2, 3, 4, 10, 11, 13, 14] and their references. When the boundary mobility is bounded, as considered here, we observe an advantage using the Russell’s method instead of HUM because the original system do not need to be transformed into a system with variable coefficients as in [4, 11]. On the other hand, the applicability of Russell’s method requires some properties of the system. Some of them are: linearity, time reversibility, local energy decay, and suitable trace theorems. With respect to local energy decay estimates for the problem considered here we use a decay estimate present in [18], which will presented in the next section. With respect to traces, the theorems we will use are presented in [19]. As seen above, the Russell’s method requires many properties of the system 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. The rest of this artile is organized as follows. In Section 2 we presents a brief summary with respect to traces, extension and decay properties. Section 3 is ded- icated to proof Theorem 1.2. In Section 4, we consider a special extension result. In Section 5, we prove Theorem 1.3. 2. Preliminaries results In this section we state some preliminaries results which are essential for applying Russell’s controllability method. We make a brief presentation about the trace, EJDE-2025/25 EXACT BOUNDARY CONTROLLABILITY FOR WAVE EQUATIONS 5 extension and local energy decay results which are fundamental ingredients in the Russell’s method. We considering Ω as the domain as defined in the introductory function whose boundary is ∂Ω = Γ0 ∪ Γ1 where Γ0 is a smooth curve. Also we consider the pair (Ω,Γ0) being a convex-complemented. So, we let Ω∗ be the a complementary convex domain associated with Ω. Let us denote Ω∗ ∞ = R2 − Ω∗. It is clear that the pair (Ω∗ ∞,Γ0) is convex complemented, because we have Ω∗ a convex compact set in R2 such that Ω∗ ∞ ⊂ R2 − Ω∗ and Γ0 ⊊ ∂Ω∗ = ∂Ω∗ ∞. So, from (1.2) it is possible to define the space H1(Ω∗ ∞). For technical reasons to apply Russell’s method under the following assumptions. (A1) Let B = B(0, R), with R > 0, be a disc where Ω ∪ Ω∗ ⊂ B. Assume that there exist a bounded linear operator P : H1(Ω) → H1(Ω∗ ∞) such that for u ∈ H1(Ω) the extension ũ = Pu satisfy the following: (1) ∂ν ũ = 0 in ∂Ω∗; (2) ũ = u in Ω; (3) ũ = 0 in R2 −B; (4) ∥Pu∥H1(Ω∗ ∞) ≤ C∥u∥H1(Ω), where C is a positive real constant independent on u. Under the above assumptions, we proof the following extension result. Lemma 2.1. Let Ω, Ω∗ and Ω∗ ∞ be as above and consider the domain Ωδ = {x ∈ Ω∗ ∞ : |x − y| ≤ δ, for y ∈ Ω}, where δ is a positive constant. Then there exists a bounded linear operator E1 : H1(Ω) → H1(Ω∗ ∞), such that for each f ∈ H1(Ω) we have E1f |Ω = f with supp(E1f) ⊂ Ωδ and ∥E1f∥H1(Ω∗ ∞) ≤ C∥f∥H1(Ω) for some constant C > 0. Proof. Let φ ∈ C∞ 0 (Ω∗ ∞) be a function such that φ = 1 in Ω and φ in Ω∗ ∞ − Ω δ 2 . By considering the operator P of Assumption (A1), let us define E1f = φPf for f ∈ H1(Ω). Note that E1 : H1(Ω) → H1(Ω∗ ∞) is linear, E1f |Ω = f in Ω and supp(E1f) ⊂ Ωδ for all f ∈ H1(Ω). Note now that E1 = Mφ ◦ P where Mφ : H1(Ω) → H1(Ω∗ ∞) the multiplication operator defined by Mφ(ϕ) = φϕ is a bounded linear operator. So, it follows that E1 is a bounded linear operator. □ Another important ingredient in the application of Russell’s controllability method is the trace regularity of the conormal derivative on time-like surfaces of the so- lutions of the initial boundary value problem to be studied. 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 Theorems 1.2 and 1.3. 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 normal unit 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 is to know what the regularity of the traces of the conormal derivative on surfaces, for this purpose we turn to [19]. Considering Ξ ⊂ RN , with N ≥ 2, [19, Theorem 2] shows that if u ∈ H1 loc(Ξ) is such that P (ξ,D)u ∈ L2 loc(Ξ) then ∂u ∂η ∈ L2 loc(Σ). 6 R. S. O. NUNES, M. R. NUÑEZ-CHÁVEZ EJDE-2025/25 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 ∈ H1 loc(Ω ∗ ∞ × R) be the solution of the initial-boundary value problem utt −∆u = 0 in Ω∗ ∞ × R u(·, 0) = u0, ut(·, 0) = u1, in Ω∗ ∞ ∂νu(·, t) = 0, on ∂Ω∗ ∞ × R (2.1) with initial data (u0, u1) ∈ H1(Ω∗ ∞)×L2(Ω∗ ∞), where supp(u0), supp(u1) ⊂ Ω∗ ∞. Let γ be a smooth hypersurface in Ω∗ ∞, with no self intersection and considers surface γ × R whose 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). Remark 2.3. If the surface γ×R in Lemma 2.2 is cylindrical then the component νt of the normal vector ν = (νx, νt) is null, that is νt = 0. So, in this case, the conormal derivative coincides with normal derivative, that is, νtut −∇u · νx = −∇u · νx. 2.1. Local energy decay. Local energy decay plays a fundamental role in the proof of the control problems proposed in this article. Particularly, we are interested in local energy decay estimates for the wave equation in exterior domains. With respect to this topic there are many paper available in the literature, to cite a few see [5, 6, 7, 12, 18, 20, 21] and references there in. In this paper, we are interested in a local decay estimate presented in [18]. To have such result let us consider the initial initial-boundary value problem utt −∆u = 0 in Ω∗ ∞ × R u(·, 0) = u0, ut(·, 0) = u1, in Ω∗ ∞ ∂νu(·, t) = 0, on ∂Ω∗ ∞ × R (2.2) Let O ⊂ Ω∗ ∞ be a bounded domain. The energy of the solution u of the (2.2) confined in O is E(t,O, u) = 1 2 ∫ O [|∇u|2 + |ut|2 + |u|2](x, t)dx. (2.3) If there exist a positive constant C (independent on initial data) and a function p(t) such that E(t,O, u) ≤ Cp(t)E(0,O, u), (2.4) with p(t) → 0, as t → +∞, we say the energy of (2.2) decays locally. In [7, 12] the authors obtain local energy decay estimates for a exterior problem like (2.2) but with null Dirichlet condition on ∂Ω∗ ∞. In this article we are interested in a local energy decay estimate but considering a null Neumann boundary condition ∂Ω∗ ∞. So, we turn to a result presented in [18, Lemma 2.1] where with a little adaptation express the following local energy decay estimate. EJDE-2025/25 EXACT BOUNDARY CONTROLLABILITY FOR WAVE EQUATIONS 7 Lemma 2.4. Let (u0, u1) ∈ H1(Ω∗ ∞) × L2(Ω∗ ∞) with supp(u0), supp(u1) ⊂ O ⊂ Ω∗ ∞, then there exist a positive real constant K, independent of u0 and u1, such that the solution u of (2.2) satisfies ∥u(·, t)∥2H1(O) + ∥ut(·, t)∥2L2(O) ≤ K(1 + t)−2 { ∥u(·, 0)∥2H1(O) + ∥ut(·, 0)∥2L2(O) } , (2.5) for t > 0 sufficiently large. The time reversibility of the wave operator has a central role to play in using Russell’s controllability method. So, it is also necessary to know the local energy decay estimates for the system (2.2) in reverse time. The next result shows a estimate for the local energy in reverse time. Lemma 2.5. Let T a positive real number and (u0, u1) ∈ H1(Ω∗ ∞)× L2(Ω∗ ∞) with supp(u0), supp(u1) ⊂ O ⊂ Ω∗ ∞. The solution u of the initial boundary value problem utt −∆u = 0 in Ω∗ ∞ × R u(·, T ) = u0, ut(·, T ) = u1, in Ω∗ ∞ ∂νu(·, t) = 0, on ∂Ω∗ ∞ × R (2.6) satisfies the estimate ∥u(·, 0)∥2H1(O) + ∥ut(·, 0)∥2L2(O) ≤ K(1 + T )−2 { ∥u(·, T )∥2H1(O) + ∥ut(·, T )∥2L2(O) } , (2.7) where K is a positive real constant independent on of the initial data (u0, u1). Proof. Let v be the solution of the problem (vtt −∆v)(·, τ) = 0 in Ω∗ ∞ × R v(·, 0) = u0, vτ (·, 0) = −u1, in Ω∗ ∞ ∂νv(·, τ) = 0, on ∂Ω∗ ∞ × R (2.8) by applying estimate (2.5) to v we obtain ∥v(·, τ)∥2H1(O) + ∥vτ (·, τ)∥2L2(O) ≤ K(1 + τ)−2 { ∥v(·, 0)∥2H1(O) + ∥vτ (·, 0)∥2L2(O) } , for τ > 0 sufficiently great. Making τ = T − t in latest inequality and observing that u(·, t) = v(·, T − t) is solution of (2.6) satisfying the estimate (2.7). □ 3. Proof of Theorem 1.2 Let Ω,Ω∗,Ω∗ ∞ and Ωδ be domains as defined in the previous section. Given an arbitrary (w0, w1) ∈ H1(Ω)× L2(Ω), according Lemma 2.1 and classical exten- sion results in L2 we can obtain bounded linear extension operator E : H1(Ω) × L2(Ω) → H1(Ω∗ ∞) × L2(Ω∗ ∞) such that the extension (w̃0, w̃1) of (w0, w1), that is (w̃0, w̃1) = E(w0, w1), satisfy supp(w̃0), supp(w̃1) ⊂ Ωδ. Let w the solution of the initial boundary value problem wtt −∆w = 0 in Ω∗ ∞ × R w(·, 0) = w̃0, wt(·, 0) = w̃1, in Ω∗ ∞ ∂νw(·, t) = 0, in ∂Ω∗ ∞ × R. (3.1) Now, for T > 0 we define the bounded linear operator ST : H1 0(Ωδ)× L2(Ωδ) → H1(Ω∗ ∞)× L2(Ω∗ ∞) 8 R. S. O. NUNES, M. R. NUÑEZ-CHÁVEZ EJDE-2025/25 such that ST (w(·, 0), wt(·, 0)) = (w(·, T ), wt(·, T )), where w is the solution of (3.1). From the decay estimate (2.5), with O = Ωδ, applied to w we obtain the estimate ∥(w(·, T ), wt(·, T ))∥2H1(Ω∗ ∞)×L2(Ω∗ ∞) ≤ K(1 + T )−2∥(w̃0, w̃1)∥2H1(Ωδ)×L2(Ωδ) , (3.2) for T > 0 sufficiently large and K is a constant independent of the data (w̃0, w̃1). In terms of the operator ST , inequality (3.2) becomes ∥ST (w̃0, w̃1)∥2H1(Ω∗ ∞)×L2(Ω∗ ∞) ≤ K(1 + T )−2∥(w̃0, w̃1)∥2H1(Ωδ)×L2(Ωδ) , (3.3) for T > 0 sufficiently large and K is a constant independent on data (w̃0, w̃1). Now we consider the cut off function ϕ ∈ C∞ 0 (Ω∗ ∞) such that ϕ ≡ 1 in Ωδ/2, and ϕ ≡ 0 outside of Ωδ. Then, for each T > 0, we solve the backward initial boundary value problem ztt −∆z = 0 in Ω∗ ∞ × R z(·, T ) = ϕw(·, T ), zt(·, T ) = ϕwt(·, T ), in Ω∗ ∞ ∂νz(·, t) = 0, in ∂Ω∗ ∞ × R, (3.4) where the function w is the solution of problem (3.1). We define the linear op- erator ST : H1 0(Ωδ) × L2(Ωδ) → H1(Ω∗ ∞) × L2(Ω∗ ∞) by ST (z(·, T ), zt(·, T )) = (z(·, 0), zt(·, 0)). Applying again the decay estimate (2.7), with O = Ωδ, for function z, we obtain ∥(z(·, 0), zt(·, 0))∥2H1(Ωδ)×L2(Ωδ) ≤ K(1 + T )−2∥(z(·, T ), zt(·, T )))∥2H1(Ωδ)×L2(Ωδ) , (3.5) for T > 0 sufficiently large and K is a constant independent on data (z0, z1). In terms of the operator ST the inequality (3.5) becomes ∥ST (z(·, T ), zt(·, T ))∥2H1(Ωδ)×L2(Ωδ) ≤ K(1 + T )−2∥(z(·, T ), zt(·, T )))∥2H1(Ωδ)×L2(Ωδ) , (3.6) for T > 0 sufficiently large and K is a constant independent on data (z0, z1). We define ṽ(·, t) = w(·, t)− z(·, t) and note that ṽ satisfies ṽtt −∆ṽ = 0 in Ω∗ ∞ × R ṽ(·, 0) = w(·, 0)− z(·, 0), ṽt(·, 0) = wt(·, 0)− zt(·, 0) in Ω∗ ∞ ∂ν ṽ(·, t) = 0, in ∂Ω∗ ∞ × R (3.7) and ṽ(·, T ) = w(·, T )− ϕw(·, T ) = (1− ϕ)w(·, T ) = 0 in Ω, ṽt(·, T ) = wt(·, T )− ϕwt(·, T ) = (1− ϕ)wt(·, T ) = 0 in Ω, since ϕ = 1 in Ω. Note that the function ṽ solves the initial boundary value problem (3.7) and has the desirable final state (ṽ(·, T ), ṽt(·, T )) = (0, 0) in Ω. Now an important step it is to know if we can obtain T > 0 such that (ṽ(·, 0), ṽt(·, 0)) extend the initial data (f, g) of the problem (1.3). That is, we wish establish solution for the equations w(·, 0)− z(·, 0) = f, wt(·, 0)− zt(·, 0) = g in Ω. This latest two equations can be rewriten as (w0, w1)−R(z(·, 0), zt(·, 0)) = (f, g) in Ω, (3.8) EJDE-2025/25 EXACT BOUNDARY CONTROLLABILITY FOR WAVE EQUATIONS 9 where R denotes the restriction to Ω. So, we want to solve (3.8) for unknown (w0, w1) ∈ H1(Ω)× L2(Ω). For this purpose we rewrite equation (3.8) in terms of the operators ST and ST . Note that (z(·, 0), zt(·, 0)) = ST (z(·, T ), ϕzt(·, T )) = ST (ϕw(·, T ), ϕwt(·, T )) = STMϕ(w(·, T ), wt(·, T )) = STMϕST (w(·, 0), wt(·, 0)) = [STMϕSTE](w0, w1), where Mϕ is the operator multiplication by ϕ. Thus, (3.8) becomes (w0, w1)−RSTMϕSTE(w0, w1) = (f, g) in Ω. (3.9) Denoting RSTMϕSTE by KT , equation (3.9) can be rewritten as (I −KT ) (w0, w1) = (f, g) in Ω, (3.10) where I is the identity operator in H1(Ω)× L2(Ω). Now, for solving equation (3.10) 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 (3.3) and (3.6) note that ∥KT (w0, w1)∥2H1(Ω)×L2(Ω) ≤ ∥STMϕSTE(w0, w1))∥2H1(Ωδ)×L2(Ωδ) ≤ K(1 + T )−2∥MϕSTE(w0, w1)∥2H1(Ωδ)×L2(Ωδ) ≤ C(1 + T )−2∥STE(w0, w1)∥2H1(Ωδ)×L2(Ωδ) ≤ CK(1 + T )−4∥E(w0, w1)∥2H1(Ωδ)×L2(Ωδ) ≤ C(1 + T )−4∥(w0, w1)∥2H1(Ω)×L2(Ω) ≤ C (1 + T )4 ∥(w0, w1)∥2H1(Ω)×L2(Ω), where C in the above inequalities represents a positive real constant which may vary from line to line. So, from the above inequalities we obtain ∥KT (w0, w1)∥H1(Ω)×L2(Ω) ≤ √ C (1 + T )2 ∥(w0, w1)∥H1(Ω)×L2(Ω), (3.11) for T > 0 sufficiently large, being C a positive constant independent of the initial data. Now, choosing a T > 0 such that √ C (1+T )2 ≤ c < 1 and for such T , KT is a contraction. After, for such T , we take the solution (w0, w1) for (3.10) and take it to the begin of the proof in order to obtain the function w, z and ṽ = w−z, where ṽ solves (3.7) and has the desirable final condition (ṽ(·, T ), ṽ(·, T )) = (0, 0). Besides, ṽ(·, 0) and ṽt(·, 0) extends f and g, respectively, from Ω to Ω∗ ∞. To complete the proof note that ṽtt − ∆ṽ ∈ L2 loc(Ω ∗ ∞ × R), so, applying the trace regularity result of Lemma 2.2 we have that the trace of conormal derivative of ṽ on surface ∂Ω× [0, T ] is well defined and it is locally square integrable. That is, ṽtνt−∇ṽ ·νx ∈ L2(∂Ω× [0, T ]). As the surface ∂Ω× [0, T ] is cylindrical follows that the component νt of the normal vector (νx, νt) is null. So, the conormal derivative coincides to normal derivative ∂ν ṽ on ∂Ω× [0, T ]. To finish the proof, we defines u := ṽ|Ω×[0,T ], the restriction of ṽ to domain Ω× [0, T ] and h := ∂ν ṽ on ∂Ω× [0, T ] and observing that the function u and h meet the conditions of Theorem 1.2. 10 R. S. O. NUNES, M. R. NUÑEZ-CHÁVEZ EJDE-2025/25 4. A special extension theorem Let us consider the domains Ω, Ω∗ ∞, and Ω̃ as defined in the previous sections. In this section, we prove an important result which is of the fundamental importance in the proof of Theorem 1.3. Such result is stated in lemma below. Lemma 4.1. Let T be a positive real number. Each (v0, v1) ∈ H(Ω) × L2(Ω) can be extended to (ṽ0, ṽ1) ∈ H(Ω∗ ∞)×L2(Ω∗ ∞) such that the solution v ∈ H1 loc(Ω ∗ ∞×R) of the problem vtt −∆v = 0 in Ω∗ ∞ × R v(·, T ) = ṽ0, vt(·, T ) = ṽ1 in Ω∗ ∞ ∂νv = 0 on ∂Ω∗ ∞ × R, (4.1) satisfies the condition v(·, 0) = 0 = vt(·, 0) in Ω̃. (4.2) Proof. Let δ be a positive real number, and Ω̃δ = {y ∈ Ω∗ ∞ : ∃x ∈ Ω̃; |x−y| < δ} be an open neighborhood of Ω̃. Given an arbitrary (w0, w1) ∈ H(Ω)×L2(Ω), according Lemma 2.1 and classical extension results in L2 we can obtain bounded linear extension operator E : H1(Ω)×L2(Ω) → H1(Ω∗ ∞)×L2(Ω∗ ∞) such that the extension (w̃0, w̃1) of (w0, w1), that is (w̃0, w̃1) = E(w0, w1), satisfy supp(w̃0), supp(w̃1) ⊂ Ω̃δ. Let w ∈ Hloc(Ω ∗ ∞ × R) the solution of the initial boundary value problem wtt −∆w = 0 in Ω∗ ∞ × R w(·, T ) = w̃0, wt(·, T ) = w̃1 in Ω∗ ∞ ∂νw = 0 on ∂Ω∗ ∞ × R, (4.3) Now, for T > 0 we define the bounded linear operator ST : H1(Ω∗ ∞)× L2(Ω∗ ∞) → 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 (2.7), with O = Ω̃δ, applied to w we obtain the estimate ∥ST (w(·, T ), wt(·, T ))∥2H1(Ω∗ ∞)×L2(Ω∗ ∞) ≤ K(1 + T )−2∥(w(·, T ), wt(·, T )∥2H1(Ωδ)×L2(Ωδ) , (4.4) for T > 0 sufficiently large and K is a constant independent on data (w̃0, w̃1). Now we consider the cut off function ϕ ∈ C∞ 0 (Ω∗ ∞) such that ϕ ≡ 1 in Ω̃δ/2, and ϕ ≡ 0 out side of Ω̃δ. Let z ∈ Hloc(Ω ∗ ∞ × R) the solution of the initial boundary value problem ztt −∆z = 0 in Ω∗ ∞ × R z(·, 0) = ϕw(·, 0), zt(·, 0) = ϕwt(·, 0) in Ω∗ ∞ ∂νz = 0 on ∂Ω∗ ∞ × R, (4.5) We define the linear operator ST : H1(Ω∗ ∞)× L2(Ω∗ ∞) → H1(Ω∗ ∞)× L2(Ω∗ ∞) by ST (z(·, 0), zt(·, 0)) = (z(·, T ), zt(·, T )). Applying again the decay estimate (2.5), with O = Ω̃δ, we obtain ∥ST (z(·, 0), zt(·, 0))∥2H1(Ω̃δ)×L2(Ω̃δ ≤ K(1 + T )−2∥(z(·, 0), zt(·, 0))∥2H1(Ω̃δ)×L2(Ω̃δ) , (4.6) EJDE-2025/25 EXACT BOUNDARY CONTROLLABILITY FOR WAVE EQUATIONS 11 for T > 0 sufficiently large andK is a constant independent on data (z(·, 0), zt(·, 0)). We define ṽ(·, t) = w(·, t)− z(·, t) and see that ṽ ∈ Hloc(Ω ∗ ∞ × R) satisfies ṽtt −∆ṽ = 0 in Ω∗ ∞ × R ṽ(·, T ) = w(·, T )− z(·, T ), in Ω∗ ∞ ṽt(·, T ) = wt(·, T )− zt(·, T ) in Ω∗ ∞ ∂ν ṽ = 0 on ∂Ω∗ ∞ × R, (4.7) and ṽ(·, 0) = w(·, 0)− ϕw(·, 0) = (1− ϕ)w(·, 0) = 0 in Ω̃, ṽt(·, 0) = wt(·, 0)− ϕwt(·, 0) = (1− ϕ)wt(·, 0) = 0 in Ω̃, since ϕ = 1 in Ω̃. Note that the function ṽ solves the homogeneous wave equation (4.7) and has the desirable final state (ṽ(·, 0), ṽt(·, 0)) = (0, 0) in Ω̃. Now, an important step it is to know if we may obtain T > 0 such that (ṽ(·, T ), ṽt(·, T )) be a extension of the initial data (v0, v1) from H(Ω)×L2(Ω) to H(Ω∗ ∞)×L2(Ω∗ ∞). That is, we wish establish solution for the equations w(·, T )− z(·, T ) = v0, wt(·, T )− zt(·, T ) = v1 in Ω. The two latest 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 the unknown (w0, w1) ∈ H(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 (z(·, 0), zt(·, 0)) = 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. As in the latest section, by 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 H(Ω)× L2(Ω). Equation (4.10) has (w0, w1) as its unknown. Note that KT is a compact linear operator. Proceeding analogously to the previous section and by considering inequalities (4.4) and (4.6) we obtain ∥KT (w0, w1)∥H1(Ω)×L2(Ω) ≤ √ C (1 + T )2 ∥(w0, w1)∥H1(Ω)×L2(Ω), (4.11) for T > 0 sufficiently large, where C is a positive independent on initial data. So, choosing a T > 0 such that √ C (1+T )2 ≤ 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 12 R. S. O. NUNES, M. R. NUÑEZ-CHÁVEZ EJDE-2025/25 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). Besides, (ṽ(·, T ), ṽt(·, T )) ∈ H1(Ω∗ ∞) × L2(Ω∗ ∞) is extension of (v0, v1) ∈ H1(Ω)×L2(Ω). By the end taken ṽ := v, we can see that v satisfy (4.1) and the condition (4.2) finalizing the proof of the Lemma 4.1. □ 5. Proof of Theorem 1.3 Let Ω, Ω∗, Ω∗ ∞ and Ω̃ as defined in the initial section. See that the Ω̃ is a domain with fixed boundary ∂Ω̃ = Γ̃ ∪ Γ0 where the pair (Ω̃,Γ0) has the convex- complemented property with respect the convex domain Ω∗. Taking (f, g) ∈ H1(Ω)×L2(Ω), let us consider the extensions (f̃ , g̃) ∈ H1(Ω∗ ∞)×L2(Ω∗ ∞) of (f, g), where supp(f̃), supp(g̃) ⊂ Ω̃. Let ũ ∈ H1 loc(Ω ∗ ∞ × R) be the solution of the initial- boundary value problem ũtt −∆ũ = 0 in Ω∗ ∞ × R ũ(·, 0) = f̃ , ũt(·, 0) = g̃ in Ω∗ ∞ ∂ν ũ = 0 on ∂Ω∗ ∞ × R, (5.1) Now, for a T > 0 sufficiently large, we take the state (ũ(·, T ), ũt(·, T )) ∈ H1(Ω̃)× L2(Ω̃) and according to Lemma 4.1, changing Ω by Ω̃ the state (ũ(·, T ), ũt(·, T )) can be extended to H1(Ω∗ ∞)×L2(Ω∗ ∞) such that the solution v ∈ H1 loc(Ω ∗ ∞ ×R) of the initial boundary value problem with extended data (ũ(·, T ), ũt(·, T )), vtt −∆v = 0 in Ω∗ ∞ × R v(·, T ) = ũ(·, T ), vt(·, T ) = ũt(·, T ) in Ω∗ ∞ ∂νv = 0 on ∂Ω∗ ∞ × R, (5.2) satisfies, at the instant t = 0, the condition v(·, 0) = 0 = vt(·, 0) in Ω̃. (5.3) Now, by considering ũ and v the solutions of (5.1) and (5.2) respectively. Defining the function ū = ũ− v see that ū(·, 0) = f̃ , ūt(·, 0) = g̃ and ū(·, T ) = 0 = ūt(·, T ) in Ω∗ ∞. Furthermore, the function u satisfy the initial-boundary value problem ūtt −∆ū = 0 in Ω∗ ∞ × R ū(·, 0) = f̃ , ūt(·, 0) = g̃ in Ω∗ ∞ ∂ν ū = 0 on ∂Ω∗ ∞ × R, (5.4) and the final condition ū(·, T ) = 0 = ūt(·, T ) in Ω∗ ∞. (5.5) Note that ūtt −∆ū ∈ L2 loc(Ω ∗ ∞ × R), so, applying the trace regularity result of Lemma 2.2 we have that the trace of conormal derivative of ū on surface ΣT ×R is well defined and it is locally square integrable. That is, ūtνt −∇ū · νx ∈ L2(ΣT × [0, T ]). So, we define u := ū|QT , the restriction of ū to domain QT = ΩT × [0, T ] EJDE-2025/25 EXACT BOUNDARY CONTROLLABILITY FOR WAVE EQUATIONS 13 and h := ūtνt−∇ū ·νx on ΣT × [0, T ] and observing that the function u and satisfy the initial boundary value problem utt −∆u = 0 in QT u(·, 0) = f, ut(·, 0) = g in Ω ∂νu = 0 on Γ0 × [0, T ] νtut −∇u · νx = h(·, t) on ΣT (5.6) with the final condition u(·, T ) = 0 = ut(·, T ) in ΩT , (5.7) finalizing the proof of the Theorem 1.3. Remark 5.1. 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Pisa, C. L. Sci. (4) 26 (1) (1998), 185–206. [20] G. Vodev; On the uniform decay of the local energy, Serdica Math. J. 25 (1999), 191-206. [21] E. C. Zachmanoglou; The decay of the initial-boundary value problem for hyperbolic equa- tions, J. Math. Anal. Appl., 13 (1966), 504-515. Ruikson S. O. Nunes UFMT- Federal University of Mato Grosso, ICET, Department of Mathematics, 78060- 900, Cuiabá, MT, Brazil Email address: ruiksonsillas@hotmail.com Miguel R. Nuñez-Chávez (corresponding author) UFMT- Federal University of Mato Grosso, ICET, Department of Mathematics, 78060- 900, Cuiabá, MT, Brazil Email address: miguel.chavez@ufmt.br 1. Introduction 2. Preliminaries results 2.1. Local energy decay 3. Proof of Theorem 1.2 4. A special extension theorem 5. Proof of Theorem 1.3 References