On the Exact Boundary Control for the Linear Klein-Gordon Equation in Non-cylindrical Domains

The purpose of this paper is to study an exact boundary controllability problem in noncylindrical domains for the linear Klein-Gordon equation. Here, we work near of the extension techniques presented By J. Lagnese in [12] which is based in the Russell’s controllability method. The control time is obtained in any time greater then the value of the diameter of the domain on which the initial data are supported. The control is square integrable and acts on whole boundary and it is given by conormal derivative associated with the above-referenced wave operator.


INTRODUCTION
Since the second half of the last century to now we have seen an increase in the interest of the mathematicians and engineers in the study of vibratory problems modeled by the wave equation. This can be confirmed by great volume of the works dealing with such problems presented in the literature. In fact, such problems have theoretical an practical importance because have wide applications in many branches of the engineering and mathematics.
An interesting part is to study vibratory problems that occur on a flexible body whose boundary deforms over time. Such phenomena generate, on the space-time scale, the non-cylindrical domains. An illustrative example is a metallic body in vibration that is placed in an environment subject to a change in the temperature. The increasing or decreasing of the temperature causes a linear expansion or contraction of the metallic body characterizing a movement of its boundary along of the time generating a non-cylindrical domain. energy decay, stabilization and control of wave equations on non-cylindrical domains have already been considered by several authors, to cite a few, see [1,2,3,4,5,6,7,8,12,13,14,15,18]. Especially the problems that deal with control processes for wave equations have been extensively studied in the last 50 years. We can find in the literature many works that brought a great contribution to the development of the control theory for hyperbolic equations. In highlight we can cite the papers ( [3,13,18]) which established important methods for the study of controllability of wave equations, but they have worked only on cylindrical domains.
When we look specifically at control problems for waves equations in non-cylindrical domains we realize that, in the literature, the number of these is very small when compared to control problems for waves in cylindrical domains. But this seems natural, since the work on cylindrical domains is simpler than on non-cylindrical domains and often to study control problems in non-cylindrical domains we need, at some point, to resort to the techniques developed to treat controllability problems on the cylindrical domains. However, we can find interesting papers in the literature that deal with exact controllability for wave equations in non-cylindrical domains with the most diverse types of controllability methods, to cite a few see [2,4,7,8,12,14,15] and citing papers.
Taking into account the importance of the theme, this work proposes to study, in the light of the ideas of [12], an exact boundary controllability problem for the linear Klein-Gordon equation in non-cylindrical domains. In order to establish the results let us consider Ω ⊂ R N , N ≥ 2, be a bounded and smooth by parts domain whose value of the its diameter will be denoted by d(Ω), and c a positive real number. Let Q be a non-cylindrical open set in R N × [0, +∞) such that the intersection of Q with hyperplane {(x,t) ∈ R N+1 , t ≥ 0} is a nonempty bounded open set Ω t in R N and such that Ω 0 = Ω. We represent the boundary of Ω t by ∂ Ω t and Γ = t≥0 ∂ Ω t × {t} is a variety that inherits from Ω the property of the smoothness by parts. Now, for T ≥ 0, we set In order to guarantee the well-posedness of the initial and boundary value problems to be considered we requires that Q T , for T ≥ 0, be contained in the conical time-like region Being U ⊂ R N an arbitrary domain, we denote by L 2 (U) and H 1 (U) the Lebesgue and Sobolev spaces, provided with theirs usual norms which will be denoted by · L 2 (U) and · H 1 (U) respectively (see [9] ). Let us also consider the space H 1 0 (U) which is the closure of C ∞ 0 in H 1 (U) provided with the norm of H 1 (U). The product space H 1 (U) × L 2 (U) is considered endowed with the product norm (·, ·) 2 H 1 (U)×L 2 (U) = · 2 H 1 (U) + · 2 L 2 (U) . The principals results of this paper are established in the two below theorems. Theorem 1.1. Let ( f , g) ∈ H 1 (Ω) × L 2 (Ω) and δ > 0 be fixed. For every T ≥ d(Ω) + δ , with exception of at most a finite of number of them, there is a extension ( f δ , g δ ) ∈ H 1 (R N ) × L 2 (R N ) of ( f , g) such that the solution of the Cauchy problem Then there is a control function h ∈ L 2 (Γ T ) such that the solution of Here (ν x , ν t ) denotes the outward unit normal vector on the surface Γ T at (x,t) point. The expression ν t u t − ∇u · ν x denotes the conormal derivative of u on Γ T at point (x,t). If Q T were a cylindrical domain we have ν t ≡ 0, so the control function h would be determined by normal derivative of u. In (1.7) one has a time dependent boundary condition. Such conditions is very important in mathematical physics when dealing with diffraction problems for wave equations that need to be limited in a region of the space and some wave signs in the boundary acquire a velocity normal to its surface (see [10] ).
We have mentioned above some papers that deal with different types of control problems for wave equations in non-cylindrical domains which use different controllability methods present in the literature. With respect the types of controllability problems the papers ( [14], [7], [2]) have worked about internal exact controllability while the papers ( [12], [8], [15], [4]) deal with exact boundary controllability. With respect the methods of controllability the references ( [7], [8], [15]) have used the HUM Method establish in [13] for obtain its desirable controllability. The papers ( [12], [4]) have worked near the Russell's controllability method which is established in [18]. By the end in ( [14], [2]) the authors obtain controllability by means of techniques based in terms of the Riemmannian metrics and geometric multipliers.
In the present paper we work in the light of Russell's controllability method in order to prove the Theorem 1.2 and also extension techniques showed in [12] in order to proof the Theorem 1.1. In the reference [12] the author used a extension technique, based in the Russell's controllability method, to obtain control for the standard wave equation in non cylindrical domains. As far as we know, there is no work in the literature that studies control for the Klein-Gordon equation in non-cylindrical domains using the methods as presented here. So, this is the contribution of this work. Finally, it is opportune to highlight the importance and usefulness of this method, because it is more easy to handle when dealing with domains that do not have a smooth geometry as proposed here, and this would be a difficulty to deal, for example, with the HUM method.
The rest of the present paper is organized in the following manner. The Section 2 is dedicated to do some essential preliminaries results. The Section 3 is devoted to prove the Theorem 1.2. The Theorem 1.1 is proved in the Section 4 and the paper is finalized with a references section.

SOME PRELIMINARIES
In this section we will highlight three important results that are essential in the proofs of the theorems proposed in the previous section. Such results are local energy decay estimates, analytic extension and suitable trace theorem to measure the regularity of the conditional derivative along of surfaces for the solution of the Cauchy problem to the Klein-Gordon equation.

Energy decay and analyticity
Considering the Cauchy problem It is known that the unique solution u of Cauchy problem (2.1)-(2.2) is such that u ∈ H 1 loc (R N ×R) and it was established in [16] explicit formulas for the solution u when t > diam(U) and x ∈ U.
Let u(.,t) be the solution of (2.1)-(2.2), of each t > 0 we define the operator solution S t : t)). The operator S t applies the initial state (u(., 0), u t (., 0)) into final state (u(.,t), u t (.,t)) and for t > d(U) such operator is compact, bounded and liner. In the proof of the Theorem 1.1 will be crucial the following result. 2) with initial data (u 0 , u 1 ) ∈ H 1 (R N ) × L 2 (R N ) compactly supported in U and ℜ the restriction operator to U. There exist positive real constants T 0 > d(U) and K, independent of u 0 and u 1 such that Tend. Mat. Apl. Comput., 21, N. 2 (2020) NUNES 375 for every t ≥ T 0 .
The proof of the above lemma is obtained by a direct manipulation of the Theorem 2.1 of [16].
Considering still u(.,t) as the solution of the Cauchy problem (2.1)-(2.2) with initial state (u(., 0), u t (., 0)) ∈ H 1 (R N ) × L 2 (R N ) compactly supported in U, another essential element for the proof of the Theorem 1.1 is the analytic extension of the map t → (u(.,t), u t (.,t)) to the sector Σ 0 = {ζ = T 1 + z, |arg(z)| ≤ π/4} as proved in [17]. For our purpose, such result can be adapted in terms of the operators S t in the next lemma.

Trace regularity
In this part we express a result on the regularity of the traces of the solution of the Klein-Gordon which it is essential to proof the Theorem 1.2. Let us begin take account some notations 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 ⊂ R N . Being Σ ⊂ Ξ an oriented smooth hypersurface which is time-like and non-characteristic with respect to P(ξ , D). Let η = (η 1 , · · · , η N ) be a unit normal to Σ. If ∑ a i j ∂ 2 ∂ ξ i ∂ ξ j is the principal part of P(ξ , D), then the expression ∂ u ∂ η = ∑ a i j ∂ u ∂ ξ i η j 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 the paper [19]. Considering Ξ ⊂ R N , with N ≥ 2, the Theorem 2 of [19] proves that if u ∈ H 1 loc (Ξ) is such that P(ξ , D)u ∈ L 2 loc (Ξ) then ∂ u ∂ η ∈ L 2 loc (Σ). Particularly, if we consider P(ξ , D) as being the Klein-Gordon operator, so its principal part will . Now, if γ is a smooth hypersurface in R N let us consider the surface γ ×R whose the unit normal vector is ν = (ν x , ν t ) where ν x = (ν 1 , · · · , ν N ). In this case the conormal derivative of u along γ × R is ∂ u ∂ ν = ν t u t − ∇u · ν x . Particularly, if we apply the trace result mentioned in the previous paragraph for the Klein-Gordon operator we obtain the following result. 2) with initial data (u 0 , u 1 ) ∈ H 1 (R N ) × L 2 (R N ). Let γ be a smooth hypersurface in R N , with N ≥ 2, with no self intersection and considers the surface γ × R which the unit normal vector is ν = (ν x , ν t ). Then the conormal derivative of u along γ × R has trace ν t u t − ∇u · ν x ∈ L 2 loc (γ × R).

PROOF OF THE THEOREM 1.2
Choose δ > 0 and T ≥ d(Ω) + δ such that Ω T ⊂ Ω. Given ( f , g) ∈ H 1 (Ω) × L 2 (Ω), take T ∈ [d(Ω) + δ , T ] for which exists an extension ( f δ , g δ ) according Theorem 1.1, and let u ∈ H 1 loc (R N × [0, +∞)) be the solution to the Cauchy problem (1.2)-(1.3) with initial data ( f δ , g δ ). Note that the solution u satisfy u(., T ) = 0 = u t (., T ) in Ω. Now, in order to obtaining the desired control function h we use the trace result available in the previous section. As we have considered the space dimension N ≥ 2 we can use the Lemma 2.3 to conclude that the trace of the conormal derivative of u is locally square integrable along of the surface Γ, that is ν t u t − ∇ u · ν x ∈ L 2 loc (Γ). Hence ν t u t − ∇ u · ν x ∈ L 2 (Γ T ). To finish the prove we define u =: u| Q T the restriction of u to the domain Q T and note that u(., T ) = 0 = u t (., T ) in Ω T . Now we extend u to a function u defined on Q T by setting u = 0 in Q T − Q T . After defines h := ν t u t − ∇u · ν x on Γ T . Note that u ∈ H 1 (Q T ) and satisfy u(·, T ) = 0 = u t (·, T ) in Ω T see that u and h satisfy the conditions of the Theorem 1.2.

PROOF OF THE THEOREM 1.1
We begin the proof with a similar construction that one made by Russell in [18]. Let (w 0 , w 1 ) ∈ H 1 (Ω) × L 2 (Ω) and Ω δ a δ -neighborhood of the domain Ω. Let E : where w 0 and w 1 are extension of w 0 and w 1 to R N respectively, both with compact support in Ω δ . Let w ∈ H 1 loc (R N × R) be the solution of the Cauchy problem Being w the solution of the Cauchy problem (4.1)-(4.2) and taking account the operator S T : as defined in the Section 2. By the Lemma 2.1, with U = Ω δ , we have the existence of positive real constants T 0 > d(Ω δ ) and K, independent of w 0 and w 1 such that is valid the estimate for every t ≥ T 0 and ℜ Ω δ is the restriction operator to Ω δ . Now, let ϕ ∈ C ∞ 0 (R N ) be a cut off function such that ϕ ≡ 1 in Ω δ 2 and ϕ ≡ 0 in R N \ Ω 3δ 2 . Note that (ϕ(.)w(., T ), ϕ(.)w t (., T )) ∈ H 1 0 (Ω δ ) × L 2 (Ω δ ). Following, we solve the Cauchy problem z(., T ) = ϕ(.)w(., T ), w t (., 0) = ϕ(.)w t (., T )) in R N , (4.5) NUNES 377 and we define the operator S T : z(., T ), z t (., T )) = (z(., 0), z t (., 0)) being z the solution of the Cauchy problem (4.4)-(4.5). It is important to highlight the relationship between the operators S T and S T as follows: Let P i be the projection of H 1 (U) × L 2 (U) onto H 1−i , i = 0, 1. Then So, from relations above we can guarantee that the operators S T have the same properties that the operators S T . So, for T > d(U) and the validity of the inequality for every t ≥ T 0 , being T 0 and K as in the inequality (4.3).
Taken w and z the solutions of the Cauchy problems (4.1)-(4.2) and (4.4)-(4.5) respectively, we define u := w − z. Note that the function u satisfy In terms of the operators S T and S T the equations above are equivalents to the equation Inserting the operator restriction to Ω, denoted by ℜ, the equation (4.11) becomes where K and C are constants independent from w 0 and w 1 . So, from the sequences of inequalities above we obtain, for T > T 0 where C is a constant dependent only K and C. Thus, by inequality (4.13) we guarantee that K T is contraction for a T > T 0 great sufficiently. This ensures the existence of T > d(Ω) great sufficiently for which the equation (4.12) has solution. However, we would like to establish a lower bound for values of T for which (4.12) is invertible. It is in this point that analytic extension expressed in the Lemma 2.2 will be useful. Because from relationships (4.6) and (4. , ζ ∈ Σ 0 }, where Σ 0 is complex sector {ζ = T 1 + z, |arg(z)| ≤ π 4 }, being T 1 any constant greater than d(Ω). After, we utilize the theorem of alternative of F. V. Atkinson (see [11] p. 370) to the effect either 1 is eigenvalue of each of the operators K ζ , ζ ∈ Σ 0 or else (I − K ζ ) −1 exists for all except at most for a finite number of values of ζ in each compact subset of Σ 0 . As observed, for a real ζ = T sufficiently large K ζ is a contraction, i.e., 1 is not eigenvalue of K ζ , hence the later possibility must be the case. So, for all T ≥ T 1 with the possible exception of a finite number of values, (I − K T ) −1 exists, concluding the proof of the Theorem 1.1. We suspect that there may be a extension ( f , g) ∈ H 1 (R N ) × L 2 (R N ) of ( f , g) ∈ H 1 (Ω) × L 2 (Ω) such that the solution of (1. . Similar result was obtained in [12] for even dimensional wave equation for Ω = B(0, 1). We intend to soon return to this subject.