On Carleman and observability estimates for wave equations on time‐dependent domains

We establish new Carleman estimates for the wave equation, which we then apply to derive novel observability inequalities for a general class of linear wave equations. The main features of these inequalities are that (a) they apply to a fully general class of time‐dependent domains, with timelike moving boundaries, (b) they apply to linear wave equations in any spatial dimension and with general time‐dependent lower order coefficients, and (c) they allow for smaller time‐dependent regions of observations than allowed from existing Carleman estimate methods. As a standard application, we establish exact controllability for general linear waves, again in the setting of time‐dependent domains and regions of control.


Introduction
In this article, we establish new Carleman estimates for the wave equation using a geometric approach. The main objective is to apply these estimates in order to derive novel observability inequalities for general linear wave equations, with the following features.
(I) The estimates apply to a general class of time-dependent domains, with moving boundaries.
(II) The estimates apply to wave equations in any spatial dimension. (III) The estimates apply to general linear waves with time-dependent lower order coefficients.
(IV) The estimates apply for a wide variety of time-dependent observation regions that are smaller than those in standard Carleman-based observability inequalities.
As a standard application of these observability estimates, we establish the exact controllability of linear waves on the same general class of time-dependent domains. Again, the region of control is allowed to be time-dependent proper subsets of regions found in classical results.
While the present paper deals only with wave equations on R 1+n (that is, on Minkowski space-time), the main Carleman estimates are proved using ideas and intuitions from Lorentzian geometry. As a result, the techniques presented here will also form the foundations for studying analogous questions for geometric wave equations on Lorentzian manifolds.

Background
In the context of evolutionary differential equations, the question of exact controllability is concerned with whether one can drive its solutions from any prescribed initial state to any desired final state at a later time, under the constraint that only some limited parameters in the system -the controls -can be set. If solutions of the differential equation represents a physical system, then the above asks whether one can completely govern the system through its controls.
1.1.1. Controllability of waves. To be more specific, let us consider a general linear wave equation on a bounded (spatial) domain Ω ⊆ R n . In this setting, the initial state and the final state each correspond to a pair of functions, representing the values of φ and ∂ t φ at given times. There are many possible choices for the control, with some common examples being the following.
• Interior controllability: An additional forcing term F on the right-hand side of (1.1).
• Dirichlet boundary controllability: Part of the Dirichlet boundary data for (1.1).
• Neumann boundary controllability: Part of the Neumann boundary data for (1.1).
Controllability of wave equations, and Problem 1.1 in particular, has been a topic of research for several decades. Here, we give a brief, and by no means exhaustive, survey of some existing research. For more complete discussions, see, for example, [46] and the references within.
Examples of early results include the works of Russell [52,53]. Modern treatments of controllability are derived from the Hilbert Uniqueness Method (HUM) of Lions [42,43]. (This is also closely related to the more abstract functional analytic framework developed in [19].) The main point is that by duality, controllability is equivalent to uniqueness properties of the adjoint differential equation. More specifically, for the present setting, in order to establish an affirmative answer to Problem 1.1, the main hurdle is to prove the observability inequality for solutions ψ of the wave equation adjoint to (1.1). Here, ∂ ν ψ denotes the Neumann boundary data for ψ, and Γ is as described in Problem 1.1. When the inequality (1.2) holds, the HUM machinery then yields exact controllability, with Dirichlet boundary controls supported in Γ. For wave equations, there is a fundamental obstruction preventing exact controllability that is due to finite speed of propagation. Indeed, some minimum amount of time is required for information from the boundary controls to travel to all of Ω. In particular, this implies a lower bound on the time span τ + − τ − (which depends on the region where the control is placed) required for any boundary controllability, and hence observability, result to hold † . Many methods have been developed for proving the crucial observability estimate (1.2). For simplicity, let us first restrict our attention to the free wave equation, ψ := −∂ tt ψ + Δ x ψ = 0. (1.3) For wave equations in one spatial dimension, observability estimates have been proved for many PDEs using Fourier methods. In the context of (1.3), these results are often based on applications of Ingham's inequality [29] and its generalizations to the Fourier series expansions of ψ. Moreover, such methods are capable of retrieving the optimal time span for control, as dictated by finite speed of propagation. For detailed discussions of Fourier methods, the reader is referred to [6]. We also remark that similar methods were recently used to prove (1.2) in higher dimensions [24].
Other proofs of (1.2) used multiplier methods, in which one integrates by parts an expression where X represents an appropriately chosen first-order operator. Using this technique, Ho [25] showed that for a sufficiently large time span τ + − τ − , the observability estimate (1.2) indeed holds for solutions ψ of (1.3), for a control region Γ of the form Γ := (τ − , τ + ) × {y ∈ ∂Ω | (y − x 0 ) · ν > 0}, (1.4) where x 0 ∈ R n is fixed, and where ν is the outward-pointing unit normal for Ω. † Note that (1.4) contains all points x ∈ ∂Ω such that the ray in R n emanating from x 0 and passing through x is exiting Ω at x. In particular, one need not apply the control on all of [τ − , τ + ] × ∂Ω. A modification of the above argument (see [43]) resulted in a lower bound on the time span, (1.5) required for the inequality (1.2) to hold, with Γ again given by (1.4). Furthermore, using 'rotated' multipliers, Osses [49] obtained analogues of (1.2) with different boundary regions Γ. Multiplier techniques can also be used to treat some wave equations of the form (1.1), as well as some more general hyperbolic equations; see [47], for instance. However, these results generally fail to recover lower bounds of the form (1.5) for the required time span. For more detailed discussions on multiplier methods and its roles in observability estimates, see [37].
Equation (1.2) has also been established using microlocal methods, yielding optimal results with regards to the requisite time span and control region. Of particular note is the seminal result of Bardos, Lebeau, and Rauch [8]. Consider now the boundary region Then, (1.2) holds for Γ as in (1.6) if and only if the geometric control condition (GCC) is satisfied: Roughly, every null geodesic in (τ − , τ + ) × Ω -with the condition that it is reflected whenever it hits the boundary (τ − , τ + ) × ∂Ω -intersects some point of Γ; see [8,13]. Note that the necessity of the GCC follows from Gaussian beam constructions that propagate along these geodesics; see [50]. The result was generalized by Burq in [12], which reduced the regularity required. A more modern and constructive proof of (1.2) along similar directions was given in [39]. Furthermore, [40] extended the above results to time-dependent subsets Γ of the boundary that satisfy the GCC; this further optimized the control regions required for (1.2) to hold.
Microlocal methods can also be adapted to some wave equations (1.1) with lower order terms, with the caveat that the coefficients (X t , X x , V ) are time-independent, or at most analytic in the time variable † . Moreover, these methods also apply to (time-independent) geometric wave operators −∂ tt + Δ h on product manifolds of the form [−τ − , τ + ] × M , where (M, h) is a Riemannian manifold with or without boundary, and where Δ h is the h-Laplace-Beltrami operator on M .
Carleman estimates represent yet another family of tools that has been useful for deriving observability. These are, roughly, weighted (space-time) integral inequalities which contain an additional free parameter λ > 0. Historically, Carleman estimates have been used extensively for establishing unique continuation properties of various PDEs, in particular with coefficients that fail to be analytic. This theory, pioneered by Carleman [15], Calderón [14], and Hörmander [26], among others, has been developed over several decades; we refer the reader to [59] for a general survey.
In the context of Problem 1.1, Carleman estimates are advantageous due to their robustness, in that they are applicable to a wide range of settings. For instance, in contrast to multiplier methods, they allow one to treat wave equations (1.1) with arbitrary lower order coefficients (with sufficient regularity), while still achieving the some control regions of the form (1.4) as well as the lower bound (1.5). In particular, by taking the free parameter λ in these estimates to be as large as necessary, one can 'absorb' away any potentially dangerous contributions from lower order terms.
For example, Carleman estimates were applied toward proving observability and controllability of wave equations (with lower order terms) in [38,64,65]. Additional adaptions of the Carleman estimate method can be found in [9,22], among many others. For further discussions, see [23].
On one hand, methods based on Carleman estimates lack the precision of microlocal methods and do not achieve the GCC in general. However, Carleman methods apply to a wider class of settings, including wave equations with time-dependent lower order coefficients, without any assumption of analyticity in time. Moreover, Carleman estimates have been extended to geometric wave equations, given additional assumptions on the (time-independent) geometry; see, for example, [20,22,61].
For this article, we are particularly concerned with lower order coefficients X t , X x , V that vary in both space and time, without any presumption of analyticity. As we wish for our results to be as widely applicable as possible, we resort to Carleman estimates methods here.
Finally, we remark that the Carleman methods applied in [9,22,38,64,65] dealt only with the case in which x 0 ∈Ω. (This was in contrast to multiplier methods, which allows for all x 0 ∈ R n .) In this article, we will also remove this restriction and consider cases with x 0 ∈ Ω. where the domains Ω τ are bounded open subsets of R n that vary smoothly with respect to τ . For example, this could represent a physical system that is itself accelerating.
In comparison to static settings, there has been a relatively small amount of research on timedependent, noncylindrical domains. Below, we briefly survey of the some existing literature.
An early study that predated the HUM is that of Bardos and Chen [7], which established interior controllability for free waves on domains U that are expanding in time. In particular, the result was proved by deriving energy decay bounds and adopting stabilization techniques. Furthermore, using geometric methods, the above result was extended to geometric (free) wave equations in [45], under additional assumptions on the background geometry.
Using the HUM, Miranda [48] established Dirichlet boundary controllability for free waves on a class of time-dependent domains with the following features.
• On the other hand, U need not be expanding nor contracting at all times.
The result is established by applying a change of variables to convert the problem into that of boundary controllability for a more general hyperbolic equation on a time-independent, cylindrical domain. The transformed problem is treated in [47] using multiplier methods and the observation that the PDE asymptotes to the standard wave equation at large times.
The problem in one spatial dimension has been further studied by several authors in recent years. For example, [17,57] (see references within for earlier works) used multiplier methods to study the case in which the boundary is given by two timelike lines 0 and (with 0 generally taken to be vertical). More recently, [54,55] considered these problems using Fourier methods. In these particular cases, the optimal time span required for control was obtained in some of the above articles. Analogous problems with more general boundaries (that are not lines) have been considered; one example is [63], which does not achieve the optimal time span.
One of the primary goals of the present article is to tackle this problem of Dirichlet boundary controllability of wave equations (with lower order coefficients varying nonanalytically in both space and time) on time-dependent domains in full generality. In particular, we make no assumptions on the shape of our domain U , aside from its boundary being timelike; in particular, U need not be expanding, contracting, or self-similar in time. To the author's knowledge, the present paper provides the first results regarding controllability of waves in moving domains in this generality.
These results will be established using a novel global Carleman estimate that is entirely supported in the exterior of a null cone. Furthermore, we obtain the best required time span for controllability that have been achieved via multiplier and Carleman estimate methods.

The main results
In this section, we state, at least roughly, the main results of this paper. More specifically, we begin by describing the main observability inequalities, as well as how they improve upon existing results. We then discuss the new Carleman estimate that is used to derive these observability estimates, as well as the main ideas involved in its derivation.
Our results will apply to spatially bounded but time-dependent domains of the form (1.7) which also have a smooth timelike boundary. We will refer to these domains throughout the paper as generalized timelike cylinders; see Definition 2.11 for a precise definition of these domains.
1.2.1. A preliminary estimate for free waves. We begin by first presenting a 'warm-up' estimate that applies only to the free wave equation on generalized timelike cylinders. While this estimate is, in multiple ways, strictly weaker than our main results, it does allow us to first discuss the effects of time-dependent domains on observability apart from the other details of the main results.
The 'warm up' estimate can be roughly stated as follows: Let U be a generalized timelike cylinder of the form (1.7), fix a point x 0 ∈ R n , and fix 'initial' and 'final' times τ ± ∈ R, with τ − < τ + . Moreover, we assume Then, for any smooth solution φ of we have the observability estimate 11) where N denotes the Minkowski outer-pointing unit normal of U , and where A precise version of Theorem 1.2, using notations developed in the article, is found in Theorem 2.23. In particular, despite its preliminary nature, Theorem 1.2 already achieves the properties (I) and (II) listed in the beginning of the introduction, at least in the setting of free waves.
The proof of Theorem 1.2 is based on Lorentzian geometric adaptations of the classical multiplier estimate found in [43]. The key points of this proof are as follows. [43] is now replaced by the (Minkowski) gradient of f * . • The proof uses the divergence theorem for Lorentzian manifolds; see [62,Appendix B.2].
For further details, the reader is referred to the proof of Theorem 2.23. We now discuss some of the main features of Theorem 1.2 and its relations to previous literature. First, observe that when U is time independent, so that R + = R − , then (1.8) reduces to the standard bound (1.5) on the time span. Moreover, (1.8) can be argued in terms of finite speed of propagation: Information placed on the boundary at time t = τ − needs time at most R − to travel to x 0 , and this then needs time at most R + to travel back to the boundary at t = τ + .
Next, since the domain U may be changing in time, the outer unit normal N may have a nonzero t-component. Here, we use N to denote the Minkowski, rather than Euclidean, normal to N . More specifically, ifN := (ν t , ν) denotes the Euclidean normal vector field to U , then N := (ν t , ν) = (−ν t , ν). (1.13) Note that the control region Γ * from (1.12) is simply the classical region (1.4), except that the condition (y − x 0 ) · ν > 0 from (1.4) is now replaced by N f * | (τ,y) > 0. In particular, observe that when U is time independent, (1.12) reduces precisely to the standard region (1.4), since (1.14) Remark 1.5. Although the standard observability estimates obtained from multipliers or Carleman estimates can be thought to be centered about the point x 0 in space, in Theorem 1.2, one would view the estimate as being centered about the event (t 0 , x 0 ) in the space-time. In fact, the function f * in (1.4) precisely measures the squared Minkowski distance to (t 0 , x 0 ). However, as was seen in (1.14), the contribution of 't 0 ' is not seen unless the domain U is changing in time.
More generally, one can compute that By the definition of N and ν t in (1.13), we can observe the following.
• If ν t > 0 at a point of ∂U , then U is expanding in time at that point.
• If ν t < 0 at a point of ∂U , then U is shrinking in time at that point.
Therefore, (1.15) implies the following general principle when t > t 0 : • where U is expanding, N f * is less positive than (x − x 0 ) · ν. Thus, the region Γ * of observation is smaller than the standard region (1.4) near points where U is expanding; • where U is shrinking, N f * is more positive than (x − x 0 ) · ν. Thus, the region Γ * of observation is larger than the standard region (1.4) near points where U is shrinking.
On the other hand, for times t < t 0 , the above relations are reversed: Γ * becomes larger wherever U is expanding, while Γ * becomes smaller wherever U is shrinking.
1.2.2. Observability estimates. From the 'warm-up' observability estimate, Theorem 1.2, one can already see (from (1.12)) the effect of the moving boundary on the region of observation. However, Theorem 1.2 fails to achieve the features (III) and (IV) at the beginning of the introduction.
• Theorem 1.2 only applies to the free wave equation (1.3), and not to general linear wave equations of the form (1.1), that is, with lower order terms.
• The region of observation can be even significantly improved from (1.12).
As mentioned before, Carleman estimate methods have been successful in handling the first point above, without any additional assumption of independence or analyticity in time for the lower order coefficients. Here, we will similarly extend Theorem 1.2 to linear wave equations of the form (1.1) -hence achieving property (III) -using our upcoming Carleman estimates.
With regards to the second point above, recall that the preliminary estimate of Theorem 1.2 is in principle centered about an event (t 0 , x 0 ) of the space-time. Another particularly novel consequence of the main Carleman estimates of this article is that the region of observation can be further restricted to the exterior D * of the null cone about (t 0 , x 0 ). In fact, the main Carleman estimate of this paper (see Theorem 3.1) is itself supported entirely on this exterior D * .
With the above in mind, we now give a rough statement of our main observability estimates. For simplicity of exposition, we avoid stating the most general cases here. Theorem 1.6. Let U be a generalized timelike cylinder of the form (1.7), fix a point x 0 ∈ R n , and fix 'initial' and 'final' times τ ± ∈ R, with τ − < τ + . In addition: • assume the bound (1.8) holds, and let t 0 ∈ (τ − , τ + ) such that (1.9) holds; • fix V, X t ∈ C ∞ (Ū ), and let X x ∈ C ∞ (Ū ; R n ); • let N denote the Minkowski outer-pointing unit normal of U , let f * be defined as in (1.12), and let Γ † denote the following subset of ∂U : Then, for any smooth solution φ of we have the observability estimate The most general observability results can be found in Theorems 4.1 and 4.5, handling the cases in which (t 0 , x 0 ) lies outside and insideŪ, respectively. A precise version of Theorem 1.6, which is a corollary of Theorems 4.1 and 4.5, can be found in Theorem 4.14. Note in particular that in contrast to previous results in the literature using Carleman estimate methods [9, 22, 38, 64, 65], we do not require that the point (t 0 , x 0 ) lies outside ofŪ in Theorem 1.6.
Observe that the assumptions of Theorem 1.6, captured in (1.7), (1.8), and (1.9), are essentially the same as in the 'warm-up' Theorem 1.2. The first point of departure is that Theorem 1.6 applies to the Dirichlet boundary problem for general linear wave equations of the form (1.1).
Remark 1.7. Theorem 1.6, as stated, applies only to smooth solutions φ of (1.17), with smooth coefficients X x , X t , V . However, as is standard, the regularities of both the solutions and the coefficients can be significantly lowered by examining more precisely the integrability and differentiability conditions required in the proofs throughout the article. Since regularity is not presently a concern, we avoid discussing these points in this article for simplicity.
The second difference between Theorems 1.2 and 1.6 is in the observation region, in particular, between Γ * in (1.12) and Γ † in (1.16). In particular, the condition τ − < t < τ + within Γ * (specifying that Γ * lies within the initial and final times) is replaced by the condition f * > 0 in Γ † (specifying that Γ † lies in the exterior of the null cone about (t 0 , x 0 )). In Lorentzian geometric terms, Theorem 1.6 improves upon existing results by further restricting the observation region to events that are not causally related to (t 0 , x 0 ). See  Remark 1.9. The assumptions (1.8) and (1.9) imply the region Γ † in (1.16) is a proper subset of Γ * in (1.12), so Theorem 1.6 represents an improvement in terms of the observation region. Note, however, that the observation region in Theorem 1.2 is exactly Γ * , whereas the observation region Y † in Theorem 1.6 must be strictly larger than Γ † (that is, any neighborhood ofΓ † ).
Compared to other results using multiplier and Carleman methods, Theorem 1.6 yields a strict improvement of the region of observation to time-independent subsets of ∂U . However, the regions Γ † are in general weaker than the geometric control condition obtained from microlocal methods (under additional assumptions of time-independence or time-analyticity). Thus, as far as the author is aware, the regions considered in Theorem 1.6 represent the best known for non-analytic wave equations, as well as for wave equations on time-dependent domains.
Remark 1.10. In addition, for general time-dependent U , the observation region Γ † in (1. 16) can be interpreted as a relativistic modification of the standard condition (x − x 0 ) · ν > 0. To explore this further, let us consider a point (τ, y) ∈ Γ † . We can then perform a Lorentz boost centered about the event (t 0 , x 0 ) to obtain a new inertial coordinate system (t , x ) on R 1+n such that In other words, (τ, y) and (t 0 , x 0 ) are simultaneous with respect to (t , x ).
Since the function f * is invariant with respect to such Lorentz boosts, then, where ν denotes the spatial component of N in the boosted (t , x )-coordinates. In other words, the condition N f * | (τ,y) > 0 is simply the standard condition (x − x 0 ) · ν > 0, but now with respect to a boosted inertial coordinate system in which (t 0 , x 0 ) and (τ, y) are simultaneous.
We also note the discrepancy between Γ † , which has the relativistic interpretation from Remark 1.10, and the actual observation region Y † from (1.18). In particular, Y † is strictly larger than Γ † , though by an arbitrarily small amount. This difference arises from two technical points.
• First, in the main Carleman estimates, we work not with f * , but rather with a perturbation f ; see Theorem 3.1 and (3.5) for details. † The reason is that the level sets of f * are only zero pseudoconvex with respect to the wave operator, and hence a Carleman estimate obtained using f * fails to control the full H 1 -norm of the solution ‡ .
• Furthermore, for the 'interior' observability estimates, where (t 0 , x 0 ) ∈ U, the result is obtained by adding together two Carleman estimates centered about two nearby points. This is needed since the weights in the Carleman estimate degenerate at the center point, hence the H 1 -norm is not controlled near there. For details, see Theorem 4.5.
Remark 1.11. In fact, when U is time independent and (t 0 , x 0 ) ∈Ū (that is, the situation treated by existing Carleman estimate methods in the literature), the observation region Y † in (1.18) can be further improved to Γ † , which by (1.14) now has the form This is precisely the portion of the standard observation region (1.4) that lies outside the null cone centered at (t 0 , x 0 ). See Theorem 4.1 and Corollary 4.12 for details and precise computations. On the other hand, when (t 0 , x 0 ) ∈ U, one does not expect that Y † can be replaced with Γ † , since in this case, Γ † violates the geometric control condition.
Remark 1.12. In the case n = 1, Theorem 1.6 recovers all the results in the existing literature, including the articles [17, 54, 55, 57, 63] mentioned previously, up to the optimal observation time § . In addition, Theorem 1.6 further extends previous results to general cases in which ∂U is given by two timelike curves, and to wave equations with arbitrary lower order coefficients.
Finally, we note that Theorem 1.6 again implies exact controllability: Corollary 1.13. Assume the definitions and hypotheses of Theorem 1.6. Then, the linear wave (1.1) is exactly controllable, with initial and final data on U ∩ {t = τ − } and U ∩ {t = τ + }, respectively, and with Dirichlet boundary control supported in Y † (all in the appropriate spaces).
Remark 1.14. The initial and final data for the wave equation need not be placed on level sets of t. In fact, with regards to Corollary 1.13, any pair of space-like cross-sections of U lying in the past and future of the null cone about (t 0 , x 0 ) would suffice; see the discussion in Section 5.2. This is an immediate consequence of (geometric) energy estimates for linear wave equations.

Carleman estimates.
As mentioned before, our main observability results, in particular Theorem 1.6, are obtained using a new Carleman estimate (derived in Section 3), which is the foundation of this article. Roughly speaking, this Carleman estimate has the following features. † Note, however, thatf can be made arbitrarily close to f * by adjusting the parameter ε within. ‡ Earlier linear and nonlinear Carleman estimates obtained from f * were derived in [4,5]. § However, Theorem 1.6 does not treat the case in which τ + − τ − is exactly equal to the optimal time.
• The estimate is supported in the exterior of a null cone about a point (t 0 , x 0 ) ∈ R 1+n . In fact, this property implies, through the standard arguments, that the resulting observability inequalities, such as Theorem 1.6, inherit this exterior support property.
• The estimate is established through Lorentzian geometric techniques. By taking advantage of methods that exploit the full space-time geometry, we obtain results that are equally applicable to domains with either fixed or moving boundaries. † The precise Carleman estimate of this paper is stated in Theorem 3.1 and proved throughout Section 3. Below, we give a brief summary of some of the ideas behind its derivation.
The Lorentzian geometric perspective toward Carleman estimates for wave equations had its roots in other applications. For example, in mathematical relativity, such estimates were used to establish rigidity properties for stationary black hole space-times; see, for example, [1,2,16,30,31].
Regarding the main Carleman estimates of this article, its ideas arose from the study of unique continuation properties of waves 'from infinity'. In [3], the following result was shown using geometric Carleman estimates: If a linear wave vanished to infinite order at a 'large enough' portion of both future and past 'null infinity' (formally, where the radiation field is manifested), then the wave must vanish locally near that portion of infinity ‡ . Furthermore, this infinite-order vanishing is necessary, as one can construct solutions of (1.3) that decay like r −k for any k ∈ N.
Next, [4] showed that in R 1+n , the infinite-order vanishing assumption can be removed provided the wave is globally regular. § Technically speaking, the infinite-order vanishing condition arose from the use of cutoff functions in the standard argument proving unique continuation from Carleman estimates. This was overcome in [4] using new global Carleman estimates for which the associated weight itself vanishes on the null cone about the origin. In particular, this vanishing weight removed the need to use a cutoff function in the unique continuation argument. In fact, a similar weight is also responsible for the exterior support property in the estimates of this paper.
Despite the global nature of the Carleman estimate in [4], it is equally applicable to finite space-time regions U ⊆ R 1+n . While this yields additional boundary terms on ∂U , the vanishing Carleman weight again ensures that no boundary terms are found on the null cone itself. A nonlinear variant of this estimate was applied in [5] to study singularity formation for focusing subconformal nonlinear wave equations. The key step with the Carleman estimate was to control the wave in the interior of a past time cone, based at a singular point, by its values on the cone itself.
Although this is closely related to boundary observability estimates, which also aim to control a wave in the interior of a cylindrical region U by its values on the boundary, the results of [4,5] unfortunately fail to imply such observability. The main reason is that these Carleman estimates make heavy use of the hyperbolic function f * from (1.12), whose level sets fail to be strictly pseudoconvex ¶ . As a result, these estimates only control the L 2 -norm in the bulk, rather than the full H 1 -norm that is required for observability and exact controllability.
In the existing literature on observability via Carleman estimates, this lack of pseudoconvexity was overcome by altering f * . More specifically, one instead considers the level sets of which are now strictly pseudoconvex. This property indeed allows for the recovery of the H 1 -norm in the bulk. However, this f c is now poorly adapted to the geometry, in particular the characteristics, of the wave operator. In particular, one cannot recover, using this f c , the aforementioned exterior support property of the Carleman estimates from [4,5].
In this paper, we instead consider different alterationsf of f * , with the following properties.
•f is well adapted to the characteristics of the wave operator. In particular, like for f * , the level set {f = 0} corresponds to the null cone about (t 0 , x 0 ).
• The level sets off are strictly pseudoconvex in the exterior of the null cone about (t 0 , x 0 ).
In particular, by making use of the abovef , one can both recover the H 1 -norm in the bulk and preserve the exterior support property in our Carleman estimates.
While these properties off can be viewed as the consequences of extensive computations, they can also explained through the perspective of conformal geometry. In this respect, the main idea in this paper is to consider instead a 'warped' metricḡ that slightly inflates the volumes of spatial spheres about the origin; see Definition 3.4. Moreover, this warped metric g can be shown to be conformally isometric to the usual Minkowski metric on R 1+n ; see Proposition 3.18.
A fairly direct computation (see Proposition 3.14) shows that the positive level sets of the function f * , with (t 0 , x 0 ) = (0, 0), are strictly pseudoconvex with respect toḡ. Then: • Since pseudoconvexity (with respect to geometric wave operators) is a conformally invariant property, the pullbackf of f * through the aforementioned conformal isometry is pseudoconvex as well; this allows for the bulk H 1 -bounds in our Carleman estimates; • Since the characteristics of geometric wave operators are also conformally invariant, this f is also as well adapted to the characteristics of the (flat) wave operator as f * is. As a result of this, we can still derive the external support property using thisf in the place of f * .
The above points provide the main ideas behind why the Carleman estimates in this paper work.
Remark 1.15. In fact, by conformal invariance, we can derive Carleman and observability estimates for wave equations with respect to other geometries that are conformally related to the Minkowski metric. However, we will not pursue details of this in the present paper. Now, regarding the proof of the main Carleman estimate, the strategy proceeds as follows.
(1) Prove a Carleman estimate for wave equations with respect to the 'warped' metricḡ.
(2) Use the abovementioned conformal isometry and invariance in order to extract a corresponding Carleman estimate for the usual (flat) wave operator on R 1+n .
Steps (1) and (2) are proved in Sections 3.2 and 3.1, respectively. We remark that although this process still involves extensive computations, the benefit is that the main steps are relatively simple. Remark 1.16. As previously mentioned, the existing literature on Carleman-based observability [9, 22, 38, 64, 65] only considered settings with (t 0 , x 0 ) ∈Ū (in the context of time-independent U ). At a technical level, this was because one required uniform positivity of |x − x 0 | in order to obtain a positive zero-order bulk term in the corresponding Carleman estimate. An interesting feature of the exterior support property in our main Carleman estimate is that one no longer requires uniform positivity for |x − x 0 | for a positive zero-order bulk. As a result, our main Carleman estimate also holds when (t 0 , x 0 ) ∈ U and hence is applicable toward 'interior' observability estimates.
Finally, we remark that this warping of the metric, in particular the difference between the Minkowski and warped metrics, is primarily responsible for the discrepancy between the Γ † and the actual observed region Y † in Theorem 1.6. Moreover, since this warping can be made arbitrarily small, the discrepancy between Γ † and Y † can be made small as well.

Other directions.
A larger aim beyond the present paper is to study to similar controllability properties for geometric wave equations, in particular to settings with timedependent geometry. As far as the author is aware, there is no literature directly addressing controllability for waves on general Lorentzian manifolds. While we do not discuss this here, our broader intention is to develop tools that can be robustly applied to geometric settings in future works.
Next, since the methods in this article do not require time-analyticity for its coefficients, they could also be used for treating nonlinear wave equations. As already mentioned, earlier versions of the main Carleman estimate have been used in [5], in the context of studying singularity formation.
Another application of the estimates in this article (and their future geometric generalizations) is toward inverse problems for wave equations in settings with time-dependent domains and moving boundaries.

Outline of the paper
The remainder of this paper will be organized as follows.
• In Section 2, we define the objects and notations that we will use throughout the paper. In particular, we describe the domains -the generalized timelike cylinders -that we will consider in our main results. We also give a precise statement of the preliminary observability estimate given in Theorem 1.2, and we give a short, simple proof of this estimate.
• In Section 3, we state and prove our main Carleman estimate, Theorem 3.1. In particular, within the proof, we introduce the 'warped' metric and establish its basic properties.
• Section 4 is dedicated to the the precise statements of the main observability inequalities of this paper, as well as to the proofs of these inequalities.
• In Section 5, we discuss some consequences of the main observability results. In particular, Section 5.1 explores the special case of one spatial dimension, while Section 5.2 deals with the exact controllability results that follow from our main inequalities.

Preliminaries
In this section, we set the notations and definitions that we will use throughout the paper. Using these definitions, we will then give a precise formulation and a proof of the preliminary observability estimate for free waves that was stated in Theorem 1.2.
Definition 2.1. Throughout, we will adopt the following standard notations for inequalities.
• We write A B to mean that there is some universal constant C > 0 such that A CB. Moreover, A a1,...,am B means that the above constant C depends on a 1 , . . . , a m .
• We write A B to mean that there is some sufficiently small and universal constant c > 0 such that A cB. Moreover, A a1,...,am B means that this c depends on a 1 , . . . , a m .

The geometric setting
Since the methods we use are Lorentz geometric in nature, we begin our discussions by setting the basic notations relating to Minkowski geometry -the background naturally associated to the wave operator. In addition, we give a precise definition of generalized timelike cylinders, the time-dependent domains on which our wave equations are set.
2.1.1. Minkowski Geometry. We begin by recalling the Minkowski metric on R 1+n : Definition 2.2. We define the following on R 1+n .
• Let t and x = (x 1 , . . . , x n ) denote the usual Cartesian coordinates on R 1+n , mapping to the first and the remaining n components of R 1+n , respectively.
• Let g denote the Minkowski metric on R 1+n : In particular, we refer to the manifold (R 1+n , g) as Minkowski space-time.
Definition 2.3. We also define the following standard functions on R 1+n .
• Let u and v denote the null coordinates: • Let f denote the hyperbolic function, Remark 2.4. The nonzero level sets of f are one-sheeted (f > 0) and two-sheeted (f < 0) hyperboloids on R 1+n , while {f = 0} is the null cone with vertex at the origin.
For our purposes, it would also be convenient to consider space-time translations of the functions from Definitions 2.3. Thus, we will also make use of the following notations: • We define the shifted time and spatial coordinates by • We define the shifted radial coordinate, null coordinates, and hyperbolic function by Definition 2.6. Recall the following standard coordinate systems on R 1+n .
• On {r = 0}, we recall the standard polar coordinates (t, r, ω) and null coordinates (u, v, ω), with ω being the angular coordinate taking values in S n−1 . In addition, we let ∂ t , ∂ r , ∂ u , ∂ v denote the coordinate vector fields with respect to these coordinate systems.
• We can also shift the above coordinate systems by any P ∈ R 1+n . More specifically, on {r P = 0}, we have the shifted polar coordinates (t P , r P , ω) and null coordinates (u P , v P , ω), with ω ∈ S n−1 now denoting the angular value about the axis {r P = 0}. Again, we let ∂ tP , ∂ rP , ∂ uP , ∂ vP denote the coordinate vector fields with respect to these coordinate systems.
Recall that the Minkowski metric can be written in terms of polar and null coordinates as whereγ is the unit round metric on the level spheres of (u P , v P ).
Definition 2.7. For tensor fields on R 1+n , we adopt the following index conventions.
• We use lowercase Greek letters (ranging from 0 to n) for space-time components in R 1+n .
• We use lowercase Latin letters (ranging from 1 to n − 1) to denote angular components, corresponding to the ω ∈ S n−1 in any of the coordinate systems in Definition 2.6.
• As is standard, we will adopt the Einstein summation notation -repeated indices in both subscript and superscript refer to sums over all possible index values.
• Unless stated otherwise, indices will be raised and lowered using g and its metric dual.
Definition 2.8. We adopt the following notations for covariant derivatives.
• Let ∇ denote the Levi-Civita connection with respect to g. In particular, for an appropriately defined function A, we write ∇A to mean the space-time differential of A.
• Let ∇ denote the gradient with respect to g; more specifically, for an appropriate scalar function h, we let ∇ h be the g-metric dual of ∇h. † • Let denote the wave operator with respect to g: • Let / ∇ denote the angular connections on the level spheres σ of (u, v), that is, the Levi-Civita connections associated with the metrics / g induced by g on the spheres σ. Similarly, we use / ∇ P to denote the angular connections on the level spheres of (u P , v P ).
Definition 2.9. Let P ∈ R 1+n . We define the domains D P and D by that is, the exteriors of the null cones about P and the origin, respectively.
Definition 2.10. Let P ∈ R 1+n and A ⊆ R 1+n .
• Let I + (P ) and I − (P ) denote the chronological future and past, respectively, of P : • Let I + (A) and I − (A) denote the chronological future and past of A: Geometric timelike cylinders. Next, we define precisely the domains (in general, time-dependent and with moving boundaries) on which we will consider our wave equations. • U ⊆ R 1+n , and ∂U is a smooth timelike hypersurface of R 1+n ; • for any τ ∈ R, the set Ω τ := {y ∈ R n | (τ, y) ∈ U} is nonempty, bounded, and open in R n ; • there is a smooth future-directed timelike vector field Z onŪ , with Z| ∂U tangent to ∂U .
In the above, we refer to Z as a generator of U .
Remark 2.12. Note that if U is a GTC, then we have the following.
• With Ω τ as in Definition 2.11, we can write • Any twoΩ τ0 andΩ τ1 are diffeomorphic, as manifolds with boundaries. For instance, one can identifyΩ τ0 withΩ τ1 by flowing along the integral curves of any generator Z of U .
Remark 2.13. The regularity of ∂U and the generator Z in Definition 2.11 could be lowered considerably without altering the main results of this paper. However, since regularity is not a main focus here, we avoid exploring optimal regularities to avoid technical baggage.
Example 2.14. Suppose Ω is bounded and open subset of R n , with a smooth boundary ∂Ω. Then, the static cylinder R × Ω is a GTC, and ∂ t is a generator of R × Ω.
Suppose U is a GTC. Given a coordinate system (y 1 , . . . , y n ) on Ω 0 (see Definition 2.11), we can lift the functions y k to U by transporting them along the integral curves of a generator Z. With respect to the coordinates (t, y 1 , . . . , y n ), we have that U R × Ω 0 . Thus, we can characterize GTCs as domains that can be reparametrized as static cylinders (in which the static direction is timelike).
Next, we describe geometrically the hypersurfaces on which one can impose Cauchy (that is, initial or final) data for a linear wave equation on a GTC: • V is a smooth space-like hypersurface of U , and ∂V ⊆ ∂U ; † • if Z is a generator of U , then every integral curve of Z hitsV exactly once.
Note that any two cross-sections V 1 and V 2 of a GTC must be diffeomorphic, since V 1 can be identified with V 2 by flowing along the integral curves of any generator. Next, we develop some basic terminology for comparing two cross-sections of a GTC.
Definition 2.16. Let U be a GTC, and let V − , V + be two cross-sections of U .
and we let ∂U (V − , V + ) denote the region of ∂U between V − and V + , The most basic examples of cross-sections are constructed from the level sets of t: • Given τ ∈ R, we define the cross-section (2.14)

Linear wave equations
The next task is to give a precise description of the linear wave equations that we will consider throughout this paper: Solve the following linear wave equation onŪ : The assumption (2.15) that the coefficients X and V in P are smooth is purely for convenience. All the results in the paper still hold, as long as X and V are sufficiently regular. For instance, our main observability results remain true when X and V are merely uniformly bounded † .
2.2.1. Energy estimates. Next, we establish standard energy estimates, one global and one local, for solutions of Problem 2.18 with vanishing Dirichlet data. Before stating and proving the energy estimates, however, we first require some notations regarding the sizes of tensorial objects: where X t , X x 1 , . . . , X x n represent the components of X in Cartesian coordinates.
The following global energy estimate can be viewed as a geometric extension of the modified energy estimates of [20,66]. As was the case in [20], these modified estimates allow us to optimize our dependence on the potential in our main observability estimates. † The assumptions for V could be further weakened (for instance, to L p -integrability) with more refinements to the proofs in this article. However, we avoid doing this here in order to avoid additional technical details. Proposition 2.21. Let U be a GTC, and let V ± be cross-sections of U with V − < V + . Moreover, consider the linear wave equation of Problem 2.18, and let Then, there exist constants C, C > 0, depending on U and V ± , such that Proof sketch. First, we fix some t ∈ C ∞ (Ū ) satisfying the following.
• ∇ t is everywhere timelike and past-directed (that is, t is a 'time coordinate').
• There exist τ ± ∈ R such that We also define the modified energy as well as the modified stress-energy tensor: Let Z be a generator of U ; we then apply the usual multiplier method to φ, with multiplier Z. First, an application of the (Lorentzian) divergence theorem on U (V − , V + ) yields (2.24) where N and N denote the outward and inward pointing unit normals to U (V − , V + ), respectively. Since Z is future timelike, then standard computations yield that Moreover, since Z and N are orthogonal on ∂U , and since φ vanishes on ∂U For the bulk term in (2.24), we expand and apply (2.16) to estimate Combining (2.24)-(2.26) and (2.28), we obtain the inequalities By varying V ± to be other level sets of t, we see that (2.29) holds for arbitrary τ ± ∈ R. The desired estimate (2.20) now follows from (2.29) and from Gronwall's inequality.
We will also need a variant of Proposition 2.21 that is localized to a null cone exterior.
. Moreover, we consider Problem 2.18, and we let Proof sketch. Again, we fix a 'time coordinate' t ∈ C ∞ (Ū ) such that: • ∇ t is everywhere timelike and past-directed; Moreover, let Q be defined as in (2.23), and define the localized energy, We only prove the case U t(P ) < V (that is, τ 0 < τ 1 ), as the remaining case V < U t(P ) can be derived analogously. We also assume V ∩ D P = ∅, since the result is trivial otherwise. Again, we let Z be a generator of U . The argument proceeds like the proof of Proposition 2.21, (2.34) where N and N denotes the outward and inward pointing unit normals to U (V − , V + ), respectively, and where the last term I denotes an integral over the null boundary U (U t(P ) , V) ∩ ∂D P . By computations similar to those in the proof of Proposition 2.21, we obtain as well as For the remaining null boundary integral, standard computations using Q imply that I 0. † Next, using (2.16), we can bound Varying V yields that (2.38) holds for any τ 1 > 0, and (2.31) follows from Gronwall's inequality.

2.2.2.
The preliminary result. We conclude this section by proving the preliminary observability estimate for free waves described in Section 1.2.1. More specifically, we establish the following:

39)
and fix t 0 ∈ (τ − , τ + ) such that Then, for any smooth solution φ of we have the observability estimate 42) where N denotes the Minkowski outer-pointing unit normal of U , and where Proof. By the translation symmetry of Minkowski space-time, we can assume without loss of generality that P = (t 0 , x 0 ) := 0 (and hence f P = f ). For convenience, we also define Note that direct computations yield The proof of (2.42) proceeds in a manner analogous to the multiplier proof in [43], except that here we use S * φ as our multiplier. First, recalling (2.41) and (2.44), we see that (2.46) For I 2 , we integrate by parts (using the Lorentzian divergence theorem [62]) to obtain In particular, for the boundary terms along ∂U τ−,τ+ in (2.48), we observed that the only nonzero components of ∇φ lie in the N -direction. Combining (2.46)-(2.48), we then obtain Now, at t = τ ± , we recall (2.51) and bound By (2.39), the first two integrands in the right-hand side of (2.53) can be bounded by Furthermore, the last term in (2.53) can be simplified via integrating by parts (using the classical divergence theorem on a domain in R n ) and recalling from (2.41) that φ vanishes on ∂U : Finally, combining (2.52) and (2.56), we conclude that In the last step of (2.57), we observed that the sign of N f (N φ) 2 in the boundary integral is simply the sign of N f , hence the part of ∂U τ−,τ+ where N f < 0 can be discarded. The desired estimate (2.42) now follows by applying the assumption (2.40) (with t 0 = 0) to (2.57). †

Carleman estimates
The objective of this section is to establish the main Carleman estimate of this paper: Theorem 3.1. Let U be a GTC, and fix P ∈ R 1+n . Moreover, assume that there exists Fix also ε, a, b > 0, and suppose the following conditions hold: Then, there exist constants C, C > 0 -which are independent of ε, a, b, R -such that given any we have the Carleman estimate

4)
where: • ζ P a,b;ε is the associated Carleman weight • N is the outer-pointing unit normal of U (with respect to g).
The proof of Theorem 3.1 consists of two main steps.
• First, by a conformal transformation, we reduce the desired estimate (3.4) to a corresponding Carleman estimate in the exterior D of a null cone, but with respect to a 'warped' Minkowski metric. This step is carried out in Section 3.1, where we discuss this warped metric in detail and then perform this conformal reduction.
• In Section 3.2, we prove the Carleman estimate in the warped Minkowski space-time.

The warped geometry
This section is devoted to discussing the warped geometries that form the backbone of the proof of Theorem 3.1. We first establish some basic properties of these geometries, and we then demonstrate how they are conformally related to the Minkowski metric. Finally, at the end of this section, we reduce the proof of Theorem 3.1 to that of establishing a corresponding Carleman estimate on the warped space-time; see Theorem 3.22.
3.1.1. The warped metric. The warped metric can be formally defined as follows: Definition 3.3. Fix a constant ε ∈ R, which we refer to as the warping factor.
Definition 3.4. We define the ε-warped Minkowski metric on R 1+n \ {r = 0} bȳ whereγ is the unit round metric on the level spheres of (u, v), and where ρ is the warped radius:ρ

7)
Remark 3.5. Note thatḡ and its associated objects depend on the warping factor ε.
We will adopt the following notational conventions regarding objects associated withḡ.
Definition 3.6. In general, objects defined with respect to the warped metricḡ are denoted with the same symbols as the corresponding objects defined with respect to the Minkowski metric g, except that these symbols will have a 'bar' over them. For example: • let∇ denote the Levi-Civita connection with respect toḡ; • let¯ :=ḡ αβ∇ αβ denote the wave operator with respect toḡ; • let/ ∇ denote the connections induced byḡ on the level spheres of (u, v).
Remark 3.7. Note thatḡ is simply the Minkowsi metric g when ε = 0. In particular, all of the general properties that we establish forḡ will also apply to g.
We now list the results of some basic computations involvingḡ: Proposition 3.8. The following properties hold.
• Theḡ-gradient∇ f of f satisfies • The nonzero components of∇ 2 f , in null coordinates, are given bȳ

9)
• f also satisfies the following identities: In the above, all indices are raised and lowered usingḡ.

11)
as well as the following wave equation: Next, we recall the region D from Definition 2.9, representing the exterior of the (Minkowski) null cone about the origin. We can then establish the following basic inequalities on D: Proposition 3.10. The following inequalities hold on D: Furthermore, if ε 0, then the following inequality holds on D: Proof. First, (3.13) follows immediately from (2.2), (2.3), and (2.8). Then, by (3.7), we have Since f > 0 on D and ε 0, taking square roots of the above yields (3.14).
Next, we derive the null convexity properties of level sets of f (with respect toḡ) whenever ε is positive † . This is most straightforwardly shown using a frame that is adapted to f . Definition 3.11. We define the following vector fields on D, both of which are normal to the level spheres of (u, v).
Proposition 3.12. T is everywhere tangent to the level sets of f , while N is everywhere normal to the level sets of f . Moreover,∇ 2 f satisfies the following: For our purposes, the null convexity is most conveniently captured in the quantity π defined below. This will be exploited in the proof of the 'warped' Carleman estimate, Theorem 3.22.
Definition 3.13. We define the following modified deformation tensor: In future computations, we will also make use of the following quantity: Proposition 3.14. The nonvanishing components ofπ are given bȳ (3.19) † Null convexity (or pseudoconvexity, in more general contexts) of these level sets can be characterized as follows:∇ XX f > 0 for all null vector fields X that are tangent to the level sets of f . In particular, observe that the preceding property is a consequence of the results forπ in Proposition 3.12.
Furthermore,w satisfies the following wave equation: Remark 3.15. In particular, one can show from the first two identities in (3.19) that for any null vector field X tangent to the level sets of f . In other words, these level sets are (ḡ)-null convex in the direction of increasing f . This null convexity is also connected to the more general notion of pseudoconvexity (with respect to¯ ) in unique continuation theory; see [26,41].
3.1.2. The conformal isometry. The next step is to describe how this warped geometry, as described in Definitions 3.3 and 3.4, is conformally related to Minkowski geometry.
In addition, for future convenience, we define the function Remark 3.17. Note that by (3.13), the right-hand side of (3.21) is well defined.
Proposition 3.18. Assume the setting of Definition 3.16. Then: • the following identities hold: We now establish a number of elementary comparisons showing that objects with respect tō g differ little from corresponding objects with respect to g: (1 + εu) n 1, (1 − εv) n 1, ξ n 1; (3.26) • the following comparisons hold on D ∩ {r < R}: • the following comparisons hold on D ∩ {r < R}: (3.28) • for any open subset Ω ⊆ D andφ ∈ C 1 (Ω), we have Proof. Since 0 < −u, v < R by (3.13), the smallness assumption on ε implies the first two comparisons in (3.26); the remaining part of (3.26) now follows from (3.

Then,Φ(U ∩ D) is an open subset of D, and its boundary in D is the hypersurfaceΦ(∂U ∩ D).
Proof. This is an immediate consequence ofΦ being a conformal isometry (see Proposition 3.18) and of the fact that conformal isometries preserve the causal geometry. Fix also ε, a, b > 0 such that the following conditions hold: Then, for any φ ∈ C 2 (Φ(U ∩ D)) ∩ C 1 (Φ(Ū ∩ D)) that is uniformly C 1 -bounded and satisfies where all integrals are with respect to volume forms induced byḡ, and where: •Φ is as given in Definition 3.16; • ζ a,b is the (warped) Carleman weight •N is the outer-pointing unit normal ofΦ(U ∩ D) (with respect toḡ).
Remark 3.23. Note that by the conclusions of Proposition 3.21, both the integral alonḡ Φ(∂U ∩ D) and the outer unit normalN toΦ(U ∩ D) are well-defined objects.
Remark 3.24. The factors f 1/2 present in the last term of (3.33) and in (3.34) could be replaced by f p , for some p > 0, provided the constants of the inequality (3.33) are also adjusted; see, for instance, a similar estimate in [4]. However, we will not need this flexibility here.
Below, we show that by assuming Theorem 3.22, we can recover our main Carleman estimate, Theorem 3.1. This effectively reduces the proof of Theorem 3.1 to that of Theorem 3.22.
Proof (Theorem 3.22 ⇒ Theorem 3.1). Assume the hypotheses of Theorem 3.1; in particular, we let R, U , a, b, ε, φ be as in its statement. Throughout the proof, we will use 'dḡ' and 'dg' to denote integrals with respect to volume forms induced byḡ and g, respectively. Also, by the translation symmetry of Minkowski space-time, we can assume without loss of generality that P = 0.
Consider the (uniformly C 1 -bounded) function We now pull each integral in (3.36) back through the diffeomorphismΦ and bound some extraneous factors. First, for the last term on the right-hand side of (3.36), we have that where we in particular note the following.
• The extra factor ξ −n−1 arises from the change of volume forms fromḡ to g.
• By (3.23), all instances of f in the left-hand side of (3.37) are replaced by ξ −1 f . Note that in particular, this results in the weight ζ a,b being replaced by ζ 0 a,b;ε ; see (3.5). • The last inequality in (3.37) is a consequence of (3.26).
By similar reasoning, we also obtain (3.38) where the factorρ −1 is also handled using (3.23). Next, writinḡ and recalling (3.25), we see (using the same reasoning as before) that For the boundary term, we observe from Proposition 3.21 thatΦ(∂U ∩ D) is mapped viā Φ −1 to ∂U ∩ D. Moreover, Proposition 3.21 implies that the push-forward ofN throughΦ −1 is precisely ξN , where N is as in the statement of Theorem 3.1. As a result, we have where in the last step, we also used (3.3).
For the remaining first-order bulk terms, we apply (3.21), (3.26), (3.28), and (3.29) to obtain for some universal constants C, C 1 , C 2 > 0. A similar computation yields for some constants C, C , C > 0. By (3.1), (3.2), and (3.13), we have In particular, (3.44) implies that the last two (negative) terms in the right-hand side of (3.43) can be absorbed into the third last (positive) term. This results in the inequality for some constants C, C > 0. Theorem 3.1 now follows from (3.45) and the identity

Proof of Theorem 3.22
It remains only to prove the warped Carleman estimate, Theorem 3.22, in order to complete the proof of Theorem 3.1. This is the topic of the present subsection. Throughout (only) this subsection, we will assume that all indices are raised and lowered usingḡ. We begin by defining some auxiliary quantities to aid with the proof:  Furthermore, for brevity, we will let denote differentiation with respect to f .
Proof. These are direct computations.
3.2.1. The pointwise identity. The first step in proving Theorem 3.22 is to establish a pointwise identity for the conjugated wave operatorL defined in (3.47): Lemma 3.28. Assume the hypotheses of Theorem 3.22, and suppose also that ψ ∈ C 2 (Φ(U ∩ D)). Then, at every point ofΦ(U ∩ D), we have the identity

49)
where the reparametrization A := A(f ) of f is given by

50)
and where the currentP :=P [ψ] is given bȳ Proof. First, we expandLψ by applying Proposition 3.8, Definition 3.13, and Definition 3.25:L where A is as in (3.50). Multiplying the above byS w ψ yields (3.52) We can now use Proposition 3.8 and Definition 3.13 to obtain Thus, lettingP we see from (3.52) and the above that Consider next the stress-energy tensor for¯ applied to ψ, and recall that∇ In addition, we note that Consequently, defining the modified current and recalling (3.17) and (3.19), we see that The desired identities (3.49)-(3.51) now follow from (3.53), (3.54), (3.56), and (3.57).

3.2.2.
A pointwise inequality. The next step is to derive from Lemma 3.28 a pointwise inequality, in which all the nondivergence terms have a definite sign.

58)
whereÑ denotes the operatorÑ Proof. Throughout the proof, we let C denote positive universal constants that may change between lines. Note that from

Reversal of conjugation.
We now establish a pointwise inequality for the original wave operator by undoing the conjugation process from Definition 3.26.

The integral estimate.
In order to complete the proof of Theorem 3.22, we integrate (3.66) overΦ(U ∩ D). However, here we must be especially careful with boundary terms.
Let φ be as in the statement of Theorem 3.22. For sufficiently small δ > 0, we define the sets on which r,ρ, f are all bounded from above and below by positive constants; see (3.13), (3.14), and (3.30). Integrating (3.66), with the above φ, over G δ and recalling (3.48), we obtain Taking a limit as δ 0 and applying the monotone convergence theorem yields provided the limit on the left-hand side exists. Next, we apply the divergence theorem to the term with the limit. Noting that {f = δ} is a timelike hypersurface of (D,ḡ), with outward (from {f > δ}) unit normal −f − 1 2S , we obtain where in the last step, we applied (3.68) and the fact that φ is uniformly C 1 -bounded. Moreover, applying the estimate (3.69), the remaining limit can be bounded as where C > 0 is some constant depending on both φ and R.
To control this limit, we foliate H δ using level sets of t and apply the coarea formula. Let Dt denote theḡ-gradient of t on H δ , with respect to the induced metric. Note thatDt and T must point in the same direction, since both vector fields are tangent to H δ and are normal to the level sets of (u, v). As a result, by (2.2) and (3.15), we see that Moreover, noting from (2.3) that then the coarea formula, along with (3.13) and (3.30), yields where we recalled in the last step that 2a > 3 2 by (3.31). Finally, combining (3.72)-(3.75) yields (3.33) and completes the proof of Theorem 3.22.

Observability estimates
In this section, we apply the Carleman estimates of Theorem 3.1 to prove our two most general observability estimates for solutions of linear wave equations on GTCs. The exterior and interior estimates are discussed in Sections 4.1 and 4.2, respectively. Some corollaries of these estimates, such as Theorem 1.6, are discussed in Section 4.3.

Exterior observability
We now discuss the first of our general observability inequalities, based on Carleman estimates centered outside the given GTC.
Theorem 4.1. Let U be a GTC in R 1+n , and fix P ∈ R 1+n \Ū and 0 < δ 1. In addition, assume U ∩ D P is bounded, and consider the setting of Problem 2.18. Moreover: • let X , V be as in (2.15), and define the constants • let N be the outward-pointing (g-)unit normal to U , and let Then, there exist C, N > 0, depending on U , such that the observability estimate, holds for any solution φ ∈ C 2 (U ) ∩ C 1 (Ū ) of (2.16) that also satisfies φ| ∂U∩DP = 0.
Remark 4.2. Note that the assumption that U ∩ D P is bounded in the statement of Theorem 4.1 implies that all of the constants M 0 , M 1 , R ± in (4.1) are finite. Remark 4.3. Note that the dependence of (4.3) on M 0 and M 1 matches those found in [20]. In particular, the dependence on M 0 is known to be sharp; see the discussions in [20].
Remark 4.4. Since r P = 0 on D P , the quantity S in (4.2) is well defined and smooth.
In principle, the proof of Theorem 4.1 is similar to the standard process of obtaining observability from Carleman estimates. The primary new technical difficulty is that the pseudoconvexity and the Carleman weight in (3.4) degenerate toward the null cone centered about the point P . As a result, one must be considerably more careful when absorbing terms (mainly, those involving ∇φ) in the Carleman estimate. In particular, various constants must be tracked more carefully.
The remainder of this section is dedicated to the proof of Theorem 4.1.

Application of the Carleman estimate.
Since U ∩ D P is bounded, it follows from (4.1) that (3.1) holds, with R := R + . In addition, choose a n large enough so it also satisfies and choose ε and b as follows (note in particular that (3.2) holds, with R + in place of R): The key step is to apply the Carleman estimate (3.4), with the above values of a, b, ε, to our given U and P . From (3.4), as well as (2.16), (3.13), (4.2), and (4.5), we see that Here, all notations are as in the statement of Theorem 3.1. Let I 1 , I 0 , I Γ denote the first, second, and third terms in the left-hand side of (4.6), respectively. Recalling (3.13) and our conditions (4.4) for a, we see that As a result, I 0 can be absorbed into the right-hand side of (4.6): We now partition U ∩ D P into , Note that on U > , we have from (3.13), (3.26), and (4.8) that Thus, shrinking domains from U ∩ D P to U > , we see from (4.7) that We now decompose I 1 as From (3.13), (4.1), and (4.4), we obtain Thus, I 1,> can be absorbed into the right-hand side of (4.9), and we have

Applications of energy estimates.
Since ∂U is timelike, then by (2.2) and (4.1), we have for any τ ∈ R with |τ − t(P )| 1 4 R − . As a result, by (2.3), (3.5), and (4.12), we have that , for the same τ as above. In particular, we see that so that by shrinking domains, applying Fubini's theorem, and recalling (4.12) again, we obtain Since (4.1) and (4.4) imply that a R + R − , we see that and hence we conclude (4.14) Applying the energy estimate (2.20) to each U τ in (4.14) yields where the constants C, K > 0 also depend on U . Hence, it follows that For I 1, , we use (3.5), (3.13), (3.26), (4.5), and (4.8) to bound We then recall (4.1), (4.4), and (4.10) to obtain Note that since ∂U is timelike, then (4.1) implies that t P | U∩DP must be bounded by R + . Consequently, applying (2.31) and Fubini's theorem to the above yields where C , K > 0 also depend on U . Combining (4.15) and (4.16) then yields We now claim that For this, we need only to show that which is a consequence of (4.4), since Having proved (4.18), we combine this with (4.17) and obtain Combining the above with (4.19) then yields Choosing a n in (4.22) so that it also satisfies the assumptions (4.4) results in (4.3).

Interior observability
In this subsection, we prove the second of our general observability inequalities, based on applying Carleman estimates that are centered within U .
Remark 4.6. Again, the assumption that U ∩ (D P1 ∪ D P2 ) is bounded implies that the constants M 0 , M 1 , R ± in (4.24) are finite. Note also the functions S i in (4.25) are again well defined and smooth.
The main difference between the exterior and interior estimates (Theorems 4.1 and 4.5, respectively) is that the latter requires the application of two Carleman estimates, about two distinct reference points (P 1 , P 2 in Theorem 4.5). The technical reason is that the weights in the right-hand side of the Carleman estimate (3.4) vanish at the center point P , which now lies in the domain.
Remark 4.7. Moreover, for any P ∈ U, the boundary region ∂U ∩ D P fails the geometric control condition. Indeed, null geodesics traveling through P (that is, along ∂D P ) fail to touch ∂U ∩ D P .
We prove Theorem 4.5 in the remainder of this subsection. Much of this proof is analogous to that of Theorem 4.1; for those parts, we omit some details and refer the reader to Section 4.1.

4.2.1.
Applications of the Carleman estimate. Fix i ∈ {1, 2}. Note from (4.24) that Let a n be large enough to also satisfy and choose ε and b as follows: Applying (3.4), with a, b, ε as above, to U and P i yields where we also recalled (2.16), (3.13), (4.25), and (4.29). Let I i 1 , I i 0 , I i Γ be the terms in the left-hand side of (4.30). Like in the proof of Theorem 4.1, we apply (3.13) and (4.28) to show that I i 0 can be absorbed into the right-hand side of (4.30): We now partition U i into , and we decompose I i 1 as Since (3.13) and (3.26) imply that on a similar argument as in the proof of Theorem 4.1, using (3.13), (4.24), and (4.28), imply that I i 1,> can be absorbed into the right-hand side of (4.31). As a result, (4.31) now becomes (4.33)

Applications of the energy estimates.
Since ∂U is timelike, then on any we have from (2.2) and (3.5) that . (4.34) In particular, we see that and, similar to the proof of Theorem 4.1, it follows from (4.28) that Next, we observe that by (4.24) and the triangle inequality, Thus, summing (4.35) over i ∈ {1, 2} and applying (2.20) yields where C, K > 0 depend on U . Moreover, analogous to the proof of Theorem 4.1, we apply (2.31), (3.5), (3.13), (4.24), (4.28), and (4.32) to estimate, for some C , K depending on U , By the same reasoning as in the proof of Theorem 4.1 (see (4.18)), we have that so that combining all the above with (4.36) and (4.37) yields The remainder of the proof again proceeds similarly to the corresponding argument in (4.40) (In the last step, we used the direct analogue of (4.19) in the current setting.) Making a suitable choice for a n in accordance with (4.28) results in (4.26).

Some corollaries
Finally, we discuss various consequences of Theorems 4.1 and 4.5.

The observation region.
Our first corollary compares the observation regions from Theorems 4.1 and 4.5 with corresponding regions in existing Carleman method results. For this, we examine the quantities S and S i in (4.2) and (4.25), whose signs determine the observation region.
Remark 4.9. Note that by (4.42), the condition N r P > 0 is equivalent to both that is, the criterion for the observation region obtained from standard Carleman estimate methods for linear wave equations on a time-independent domain; see, for instance, (1.4) and [38,43,64].
Using Proposition 4.8, we can now reformulate Theorems 4.1 and 4.5 so that the regions of observation are characterized in terms of the aforementioned angle θ ν,P : Theorem 4.10. Let U be a GTC, and fix P ∈ R 1+n \ ∂U and 0 < δ 1. In addition: • let N be the outward-pointing (g-)unit normal to U , let ν t and ν be defined as in (4.41), and let θ ν,P : ∂U → (−π, π] denote the angle between ν and x P ; • consider the setting of Problem 2.18, and let X , V be as in (2.15); • assume U ∩ D P is a bounded subset of R 1+n , and let M 0 , M 1 be as in (4.1); • let Γ P,δ denote the following boundary region: Then, for any solution φ ∈ C 2 (U ) ∩ C 1 (Ū ) of (2.16) that also satisfies φ| ∂U∩DP = 0: • if P ∈Ū , then we have the observability inequality • if P ∈ U, then for any open subset Y P,δ of ∂U containingΓ P,δ , we have that Remark 4.11. In particular, recalling Remark 4.9, we can make, modulo restrictions to D P , the following comparison between the region Γ P,δ in Theorem 4.10 and the classical region θ P,δ > 0.
• At points where t P ν t > 0 (that is, where U expands away from P ), the observation region Γ P,δ is smaller than the corresponding region cos θ ν,P > 0 in classical results.
• At points where t P ν t < 0 (that is, where U shrinks away from P ), the observation region Γ P,δ is larger than the corresponding region cos θ ν,P > 0 in classical results.
• At points where t P ν t = 0, the observation region Γ P,δ matches the classical results.
Proof. First, we consider the case P ∈Ū . Applying Theorem 4.1 to U , P , and δ yields for φ as in the assumptions of the theorem, and with Γ + as defined in (4.2). Thus, to complete the proof in the case, it suffices to show that this Γ + is contained in Γ P,δ . Let R + be as in (4.1). Applying (4.42) and (4.43), with ε := δ 2 R −1 + , we see that on Γ + , or equivalently, Using that |t P | r P R + on ∂U ∩ D P , we observe that: • if t P ν t < 0, then (4.47) implies Thus, comparing the above with (4.44), we conclude that Γ + ⊆ Γ P,δ , as desired. Next, consider the case P ∈ U, and let Y P,δ be as assumed. Since U ∩ D P is bounded,Γ P,δ is compact. Thus, by continuity, we can find nearby P 1 , P 2 ∈ U, with t(P 1 ) = t(P 2 ) = t(P ), with where Γ Pi,δ is defined as in (4.44), but with P replaced by P i . Applying Theorem 4.5 to U , (P 1 , P 2 ), and δ yields for φ as in the assumptions of the theorem, and with Γ 1 + , Γ 2 + as defined in (4.25). † Consequently, to complete the proof, it suffices to show that Γ 1 + ∪ Γ 2 + is contained in Y P,δ . For this, we apply (4.42) and (4.43) as before in order to obtain Combining this with (4.48) yields, as desired,

Static domains.
Next, we consider the special case of time-independent domains mentioned in Example 2.14. Applying Theorem 4.10 to this setting results in the following: In addition: • fix x 0 ∈ R n , as well as τ ± ∈ R satisfying • consider the setting of Problem 2.18, and let X , V be as in (2.15); • let ν : ∂Ω → R n represent the outer unit normal to Ω in R n , and define Then, for any solution φ ∈ C 2 (U ) ∩ C 1 (Ū ) of (2.16) that also satisfies φ| ∂Uτ − ,τ + = 0: • if x 0 ∈Ω, then we have the observability inequality (4.52) • if x 0 ∈Ω, then for any open subset Y x0;τ± of ∂U containingΓ x0;τ± , (4.53) Remark 4.13. In particular, we can directly compare the results in Corollary 4.12 with those from classical Carleman-based methods. Recalling Remark 4.9, we see that: • when x 0 ∈Ω, the observation region Γ x0,τ± is simply the observation region from classical results, but further restricted to the null cone exterior D P ; • when x 0 ∈Ω, the observation region Y x0,τ± is strictly larger than the standard observation region restricted to D P , though by an arbitrarily small amount. This arises from the fact that the interior estimate (4.26) must be applied at two different points.
Proof. First, we observe that ν in the theorem statement coincides with the ν defined in Theorem 4.10. In addition, here ν also represents the Minkowski outer normal to U .
Let us first assume that x 0 ∈ ∂Ω, and hence P ∈ ∂U . We now apply Theorem 4.10 to the above U and P , along with any fixed 0 < δ 1. We claim that Γ x0;τ± coincides with Γ P,δ , as defined in (4.44); this is a consequence of the following observations.
The application of Theorem 4.10 now splits into two cases.

A unified estimate.
We conclude with a precise version of the result roughly stated in Theorem 1.6. While this is slightly weaker (in terms of the observation region) than Theorems 4.1 and 4.5, it unifies the interior and exterior estimates and provides a cleaner statement.
Theorem 4.14. Let U ⊆ R 1+n be a GTC, and fix x 0 ∈ R n and τ ± ∈ R such that • In addition, choose t 0 ∈ (τ − , τ + ) so that • consider the setting of Problem 2.18, and let X , V be as in (2.15); • let N denote the outer-pointing (g-)unit normal of U , let (4.56) and let Y † be a neighborhood ofΓ † in ∂U .
Then, for any solution φ ∈ C 2 (U ) ∩ C 1 (Ū ) of (2.16) that also satisfies φ| ∂Uτ − ,τ + ∩DP = 0, Proof. We divide the proof into cases depending on whether P , as defined in (4.56), lies in U. First, if P ∈Ū , we apply Theorem 4.1 to U , P , and some 0 < δ 1 to be fixed later; this yields with Γ + as in (4.2). By continuity and (4.1), taking δ arbitrarily small makes S (as defined in (4.2)) arbitrarily close to N f P . Thus, by taking 0 < δ 1 sufficiently small (depending on Y † ), we have that Γ + ⊆ Y † . Recalling (2.20) and (4.58) then yields, as desired, Next, if P ∈ ∂U , then we apply Theorem 4.1 to a point P near P that is not inŪ . Again, by continuity, if P is sufficiently near P and δ is sufficiently small, then the resulting control region Γ + (now associated with P ) lies inside Y † , and (2.20) yields the desired bound: Finally, when P ∈ U, we apply Theorem 4.5, with respect to some 0 < δ 1 and distinct points P 1 , P 2 ∈ U with t(P ) = t(P 1 ) = t(P 2 ). By continuity, if δ is sufficiently small, and if P 1 , P 2 are sufficiently close to P , then Γ 1 + ∪ Γ 2 + (as defined in (4.25)) lies in Y † , and the result follows from (2.20) and (4.26) in a manner analogous to the preceding two cases.

Consequences and applications
In this section, we provide further discussions relating to the observability results in Section 4.

The case n = 1
The first objective is to take a closer look at GTCs in one spatial dimension. Here, we extend known results in the literature; we show, in full generality, that one can recover observability for linear waves on time-dependent domains up to the optimal required time span.
More specifically, throughout this subsection, we will study the following setting: Definition 5.1. Fix two curves 1 and 2 in R 1+1 , parametrized as 1 (τ ) := (τ, λ 1 (τ )), 2 (τ ) := (τ, λ 2 (τ )), (5.1) where λ 1 , λ 2 : R → R are smooth and satisfy, for all τ ∈ R, We now let U be the region bounded by 1 and 2 , that is, In particular, the second and third conditions in (5.2) imply that 1 and 2 are timelike, and hence ∂U is timelike as well. We also observe the following basic facts: Proposition 5.2. Let the region U ⊆ R 1+1 be as in Definition 5.1. Then, U is a GTC, and the outward-pointing (g-)unit normal N to U satisfies  (2.15). In addition, assume: • there exists T − > 0 such that the forward, leftward null ray emanating from the point (τ − , λ 2 (τ − )) ∈ 2 intersects 1 at time τ − + T − , that is, • there exists T + > 0 such that the forward, rightward null ray emanating from the point • the following relation holds: Then, for any solution φ ∈ C 2 (U ) ∩ C 1 (Ū ) of (2.16) that also satisfies φ| 1∪ 2 = 0, we have Remark 5.5. Theorem 5.3 states that, with initial data at t = τ − , we have observability from 2 if the time span is strictly greater than T + + T − . Moreover, this time span T + + T − is precisely what is required by the geometric control condition and hence is optimal.
5.1.2. Two-sided observation. Next, we consider the case in which both 1 and 2 are observed. (In particular, this corresponds to the problem of imposing controls on both 1 and Theorem 5.6. Let U be as in Definition 5.1, and fix τ − , τ +,1 , τ +,2 ∈ R. Consider the setting of Problem 2.18, in the case n = 1, and let X , V be as in (2.15). In addition, assume: • there exists T 1 > 0 such that the forward, leftward null ray emanating from the point (τ − , λ 2 (τ − )) ∈ 2 intersects 1 at time τ − + T 1 , that is, • there exists T 2 > 0 such that the forward, rightward null ray emanating from the point (τ − , λ 1 (τ − )) ∈ 1 hits 2 at the time τ − + T 2 , that is, • the following relations hold: Then, for any solution φ ∈ C 2 (U ) ∩ C 1 (Ū ) of (2.16) that also satisfies φ| 1∪ 2 = 0, we have Remark 5.7. Again, like in the preceding one-sided setting, the time span max(T 1 , T 2 ) implied by Theorem 5.6 exactly matches the sharp value dictated by the geometric control condition.
Proof. Let t 0 ∈ R be such that In other words, t 0 is the time at which the forward, leftward null ray from (τ − , λ 2 (τ − )) ∈ 2 and the forward, rightward null ray from (τ − , λ 1 (τ − )) ∈ 1 intersect. We now set the point P to be slightly above this intersection point of the two null rays described above: By continuity, choosing d sufficiently small, we have P ∈ U , and: • the forward, leftward null ray from P hits 1 before time τ +,1 ; • the forward, rightward null ray from P hits 2 before time τ +,2 .
We now apply Theorem 4.10, with U = U and P as above, and with 0 < δ 1. Similar to the proof of Theorem 5.3, letting θ ν,P be as in the statement of Theorem 4.10, we see that Therefore, we obtain from (5.12)-(5.14), (5.17), and our choices of δ and P that for some ε > 0, with Γ P,δ as in (4.44). Finally, note that the desired observation region is an open subset of ∂U containingΓ P,δ . Thus, (5.15) follows from (4.46) and the above.
Proof. This is completely analogous to the proof of Corollary 5.8.
Remark 5.11. Again, let us assume, on top of Corollary 5.10, that 1 is a vertical line: Then, the optimal times for one-sided and two-sided observability, respectively, reduce to

Exact controllability
It is well known that observability inequalities are necessary for establishing exact controllability properties; see the discussion in Section 1.1. Here, we briefly describe this connection in the context of time-dependent domains with moving boundaries.

5.2.1.
Well-posedness. The first step in this discussion is to precisely define the relevant spaces and norms. In the subsequent definitions, we accomplish this in the context of GTCs: Definition 5.12. Let M be a submanifold of (R 1+n , g), and suppose its induced metric is either Riemannian or Lorentzian. We then define the following norms and spaces. where the integral is defined with respect to the metric induced by g.
In order to state the usual well-posedness theorems for wave equations, we will need to define additional functional spaces on cross-sections of GTCs.
Definition 5.13. Let U be a GTC, and let V be a cross-section of U .
• For any φ ∈ C ∞ 0 (V), we define where γ is the (Riemannian) metric on V induced by g, where the integrals on the right-hand side are with respect to γ, and where Dφ denotes γ-gradient of φ.
We now state the standard existence and uniqueness results for the linear wave equations of Problem 2.18 that will be relevant to Dirichlet boundary controllability (see also [21,44]).
Theorem 5.14. Let U be a GTC in R 1+n , let Z be a generator of U , let N be the outer (g-)unit normal of U , and let V be a cross-section of U . Also, consider the setting of Problem 2.18. Then: • Given any initial data (φ 0 , φ 1 ) ∈ H 1 0 (V) × L 2 (V), there exists a unique φ ∈ L 2 loc (U ) that solves the wave equation (2.16) and satisfies, in a trace sense, Furthermore, for any cross-sections V ± such that V − < V < V + , • Given any initial data (φ 0 , φ 1 ) ∈ L 2 (V) × H −1 (V) and boundary data φ b ∈ L 2 (∂U ), there is a unique solution φ ∈ L 2 loc (U ) of (2.16) that satisfies, in a trace sense, Furthermore, for any cross-sections Remark 5.15. A technical issue inherent to Theorem 5.14 (and more generally to the theory of hyperbolic PDEs) is establishing the precise sense that φ is a solution of Problem 2.18.
, the solution φ can be interpreted as a weak solution of (2.16), via integrations by parts and the Hilbert space structures of the spaces H 1 0 (V ± ). This is described, for a slightly simpler class of hyperbolic PDEs, in [21].
• More generally, as long as we assume X and V are smooth, then φ, in all the above, can also be interpreted as a distributional solution of (2.16).

Controllability and observability.
Using the well-posedness results of Theorem 5.14, we can now give a precise statement of the exact controllability problem we will consider: Definition 5.16. Let U be a GTC in R 1+n , let Z be a generator of U , and fix cross-sections V ± of U with V − < V + . Moreover, we consider the setting of Problem 2.18 on U .
there exists φ b ∈ L 2 (∂U ), supported in Γ, such that the solution of (2.16) satisfying † The following theorem summarizes, again in the context of GTCs and Problem 2.18, the connection, via the HUM (see [43]), between observability and exact controllability.
where ψ denotes the solution of the adjoint problem † such that the following hold.
As is standard, we can now combine Theorem 5.17 with our main observability results in order to establish exact Dirichlet boundary controllability for Problem 2.18 on time-dependent domains. Below, we demonstrate this for the unified observability estimate of Theorem 4.14.
Then, the equation (2.16) is exactly controllable on U τ−,τ+ , with control on Y † . More generally, under the above assumptions, given any cross sections V ± of U that satisfy Proof. Consider a solution ψ of the adjoint problem (5.32), with V − := U τ− and Z being any generator of U . Note the wave equation in (5.32) is of the same form as those in Problem 2.18. Thus, whenever ψ is smooth, we can apply Theorem 4.14 to obtain the observability estimate Moreover, a standard approximation argument combined with (5.36) yields the inequality for all solutions ψ generated from initial data (ψ − 0 , ψ − 1 ) ∈ H 1 0 (U τ− ) × L 2 (U τ− ), that is, solutions generated from the first part of Theorem 5.14. Applying Theorem 5.17 along with (5.37) yields that (2.16) is indeed exactly controllable on U τ−,τ+ , with control on Y † .
Finally, for general cross-sections V ± , we combine the observability inequality (5.36) with the energy estimate (2.20) and an approximate argument to obtain (5.31) for all solutions ψ arising from initial data ( . By Theorem 5.17, we again conclude that (2.16) is exactly controllable on U (V − , V + ), with control on Y † .
Remark 5.19. Note that Theorem 5.17 and Corollary 5.18 allow for geometric extensions of controllability results, where the initial and final data can be placed on cross-sections of U that are not level sets of t. As before, one still needs to impose the Dirichlet boundary data on a sufficiently large portion of ∂U in order to achieve exact controllability.
Finally, we note that all the previous results in one spatial dimension -Theorems 5.3 and 5.6, Corollaries 5.8 and 5.10 -also lead to corresponding exact controllability results.

Appendix. Details and computations
In this appendix, we provide -for interested readers' convenience -additional proofs, computations, and details that were omitted in the main sections of this article.
A.1. Preliminary computations. First, a number of computations throughout this appendix will make use of the following null coordinate computations for the warped metric: Lemma A.1. Let ε ∈ R andḡ be defined as in Definition 3.3 and 3.4. Then: • the nonzero components ofḡ andḡ −1 in the null coordinates (u, v, ω) are given bȳ • the nonzero Christoffel symbols ofḡ in the null coordinates (u, v, ω) are given bȳ A.2. Proof of Proposition 3.8. To derive (3.8), we use (A.1) to obtain that and that∇ Next, using (A.2), we obtain Finally, for (3.20), we simply apply (3.12): A.6. Proof of Proposition 3.18. The identities in (3.23) follow from direct computations. First, For the remaining identity, we combine (2.2) and (3.21) with the above to obtain Next, we compute the derivative of u •Φ of v •Φ: In particular, by (3.13), both derivatives are positive and uniformly bounded from below. This implies u •Φ and v •Φ define invertible functions of u and v, respectively, with smooth inverses.
It remains to show the conformal identity (3.24). For this, we first note that the above derivatives imply that the push-forwards of the null coordinate vector fields satisfy and hence the corresponding pullbacks satisfȳ (Moreover,Φ by definition leaves the angular components unchanged.) As a result, by (3.23), Recalling Lemma A.1 (with ε = 0), we compute g αβ ∇ α ξ∇ β ξ = −∂ u ξ∂ v ξ = ε 2 (1 − εv)(1 + εu) = ε 2 ξ, † The formula found in [62] only takes into account the conformal factor ω, and not the presence of the diffeomorphismΦ. However, the formula withΦ is similar, as one only has to appropriately map points through Φ. as well as Thus, combining all the above with (3.23) yields Finally, recall that the standard formula for how the wave operator changes under conformal transformations (again, see [62,Appendix D], for instance) is given by Substituting our identity forR into the above results in (3.25).
In these new coordinates, (2.16) takes the form of a second-order hyperbolic PDE, for some smooth coefficients A ij , B i , X i , Y , V , on a static cylinder of the form R × Σ. Treating (φ 0 , φ 1 ) as initial data on Σ and φ b (or zero in the first statement of Theorem 5.14) as boundary data on R × ∂Σ, we can now apply the standard well-posedness theory for linear second-order hyperbolic equations; see [21,44]. More specifically, invoking this well-posedness theory in H 1 0 × L 2 and L 2 × H −1 yields existence in both cases of Theorem 5.14. The above also yields the energy estimates (5.25) and (5.27), in the case that V ± are also level sets of t. (The H 1 0 × L 2 case also furnishes a 'hidden regularity, in the form an L 2 -bound on the Neumann data N φ.) Moreover, (5.25) and (5.27) imply that φ ∈ L 2 loc (U ). Suppose now that φ 1 and φ 2 are two solutions of (2.16) (in either statement of Theorem 5.14), and consider the difference φ := φ 2 − φ 1 , which also solves (2.16), lies in L 2 loc (U ), and has zero traces on V and ∂U . Applying a standard unique continuation argument with respect to any 'time' coordinate t, as described above, yields that φ vanishes on U , establishing uniqueness.
Finally, to establish (5.25) and (5.27) in general, we repeat the above argument for another system of coordinates (t,ỹ 1 , . . . ,ỹ n ) such that V and V ± are level sets oft. The above uniqueness argument ensures that the solutions obtained in the two coordinate systems coincide.
As a result, it remains only to prove Theorem 5.17 in the case (φ + 0 , φ + 1 ) = (0, 0). Toward this end, the following lemma provides the main identity characterizing null controllability: where a − ∈ C ∞ (V − ), where the vector field B − is everywhere tangent to V − , and where div V− denotes the divergence on V − with respect to the metric induced by g. In addition, fix and assume φ b is supported in Γ. Then, φ b drives (2.16) from (φ − 0 , φ − 1 ) to (0, 0) if and only if for any ( where ψ denotes the corresponding solution (see Theorem 5.14) of (5.32).
Proof. For convenience, we also let N + denote the future-pointing unit normal to V + , and we set X + := g(X , N + ), N + := a + Z + B + , b + := div V+ B + , (A.6) with a + ∈ C ∞ (V + ), B + a vector field tangent to V + , and b + the induced divergence of B + on V + . Furthermore, we let φ be the solution (see again Theorem 5.14) to (2.16) which also satisfies By standard approximation arguments, it suffices to only consider the case in which φ and ψ are both everywhere smooth. Recalling (2.16) and integrating by parts, we see that In the above, we also used that ψ vanishes on ∂U (V − , V + ).