Control from an Interior Hypersurface

We consider a compact Riemannian manifold M (possibly with boundary) and Σ ⊂ M \ ∂M an interior hypersurface (possibly with boundary). We study observation and control from Σ for both the wave and heat equations. For the wave equation, we prove controllability from Σ in time T under the assumption (TGCC) that all generalized bicharacteristics intersect Σ transversally in the time interval (0,T ). For the heat equation we prove unconditional controllability from Σ. As a result, we obtain uniform lower bounds for the Cauchy data of Laplace eigenfunctions on Σ under TGCC and unconditional exponential lower bounds on such Cauchy data.


Introduction
Let (M, g) be a compact n dimensional Riemannian manifold possibly with boundary ∂M and denote ∆ g the (non-positive) Laplace-Beltrami operator on M. We study the observability and controllability questions from interior hypersurfaces in M.
To motivate the more involved developments in control theory, let us start by stating (slightly informally) the counterpart of our observability/controllability results for lower bounds for eigenfunctions, i.e. solutions to (−∆ g − λ 2 )φ = 0, φ| ∂M = 0. (1.1) In the particular case Σ is a compact interior hypersurface, then it is an interior hypersurface with Σ 0 = Σ. Since M is endowed with a Riemannian structure, the coorientability assumption is equivalent to that of having a smooth global vector field ∂ ν normal to Int(Σ 0 ). Note that the coorientability condition can be slightly relaxed, see the discussion in Section 1.5 below.
Given an interior hypersurface Σ, the main goal of this paper is to study the controllability of some evolution equations with a control force of the form where the distributions f 0 δ Σ and f 1 δ Σ are defined by (1.5) In this expression σ denotes the Riemannian surface measure on Σ induced by the metric g on M. This contrasts with usual control problems for PDEs, for which the control function appears in the equation: • either as a localized right handside (distributed or internal control) 1 ω f , where ω is an open subset of M, and typically, the control function f is in L 2 ((0, T ) × ω); • or, in case ∂M ∅, as a localized boundary term, e.g. under the form u| ∂M = 1 Γ f , where Γ is an open subset of ∂M, and typically, the control function f is in L 2 ((0, T ) × Γ) (here, u denotes the function to be controlled).
Concerning the wave equation, the main result is the Bardos-Lebeau-Rauch Theorem [BLR92,BG97] providing a necessary and sufficient condition for the exact controllability with such control forces (see also e.g. [DL09,LL16,LLTT16] for recent developments). Concerning the heat equation, the question of nullcontrollability with internal or boundary control was solved independently by Lebeau-Robbiano [LR95] and Fursikov-Imanuvilov [FI96]. The aim of the present paper is threefold: • Formulating a well-posedness result as well as an analogue of the Bardos-Lebeau-Rauch Theorem, for the wave equation with control like (1.4) (see Section 1.1); • Formulating an analogue of the Lebeau-Robbiano-Fursikov-Imanuvilov Theorem for the heat equation with control like (1.4) (see Section 1.2); • Formulating general lower bounds for restrictions on Σ of eigenfunctions on M (see Theorem 1.1 above and Section 1.3). These are analogues of the observability inequalities used to prove the above controllability statements and are of their own interest.
Next, we define the glancing and the elliptic sets for above Σ as G = Char( ) ∩ ι(T * (R × Int(Σ)), where ι : T * (R × Int(Σ 0 )) → T * (R × M) (1.8) is the inclusion map. A more explicit expression of these sets in normal coordinates is given in Section 2.3 below.
Roughly speaking, the elliptic set E (resp. E Σ ) consists in points (t, x, τ, ξ) in the whole phase space (resp. in tangential phase space to Σ) such that x ∈ Int(Σ) in which no "ray of optics" for lives. The glancing set G (resp. G Σ ) consists in points (t, x, τ, ξ) in the whole phase space (resp. in tangential phase space to Σ) such that x ∈ Int(Σ), through which "rays of optics" for may pass tangentially. The complement of G ∪ E in the characteristic set of above R × Int(Σ) is the set of point through which "rays of optics" for may pass transversally. With these definitions in hand, our well-posedness result may be stated as follows.
Theorem 1.4. For all (v 0 , v 1 ) ∈ L 2 (M)×H −1 (M) and for all f 0 ∈ H −1 comp (R * + ×Int(Σ)) and f 1 ∈ L 2 comp (R * + ×Int(Σ)) such that there exists a unique v ∈ L 2 loc (R * + ; L 2 (M)) solution of (1.6). We refer e.g. to [Ler10,Definition 1.2.21] for a definition of the H s wavefront set WF s of a distribution. The wavefront condition states roughly that ( f 0 , f 1 ) should have improved (namely H − 1 2 × H 1 2 ) microlocal regularity near the glancing set G Σ (when compared to overall the H −1 (R × Σ) × L 2 (R × Σ) regularity) for the Cauchy problem to be well-posed. A more precise version of this result is given in Theorem 3.7 below (where, in particular, the meaning of "solution" is made precise in the sense of transposition, see [Lio88]). This wavefront set condition on f 0 , f 1 is far from sharp because we use a very rough analysis of solutions to the free wave equation near G. A more detailed analysis near G, similar to that in [Gal16], would yield sharper regularity requirements.
With this well-posedness result and the definition of T GCC, we now give a sufficient condition for the null-controllability of (1.6) from Σ. so that the solution to (1.6) has v ≡ 0 for t ≥ T .
Here, WF stands for the usual C ∞ wavefront set. Theorem 1.5 follows from an observability inequality given in Theorem 4.1 below.
Of course, it is classical to check that a necessary condition for controllability to hold is that all generalized bicharacteristics intersect T * (0,T )×Σ (R × M). As for the well-posedness problem, the issue of rays touching R × Σ only at points of G Σ is very subtle, and will be addressed in future work. See the discussion in Section 1.4 below.

Controllability from a hypersurface for the heat equation
We next consider the controllability of the heat equation from a hypersurface, namely (1.10) Well-posedness in the sense of transposition follows from the standard parabolic estimates, and is proved in Section 5.1. We only state a null-controllability result for (1.10). such that the solution v of (1.10) satisfies v| t=T = 0.
Note that we also provide an estimate of the cost of the control as T → 0 + , similar to the one in case of internal/boundary control [FI96,Mil10].

Eigenfunction Restriction Bounds
As usual, the above two control results (or rather, the equivalent observability estimates) have related implication concerning eigenfunctions, stated in Theorem 1.1 above. We now formulate these results under the (stronger) form of resolvent estimates. Below, we write λ := (1 + |λ| 2 ) 1/2 . Theorem 1.7 (Universal lower bound for eigenfunctions). Assume M is connected and Σ is a nonempty interior hypersurface. Then there exist C, c > 0 so that for all λ ≥ 0 and all u ∈ H 2 (M) ∩ H 1 0 (M) we have (1.11) As far as the authors are aware, estimates (1.2)-(1.11) are the first general lower bounds to appear for restrictions of eigenfunctions. Moreover, these estimates are sharp in the sense that simultaneously neither the growth rate e cλ nor the presence of both u and ∂ ν u can be improved in general. This is demonstrated by the following example.
with variables (z, θ), endowed with the warped product metric Assume that R is smooth, that This result is proved in Appendix B.
We expect that the symmetry in this example is the obstruction for removing one of the traces in the right handside of (1.2), and formulate the following conjecture.
Conjecture 1. Let (M, g) be a Riemannian manifold and Σ an interior hypersurface with positive definite second fundamental form. Then there exists C, c, Note that if Σ has positive definite second fundamental form, then it is geodesically curved and in particular, not fixed by a nontrivial involution. This prevents the construction of counterexamples via the methods used to prove Lemma 1.8.
Under the geometric control condition T GCC the estimate (1.11) can be improved.
Theorem 1.9 (Improved lower bound for eigenfunctions under T GCC). Assume that the geodesics of M have no contact of infinite order with ∂M and that Σ satisfies T GCC. Then there exists C > 0 so that for all λ ≥ 0 and u ∈ H 2 (M) ∩ H 1 0 (M), we have (1.12) Conjecture 2. Let (M, g) be a Riemannian manifold and Σ an interior hypersurface with positive definite second fundamental form satisfying T GCC. Then there exists C, c, Other known lower bounds come from the quantum ergodic restriction theorem and apply to a full density subsequence of eigenfunctions rather than to the whole sequence [TZ12,TZ13,DZ13,TZ17]. These hold under a ergodicity assumption on the geodesic (or the billiard) flow, together with a microlocal asymmetry condition for the surface Σ. This assumption states roughly that the measure of the set of geodesics through Σ whose tangential momenta agree at adjacent intersections with Σ is zero. In another direction, the work of Bourgain-Rudnick [BR12,BR11,BR09] shows that on the torus T d , d = 2, 3 for any hypersurface Σ with positive definite curvature, (1.3) holds with the normal derivative removed from the left hand side. While the results of Bourgain-Rudnick do not hold on a general Riemannian manifold, we expect that either of the terms in the left hand side of (1.2) can be removed whenever Σ is not totally geodesic (which is even weaker that Σ having positive definite second fundamental form).

Weakening Assumption T GCC
One might hope that Theorem 1.9 and its analog for the wave equation (the control result of Theorem 1.5 above and the observability inequality of Theorem 4.1 below) when the assumption T GCC is replaced by the (weaker) assumption that every generalized bicharacteristics of intersects T * ((0, T ) × Int(Σ)) (1.13) (rather than T * ((0, T ) × Int(Σ)) \ G Σ ). The following example shows that this is more subtle (see Appendix C for the proof).
Proposition 1.10. Assume M = S 2 and Σ is a great circle. Then there exists a sequence (λ j , φ j ) satisfying (−∆ g − λ 2 j )φ j = 0 together with λ j → +∞ and 6 In particular, this shows that Theorem 1.9 and associated observability inequality for the wave equation cannot hold under only (1.13). Moreover, the proof shows that φ j is microlocalized λ −1 j close to the glancing set on Σ, this calculation suggests that one must scale the normal derivative and restriction of an eigenfunction as in [Gal16] to obtain an analog of Theorem 1.9 under (1.13). More precisely, Conjecture 3. Suppose that Σ is a compact interior hypersurface. Then there exists C > 0 so that if (λ, φ) satisfies (1.1), then Suppose moreover that Σ satisfies (1.13). Then there exists C > 0 so that if (λ j , φ j ) satisfies (1.1), then where ∆ Σ is the Laplace-Beltrami operator on Σ induced from (M, g), and the operator (1 + λ −2 ∆ Σ ) + is defined via the functional calculus, see also [Gal16, Section 1].

Finite unions of hypersurfaces
In all of our results, one may replace Σ by any finite union of cooriented interior hypersurfaces (1.14) Then, all above results generalize with the sole modification that generalized bicharacteristics need only intersect one of the Σ i 's transversally. This furnishes several simple examples for which our controllability/observability results for waves holds. Take e.g.
This remark can also be used to remove the coorientability assumption. If the interior hypersurface Σ is not coorientable, we can cover it by a union of overlapping cooriented hypersurfaces Σ = ∪ m i=1 Σ i and control from Σ by a sum like (1.14). In this context, we still obtain controllability results with controls supported by the hypersurface Σ, but the form of the control is changed slightly.

Sketch of the proofs and organization of the Paper
We start in Section 2 with the introduction of coordinates, some geometric definitions and Sobolev spaces on Σ.
Section 3 is devoted to the proof of (a slightly more precise version of) the well-posedness result of Theorem 1.4. The definition of solutions in the sense of transposition follows [Lio88]. The well-posedness result relies on a priori estimates on an adjoint equation -the free wave equation. The well-posedness statement then reduces to the proof of regularity bounds for restrictions on Σ. This is done in Section 3.1. Namely, we show that if u is an H 1 solution to u = 0, then the restriction (u| Σ , ∂ ν u| Σ ) belong to H 1 2 × H − 1 2 overall R × Σ, and have the additional (microlocal) regularity H 1 × L 2 everywhere except near Glancing points (G Σ ). This fact is already known (see e.g. [Tat98]) but we rewrite a short proof for the convenience of the reader. Then, Section 3.2 is aimed at defining the appropriate spaces for the statement of the precise version of the well-posedness result. These are needed in particular to state the stability result associated to well-posedness, as well as to formulate the duality between the control problem and the observation problem. They are (loc and comp) Sobolev spaces on R × Σ that have different regularities near and away from the Glancing set G Σ . With these spaces in hand, we define properly solutions of (1.6) and prove well-posedness in Section 3.3.
Section 4 is devoted to the proof of the control result of Theorem 1.4. Before entering the proofs, we briefly explain how Theorem 1.9 is deduced from the observability inequality of Theorem 4.1. Firstly, we prove in Section 4.1 that the condition T GCC implies a stronger geometric statement. Namely, using the openness of the condition and a compactness argument, we prove that all rays intersect in (ε, T − ε) an open set of Σ "ε-transversally" (i.e. ε far away from the glancing region) for some ε > 0. Secondly, this condition is used in Section 4.2 to prove an observability inequality, stating roughly that the observation of both traces (u| Σ , ∂ ν u| Σ ) of microlocalized ε far away from the glancing region in the time interval (ε, T − ε) determines the full energy of solutions of u = 0 (in appropriate spaces). The proof proceeds as in [Leb96] by contradiction, using microlocal defect measures. It contains two steps: first, we prove that the strong convergence of a sequence (u k | Σ , ∂ ν u k | Σ ) → 0 near a transversal point of Σ implies the strong convergence of the sequence u k in a microlocal neighborhood of the two rays passing through this point (using the hyperbolic Cauchy problem). Then, a classical propagation argument (borrowed from [Leb96,BL01] in case ∂M ∅) implies the strong convergence of (u k ) everywhere, which yields a contradiction with the fact that the energy of the solution is normalized. This observability inequality contains, as in the usual strategy of [BLR92], a lower order remainder term (in order to force the weak limit of the above sequence to be 0). The latter is finally removed in Section 4.3 by the traditional compactness uniqueness argument of [BLR92], concluding the proof of the observability inequality. Finally, in Section 4.4, we deduce the controllability statement Theorem 1.5 (or its refined version Theorem 4.8) from the observability inequality (Theorem 4.1) via a functional analysis argument. The latter is not completely standard, since we do not know whether the solution of the controlled wave equation 1.6 has the usual C 0 (0, T ; L 2 (M)) ∩ C 1 (0, T ; H −1 (M)) regularity, but only prove L 2 (0, T ; L 2 (M)). As a consequence, we cannot use data of the adjoint equation at time t = T as test functions. The test functions we use are rather forcing terms F in the right hand-side of the adjoint equation, that are supported in t ∈ (T, T 1 ), that is, outside of the time interval (0, T ). Also, we construct control functions having H N regularity near G Σ and prove that they do not depend on N, yielding the statement with the C ∞ wavefront set.
Section 5 deals with the case of the heat equation and the universal lower bound of Theorem 1.7, in the spirit of the seminal article [LR95]. First, Section 5.1 states the well-posedness result, in the sense of transposition. Again, it relies on the regularity of restrictions to Σ of solutions of the adjoint free heat equations. The latter are deduced from standard parabolic regularity combined with Sobolev trace estimates. Then, to prove observability/controllability, we proceed with the Lebeau-Robbiano method [LR95]. The starting point is a local Carleman estimate near Σ, borrowed from [LR97], from which we deduce in Section 5.2 a global interpolation inequality for the operator −∂ 2 s − ∆ g . Theorem 1.7 directly follows from this interpolation inequality. To deduce the observability of the heat equation, we revisit slightly (in an abstract semigroup setting) the original Lebeau-Robbiano method (as opposed to the simplified one [LZ98,Mil06,LRL12], relying on a stronger spectral inequality) in Section 5.3. The interpolation inequality yields as usual an observability result for a finite dimensional elliptic evolution equation (i.e. cutoff in frequency), from which we deduce observability for the finite dimensional parabolic equation, with precise dependence of the constant with respect to the cutoff frequency and observation time. The latter argument simplifies the original one by using an idea of Ervedoza-Zuazua [EZ11b,EZ11a]. The observability of the full parabolic equation is finally deduced using the iterative Lebeau-Robbiano argument combining high-frequency dissipation with low frequency control/observation. We in particular use the method as refined by Miller [Mil10]. We explain in Section 5.4 how the heat equation observed by/controlled from Σ fits into the abstract setting.
Appendix A contains some background information on pseudodifferential operators used in Sections 3 and 4. for the wave equation. Appendix B proves Proposition 1.8, i.e. constructs an example showing that Theorem 1.7 is sharp. Finally, Appendix C gives a proof of Proposition 1.10.
so that the differential operator −∆ g takes the form where c(x, D) is a first order differential operator and r(x 1 , x , D x ) is an x 1 -family of second-order elliptic differential operators on Int(Σ 0 ), i.e. a tangential operator, with principal symbol r( In these coordinates, note that we have in particular |x 1 (p)| = d(p, Σ), ∂ ν = ∂ x 1 (up to changing x 1 into −x 1 ), as well as where −∆ Σ 0 is the Laplacian on Int(Σ 0 ) given by the induced metric on Σ 0 . We also recall that With a slight abuse of notation, we shall also denote by (x 1 , x ) ∈ R × R n−1 (and (ξ 1 , ξ ) ∈ R × R n−1 associated cotangent variables) local coordinates in a neighborhood of a point in Int(Σ 0 ). In these coordinates, the Hamiltonian vector field of is given by and generates the Hamiltonian flow of (these coordinates beeing away from the boundary R × ∂M).

The compressed cotangent bundle over M
This section is independent of the hypersurface Σ and is only aimed at defining, in case ∂M ∅, the space Z on which the Melrose-Sjöstrand bicharacteristic flow is defined, as well as some properties of the flow. In case ∂M = ∅, this set is Char( ) ⊂ T * (R × M) \ 0, the flow is the usual bicharacteristic flow of , and this section not needed and may be skipped. We refer to [MS78], [Leb96, Appendix A2] for more complete treatments.
We first embed M →M into a manifold,M, without boundary and write be the natural "compression" map. In any coordinates (x , x n ) on M where x n defines ∂M and x n > 0 on M, j has the form j(t, x, τ, ξ) = (t, x, τ, ξ , x n ξ n ). (2. 3) The map j endows b T * (R × M) with a structure of homogeneous topological space. We then write and the associated sphere bundle, which, endowed with the induced topology, are locally compact metric spaces.
Away from the boundary, j is a bijection and we shall systematically identify b T * (R × Int(M)) with T * (R × Int(M)) and Z ∩ b T * (R × Int(M)) with Char( ) ∩ T * (R × Int(M)). This will be the case in particular near the hypersurface R × Σ.
Under the assumption that the geodesics of M have no contact of infinite order with ∂M, and with Z as in (2.4), the (compressed) generalized bicharacteristic flow for the symbol 1 2 (−τ 2 + |ξ| 2 g ) is a (global) map ϕ : R × Z → Z, (s, p) → ϕ(s, p) ( • ϕ coincides with the usual bicharacteristic flow of (i.e. the Hamiltonian flow of σ( )) in the interior Char( ) ∩ T * (R × Int(M)); • ϕ satisfies the flow property ϕ(t, ϕ(s, p)) = ϕ(t + s, p), for all t, s ∈ R, p ∈ Z; (2.7) • ϕ is homogeneous in the fibers of Z, in the sense that where M λ denotes multiplication in the fiber by λ > 0; Hence, it induces a flow on SẐ.
With these definitions, T GCC can be written as

Spaces on interior hypersurfaces
In case Σ is a compact internal hypersurface, then the Sobolev spaces H s (Σ) have a natural definition. Here, we give a definition adapted to the case ∂Σ ∅.
Definition 2.2. Let S be an interior hypersurface of a d dimensional manifold X, and let S 0 be an extension of S (see Definition 1.2). Given s ∈ R, we say that u ∈H s (S ) (extendable Sobolev space) if there exists u ∈ H s comp (S 0 ) such that u| S = u. To put a norm onH s (S ), let χ ∈ C ∞ c (Int(S 0 )) such that χ = 1 in a neighborhood of S . We denote by (U j , ψ j ) j∈J an atlas of S 0 such that for all j ∈ J (2.11) The definition of the normH s (S ) depends on S 0 , χ, the choice of charts (U j , ψ j ) and the partition of unity (χ j ). One can however prove that, once S 0 and χ are fixed, two such choices of charts (U j , ψ j ) and partition of unity (χ j ) lead to equivalent normsH s (S ). In what follows, (U j , ψ j , χ j ) shall be traces on S 0 of charts and partition of unity on X. In case S is a compact interior hypersurface, then the spacesH s (S ), · Hs (S ) coincides with usual H s (S ) space.

Regularity of traces and well-posedness for the wave equation
The ultimate goal of the present section is to prove the well-posedness result for (1.6), see Theorem 1.4. Defining solutions by transposition as in [Lio88], this amounts to proving regularity of traces on Σ of solutions to the free wave equation.

Regularity of traces
We start by giving estimates on the restriction to Σ of a solution to (3.1) These bounds, indeed stronger bounds, can be found in [Tat98], but we choose to give the proof of the simpler estimates here for the convenience of the reader. They are closely related to the semiclassical restriction bounds from [BGT07, Tac10, Tac14, CHT15, Gal16].
2) To prove Proposition 3.1 we need the following elementary lemma.
Lemma 3.2. Suppose that S is an interior hypersurface of the d dimensional manifold X (in the sense of Definition 1.2) and P ∈ Ψ m phg (Int(X)) is elliptic on the conormal bundle to Int(S 0 ), N * Int(S 0 ). Then for any s ∈ R, k ≥ 0 and > 0, there exists C = C( , k, s) > 0 so that for all u ∈ C ∞ (M), Proof. We start by proving the case k = 0. In case s > 0, the stronger inequality u| S Hs (S ) ≤ C u H s+k+1/2 (X) holds as a consequence of standard trace estimates [Hör85, Theorem B.2.7] (that theH s (S ) norm is the appropriate one in case S is not compact is made clear below). We now assume that s ≤ 0, and estimate each term in the definition (2.11) of u| S Hs (S ) in local charts. For this, we use charts (Ω i , κ i ) i∈I of Int(X) such that S 0 ⊂ i∈I Ω i and such that (Ω i ∩ S 0 , κ i | S 0 ) i∈I satisfy the assumptions of Definition 2.2. In a neighborhood of S 0 , we have u = iχi u (where (χ i ) is now a partition of unity of S 0 associated to Ω i , and hence, (χ i | S 0 ) satisfies the assumptions of Definition 2.2), and estimating u| S Hs (S ) amounts to estimating each We may now work locally, where S is a subset of {x 1 = 0}, and estimate the trace of z =χw. Let , 0 ≤ χ ≤ 1 and fix δ > 0 small enough so that P (which, by abuse of notation, we use for the operator in local coordinates) is elliptic on a neighborhood of and let χ δ (ξ 1 , ξ ) = χ 2|ξ | δξ 1 for ξ 1 0 and χ δ (0, ξ ) = 0. Then, we have We now estimate each term. With the Cauchy Schwarz inequality, the first term is estimated by Again with the Cauchy Schwarz inequality, the second term is estimated by with C finite as soon as > 0, and ξ 2s−2 ≤ 1 since s ≤ 0. Combining the last three estimates and recalling that z =χw yields Now, according to the definition of χ δ , the operator P is elliptic on a neighborhood of supp(χ) × supp(χ δ ), a classical parametrix construction (see for instance [Hör85, Theorem 18.1.9]) implies, for any N ∈ N, whereχ is supported in the local chart and equal to one in a neighborhood of supp(χ). Recalling that w is the localization of u, and summing up the estimates (3.3)-(3.4) in all charts yields the sought result for k = 0. We now show that the k = 0 case implies the k > 0 case. LetP ∈ Ψ m phg (Int(X)) be elliptic on N * (Int(S 0 )) with WF(P) ⊂ {σ(P) 0} (see e.g. Appendix A for a definition of WF(A) for a pseudodifferential operator A). Then, applying the case k = 0 to the operatorP, we obtain Since P is elliptic on WF(P), by the elliptic parametrix construction, we can find E 1 ∈ Ψ k phg (Int(X)), and 2 , which, combined with (3.5), yields the result.
We now proceed with the proof of Proposition 3.1.
We proceed by making a microlocal partition of unity on a neighborhood T * (R × K). It suffices to obtain the estimate for χ supported in a conic neighborhood of an arbitrary point, q 0 = (t 0 , τ 0 , x 0 , ξ 0 ) in T * (R × K). We will focus on four regions: q 0 Char( ) (an elliptic point); q 0 ∈ Char( ) but away from Σ; q 0 ∈ T * R×Σ (R × M) ∩ T 0 (a transversal point); and q 0 ∈ T * R×Σ (R × M) ∩ G 0 (a glancing point). In all regions, we shall use that given χ ∈ S 0 phg a cutoff to a conic neighborhood, U of q 0 , we have 13 First start with q 0 in the elliptic region: q 0 Char( ). Shrinking the neighborhood if necessary, the microlocal ellipticity of near q 0 with (3.7) yields Hence, rough trace estimates imply and boundedness of A proves (3.6) in this case.
Second, suppose that q 0 ∈ Char( ) but x 0 Σ, then clearly there is a neighborhood U of q 0 and χ elliptic at q 0 with supp χ ⊂ U so that and again boundedness of A proves (3.6).

Microlocal spaces on the hypersurface
This section is aimed at defining the appropriate spaces for the statement of the well-posedness and control results in the present context. All along the section, a sequence S = (ε j ) j∈N , ε j → 0 is fixed and ε, ε ∈ S. This precision is sometimes omitted for concision. Fix a family of interior hypersurfaces Σ ε with (3.9) Let Γ ⊂ T * (R × Int(Σ)) \ 0 be a closed and conic set. (3.10) We define spaces adapted to Γ, i.e. measuring different regularities near and away from Γ. In the applications below, we shall take Γ = G Σ for the study of the Cauchy problem and Γ = E Σ ∪ G Σ = E Σ for the study of the control problem.
To this end, let ε → Γ ε , ε ∈ S, be a family of closed conic subset of T * R × Int(Σ) \ 0 such that Γ ε is closed and conic for any ε, Next, fix a family of cutoff functions and a family of cutoff operators (3.13) Note that once Γ will be fixed, (see Sections 3.3 and 4), a more explicit expression for the symbol of the operators B Γ ε will be given. Next, we define for k ≥ s, the Banach space with topology given by the seminorms · H s,k comp,Γ,ε (Σ T ) (taken for a sequence of ε going to zero). Functions/distributions in the space H s,k comp,Γ (Σ T ) are H s overall and microlocally H k (k ≥ s) on Γ. In case k = s, we simply have H s,k comp,Γ (Σ T ) = H k comp ((0, T ) × Int(Σ)). Similarly, we define for k ≤ s, the vector space endowed with the seminorm We define as well the Fréchet space with topology given by the seminorms · H s,k loc,Γ,ε (Σ T ) . Functions/distributions in the space H s,k loc,Γ (Σ T ) are locally H k overall and microlocally H s (s ≥ k) outside of Γ. Remark again that in case k = s, we simply have extends uniquely as a continuous sesquilinear map (3.14) Fix any ε ∈ S and χ ∈ C ∞ c ((0, T ) × Int(Σ)). Then there exists ε 0 > 0 depending only on ε such that for This, together with (3.14) proves that µ ∈ H −s,−k loc,Γ (Σ T ), and hence the lemma.

Definition of solutions and well-posedness
Observe that G Σ andĒ Σ := G Σ ∪E Σ satisfy (3.10) and for k ≥ s, we therefore have Fréchet spaces H s,k . With these definitions in hand, we can reformulate Proposition 3.1 as follows: For any T > 0, the map We can now study the well-posedness for the control problem (1.6). We first recall that, given Then, we have the identity Definition 3.6. Given , we say that v is a solution of (1.6) if v ∈ L 2 ((0, T ); L 2 (M)) and for any F ∈ L 2 ((0, T ); L 2 (M)), we have .
where u is the unique solution to (3.16) Note in particular that taking F ∈ C ∞ c ((0, T ) × Int(M)) implies that such a solution is a solution of the first equation of (1.6) in the sense of distributions.
Remark 3.8. Note that, given two different times T < T , an initial data (v 0 , v 1 ) and control functions f 0 , f 1 compactly supported in (0, T ) ⊂ (0, T ), the above definition/theorem yield two different solutions: one defined on (0, T ) and one defined on (0, T ). However, one can observe that these two solutions coincide by extending all test functions F ∈ L 2 ((0, T ); L 2 (M)) by zero on (T, T ) to obtain test functions in L 2 ((0, T ); L 2 (M)). With this in mind, Theorem 1.4 is a direct consequence (and a simplified version) of Theorem 3.7. , and prove that it is as a continuous linear form on L 2 (0, T ; L 2 (M)), with appropriate norm. We have .
From the definition of the spaces in Section 3.2, there exists ε > 0 such that comp,G Σ ,ε (Σ T ) and hence, we obtain from Lemma 3.3, Coming back to , we have obtained the existence of ε ∈ S, C ε > 0 such that Hence, is a continuous linear form on L 2 (0, T ; L 2 (M)). There is thus a unique v ∈ L 2 (0, T ; L 2 (M)) such that (F) = T 0 (v(t), F(t)) L 2 (M) dt for all F ∈ L 2 (0, T ; L 2 (M)), that is precisely the definition of a solution of (1.6) in Definition 3.6. This solution moreover satisfies, for , which is the continuity statement. This concludes the proof of the Theorem.

Observability and controllability for the wave equation
The aim of this section is to study the observability of (3.1) from Σ. In particular, we prove  (3.9), for all N > 0, there exists c N > 0 so that for any solution u to (3.1), we have (4.1) Let us briefly explain why the observability inequality of Theorem 4.1 implies Theorem 1.9.
Proof of Theorem 1.9. We apply Theorem 4.1 to the function u(t, x) = e itλ v(x) with v ∈ H 1 0 (M) ∩ H 2 (M). First observe that A δ is bounded on L 2 and hence Observe also that there exists δ 0 > 0 so that ϕ δ 0 D t is elliptic on WF(A δ ) and therefore, Note also that u = e itλ (−∆ g − λ 2 )v and hence the right hand side of (4.1) is bounded by Finally, noticing that (u| t=0 , ∂ t u| t=0 ) = (v, iλv), , finishing the proof of Theorem 1.9.

The geometric assumption T GCC
To prove Theorem 4.1 we start with a dynamical lemma where we show that the a priori weaker assumption GC-(0,T ) implies the stronger assumption Recall that Z is as in (2.4).
Proof. We define Z ± 1 := Z ∩ {τ = ±1, t = 0}. We shall show that Assumption GC-(0,T ) implies the existence of ε > 0 such that We first show that (4.2) implies the lemma. With the identification b T * (R × M) Let M λ be multiplication in the fiber by λ > 0. Then, According to the homogeneity of ϕ, see (2.8), and the flow property (2.7), we have We now prove (4.2), writing explicitly the argument for Z − 1 . The case of Z + 1 is handled similarly. Notice first that since ϕ is the generalized bicharacteristic flow for 1 2 (−τ 2 + |ξ| 2 g ), we have for p ∈ Z − 1 , t(ϕ(s, p)) = s. This, together with Assumption GC-(0,T ) implies that for each p ∈ Z − 1 , we have Therefore, for each p ∈ Z − 1 , there exists ε p > 0 and s p ∈ (ε p , T − ε p ) such that Let β be a defining function for Σ 0 near ϕ(s p , p), and consider g(s, q) = β • π 0 • ϕ(s, q) for (s, q) in a neighborhood N p of (s p , p), where π 0 : T * (R × Int(M)) → R × Int(M) is the canonical projection. By [MS78,Theorem 3.34], the Melrose-Sjöstrand generalized bicharacteristic flow ϕ is continuous and so g is continuous on N p . Moreover, since Σ is an interior hypersurface, there exists δ p so that is C 1 for q in a neighborhood of p since ϕ coincides with the usual bicharacteristic flow of near ϕ(s p , p).
according to Remark 2.1. Hence by the implicit function theorem [Kum80], the equation g(s, q) = 0 defines a continuous function s = s(q) near q = p. In particular, set Then there is a neighborhood, U p of p and a continuous function s : U p → R with s(p) = s p , such that ϕ s(q) (q) ∈ T ε p /2 ∩ T * (ε p /2,T −ε p /2)×Σ εp /2 (R × M) and |s(q) − s p | < δ 0 for all q ∈ U p . Since Z − 1 = j(Char( ) ∩ {τ = −1, t = 0}) is compact, we may extract from the cover Z − 1 ⊂ p∈Z − 1 U p a finite cover {U p i } n i=1 . Then taking ε = min 1≤i≤n ε p i /2, we have that for all p ∈ Z − 1 , In particular, (4.2) holds, which concludes the proof of the lemma.
(4.4) 20 We begin with two preliminary lemmas. We again work in fermi normal coordinates near Σ. A more general of version of the following Lemma is given in [Hör85, Lemma 23.2.8], but we decided to include a short proof in this particular context for the sake of readability.
We now turn to the proof of Proposition 4.3. We follow the general structure of proof introduced by Lebeau in [Leb96], using the microlocal defect measures of Gérard [Gér91] and Tartar [Tar90]. Note that from the quantitative estimate of Lemma 4.5, and in case ∂M = ∅, "constructive proofs" (i.e. using no contradiction argument, and hence no defect measures) of Proposition 4.3 are possible, see [LL17] or [LL16].
Let us first show that µ ≡ 0. Notice that Lemma 4.2 implies there exists ε > 0 so that Assumption GC-(ε,T) holds. We first prove that µ = 0 on a neighborhood of T ε ∩ T * (ε,T −ε)×Σ ε (R × M). Then, since µ is invariant under the generalized bicharacteristic flow ϕ(s, ·) defined in (2.6) (which passes to the quotient space SẐ according to homogeneity (2.8)), see [Leb96,BL01], Assumption GC-(ε,T) implies µ ≡ 0 (note that it is sufficient that . Then for δ < ε, we have σ(A δ )(π(q 0 )) = 1. Therefore, Lemma 4.5 applies with Op(b 0 ) = A δ and hence forχ supported close enough to q 0 , Now, the right hand side tends to 0 by assumption. Thus, pseudodifferential calculus together with (4.15), imply the existence of a conic neighborhood U of q 0 so that µ(U/R * + ) = 0. Since this is true for any q 0 ∈ T ε ∩ T * [ε,T −ε]×Σ ε (R × M), there is a conic neighborhood U 1 of T ε ∩ T * (ε,T −ε)×Σ ε (R × M) so that µ(U 1 /R * + ) = 0. Invariance of µ and Assumption GC-(ε,T) imply that µ vanishes identically, which precisely means (4.16) Now, we denote E k (t) := ∇u k (t, ·) 2 L 2 (M) + ∂ t u k (t, ·) 2 L 2 (M) , and observe from (4.13)-(4.14) that E k (0) → 1. Moreover, for all s 1 , s 2 ∈ [0, T ], we have In particular, since this convergence is uniform in s 1 , s 2 , Together with (4.16), this yields 0 < T ← There are different ways of writing the compactness-uniqueness argument of [BLR92] (both reducing the problem to a unique continuation property for Laplace eigenfunctions). The first one is the precise argument of [BLR92]: it uses again the geometric condition together with the propagation of wavefront sets (see also [LLTT16]). A second form seems to be due to [BG02]: it is a bit longer but uses only that the observation region is time invariant. We write this version of the proof in the present context. We first need a weak unique continuation property from a hypersurface. This is a weak version of Theorem 1.7, but we chose to give a proof since it is much less involved. Note that no compactness is assumed and no boundary conditions are prescribed here.
we have v ∈ L 2 (Ω), with, moreover (∂ ν pointing towards Ω + ) This follows from the jump formula written in Fermi coordinate charts ( We next define for any T > 0 and ε > 0 the set of invisible solutions from [ε, T − ε] × Σ ε where Σ ε is as in (3.9): N(ε, T ) = (u 0 , u 1 ) ∈ H 1 0 (M) × L 2 (M) such that the associated solution of (3.1) with We have the following lemma, which is a consequence of Proposition 4.3.
Lemma 4.7. Suppose GC-(0,T) holds. Then there exists ε 0 > 0 such that for all 0 < ε < ε 0 , we have We denote by A the generator of the wave group, namely so that the wave equation (3.1) with F = 0 may be rewritten as Proof.
Step 3: reduction to unique continuation for Laplace eigenfunctions: end of the proof. Since N(ε, T ) is a finite dimensional subspace of D(A), stable by the action of the operator A, it contains an eigenfunction of A. There exist µ ∈ C and U = (u 0 , u 1 ) ∈ N(ε, T ) such that AU = µU, that is, given the definition of A in (4.17), −∆ D u 0 = −µ 2 u 0 and u 1 = −µu 0 . Hence u 0 is an eigenfunction of the Laplace-Dirichlet operator on M, associated to −µ 2 ∈ R + , i.e. µ = iλ, λ ∈ R. The associated solution to (3.1) is u(t, x) = e iλt u 0 , and U 0 ∈ N(ε, T ) implies ∂ ν u 0 | Σ = u 0 | Σ = 0. This, together with the fact that u 0 is a Laplace eigenfunction and Lemma 4.6 proves that u 0 = 0 and then U = 0. This proves that N(ε, T ) = {0}.
From Lemma 4.7, we can now conclude the proof of Theorem 4.1.

Controllability of the Wave Equation
Theorem 1.5 is a straightforward corollary of the following theorem. Recall thatĒ Σ = E Σ ∪ G Σ .
Proof. Fix 0 < T < T 1 . Then define Now, suppose that Assumption GC-(0,T ) holds and let A δ as in Theorem 4.1. For ε > 0 small BĒ Σ ε is elliptic on WF(A δ ) and hence using the elliptic parametrix construction we write , and G ∈ Ψ 0 phg ((0, T ) × Int(Σ)). Therefore Theorem 4.1 implies that there exists ε > 0 small enough depending only on (Σ, T ) and for all N ∈ N, there exists C N > 0 so that (4.25) Let (v 0 , v 1 ) ∈ H −1 (M) × L 2 (M) and define the linear functional N : ran(K) → C by Since N is a continuous linear functional defined on a subspace of H 1,−N loc,Ē Σ (Σ T ) × H 0,−N loc,Ē Σ (Σ T ) by the Hahn-Banach theorem N extends to a continuous linear functional on the whole space (still denoted N ) with Thus, by Lemma 3.4, there exists ( f 0,N , f 1,N ) and hence for some ε > 0, Let v be the unique solution to given by Definition 3.6 and Theorem 3.7. Then for any F ∈ L 2 ([T, is dense. So, in particular, N extends to a linear functional on H 1,−k loc,Ē Σ (Σ T ) × H 0,−k loc,Ē Σ (Σ T ) by density. This yields . This implies that f 0,k = f 0,N and f 1,k = f 1,N and hence that which concludes the proof of the theorem. Lemma 5.1. Given T > 0, assume that the Then, we have the identity Also, we have the following "admisibility result" (regularity of traces).
Lemma 5.2. Given T > 0, there is C > 0 such that for all F ∈ L 2 ((0, T ) × M),ũ ∈ H 1 0 (M) and u associated solution of Proof. This is a direct consequence of the regularity theory for the heat equation where u is the unique solution to i.e. u(s) = e (t−s)∆ũ .
The following result is a direct consequence of (a slight variation on) [Cor07, Theorem 2.37] and the admissibility estimate of Lemma 5.2.

Global interpolation inequality and universal lower bound for traces of eigenfunctions
We follow the general Lebeau-Robbiano method [LR95] and use moreover a Carleman estimate of [LR97]. We refer to [LRL12] for an exposition of these works. The global strategy [LR95] is the following: Also, the unique continuation estimate for eigenfunctions of Theorem 1.7 can be deduced from the global interpolation estimate. The present section proves steps 1, 2, 3. The next section is devoted to that of steps 4, 5, 6.
In the following, for α > 0, we set If we were considering a second order elliptic operator Q on a manifold Y S with smooth boundary, and with Dirichlet condition on the whole ∂Y S , this estimate would simply read v H 1 (Y S ) ≤ C Qv L 2 (Y S ) + ψv| Σ 0 L 2 (Σ 0 ) + ψ∂ ν v| Σ 0 L 2 (Σ 0 ) . The proof of Theorem 5.5 follows from arguments of Lebeau and Robbiano [LR95,LR97]. The idea is that such interpolation inequalities follow locally from Carleman estimates, and then propagate well. Hence, our task is only (i) to deduce from a local Carleman estimate near Σ β that the traces at the boundary "control" a small nonempty open set near Σ β (i.e. that (5.3) holds with, in the l.h.s. the local H 1 norm in this set) (ii) to use a global interpolation inequality implying that such a small set "controls" the H 1 (Y β ) norm, and then put the two inequalities together.

However
For the second point (ii), we can start from the following result of [LR95, Section 3, Estimate (1)].
Theorem 5.6. Let U ⊂ Y S be any nonempty open set, then there is C > 0 and δ 0 ∈ (0, 1) such that we have for all v ∈ H 2 (Y S ) such that v| (−S ,S )×∂M = 0.
To prove (5.5), we shall take m ∈ Σ a point for which ψ(m) 0, and assume that the set U is a small neighborhood of m intersected with a single side of Σ. Also, we shall say that ∂ ν is pointing towards U. We now work in the local Fermi normal coordinates near m ∈ Σ, described in Section 2.1. The operator Q = −∂ 2 s − ∆ g , still denoted by Q in these coordinates, is given, modulo conjugation by a harmless exponential factor, by where • (s, x ) are the variables in (−S , S ) × Σ, ξ s ∈ R is the cotangent variable associated to s; • variables are in a neighborhood of zero in the half space R n+1 • ∂ ν is given by ∂ x 1 in these coordinates. Now, the proof of (5.5) relies on the following Proposition [LR97, Proposition 1]. Here, the variable s does not play a particular role: hence, in what follows, we only write (with a slight abuse of notation) x ∈ R n+1 for the overall variable, and accordingly q = q(x, ξ) = q(s, x 1 , x , ξ s , ξ 1 , ξ ). We also use the notation q ϕ (x, ξ) = q(x, ξ + idϕ(x)).
Then, we have h e ϕ/h u 2 L 2 (R n+1 h 4 e ϕ/h Qu 2 L 2 (R n+1 The end of proof of Theorem 5.6 is then similar to [LR95] or [LZ98,Appendix]. End of the proof of Theorem 5.6. We first fix R > 0 small enough such that B(0, R) is contained in the coordinate chart and that the set B(0, R) ∩ {x 1 = 0} (where the observation shall take place) is contained in the set {ψ > 0} (where ψ is the cutoff function appearing in (5.3)). Second, we define the weight function ϕ(x) = e −µ|x−x a | − e −µ|x a | , where µ > 0 (large, to be chosen) and, for a ∈ (0, R), we have x a = (0, · · · , 0, −a) R n+1 + . Hence, ϕ is smooth and satisfies ∂ x 1 ϕ 0 on K = R n+1 + ∩ B(0, R). According to classical computations (see e.g. [LRL12, Lemma A.1]), ϕ satisfies the Hörmander subellipticity condition on K for µ large enough (depending on R and a, and fixed from now on).
From Theorem 5.5, we deduce a proof of Theorem 1.7.
Proof of Theorem 1.7. For a non identically vanishing function ψ such that supp(ψ) ⊂ Σ β , we apply Theo- and we estimate each remaining term. First, we have We may assume that (−∆ g − λ 2 )u L 2 (M) ≤ u L 2 (M) since otherwise the inequality (1.11) holds trivially, and therefore obtain v 2 H 1 (Y S ) ≤ Ce cλ u 2 L 2 (M) . Third, we have Plugging the above three inequalities in (5.7) and dividing by u 2(1−δ) L 2 (M) (if non zero) yields the sought result.

From interpolation inequality to observability in an abstract setting: the original Lebeau-Robbiano method revisited
In this section, we explain how to deduce the observability estimate for the heat equation from the interpolation inequality of Theorem 5.5. This follows the Lebeau-Robbiano method introduced in [LR95] in its original form (used also in [Léa10]), as opposed to the simplified version (see e.g. [LZ98,LRL12]) which uses the stronger spectral inequality [JL99,LZ98] (which we do not prove in the present context). We explain how this method can be simplified using [Mil10,EZ11b,EZ11a]. We consider an abstract setting containing the above particular situation of the heat equation. Most results presented here still hold in the much more general abstract setting of [Mil10]. In Section 5.4 below, we explain how the problem of the heat equation controlled by a hypersurface is put in this general framework.
We denote by H (with norm · ) and K (with norm · K ) two Hilbert spaces, namely the state space and the observation space. We denote by A : D(A) ⊂ H → H a non-positive selfadjoint operator on H, with compact resolvent. We denote by (φ j ) an orthonormal basis of eigenfunctions associated to the eigenvalues λ 2 j ≥ 0 of −A (we keep the notation used for the Laplace operator) and set E λ := span{φ j , λ j ≤ λ}, λ > 0. (5.8) The operator A generates a contraction semigroup (e tA ) on H. We denote by B ∈ L(D(A); K) the observation operator. We say that B is an admissible observation for (e tA ) if there is T > 0 and C adm,T > 0 such that Be tA y L 2 ((0,T ),K) ≤ C adm,T y , for all y ∈ D(A). (5.9) On account of the semigroup property, (5.9) holds for all T > 0 if it holds for some T (see [Cor07, Section 2.3]). Hence, under the above admissibility assumption, for any T > 0, the map u 0 → (t → Be tA u 0 ) extends uniquely as a continuous linear map H → L 2 (0, T ; K), which we shall still denote Be tA .
In our next lemma, we use the notation, for s ∈ N and τ > 0, Lemma 5.8. Let S > β > 0 and ϕ ∈ C ∞ c (−S , S ). Assume there is C > 0 and δ ∈ (0, 1) such that for all v ∈ H 2 S , we have v H 1 (5.10) Then, there exists S , C, c > 0 such that Note that in the formula (5.11), we extend cosh(s ) by continuity by Id (resp. by s Id) on ker(A). Thus, denoting by Π 0 the orthogonal projector on ker(A) and Π + = Id −Π 0 , (5.11) can be rewritten more explicitely by Hence v(s) in (5.11) is the unique solution to Proof of Lemma 5.8. Note first that with v in (5.11), we have (−∂ 2 s − A)v(s) = 0 so that, in (5.10), it suffices to estimate v H 1 S from above and v H 1 β from below. For (v 0 , v 1 ) ∈ E λ × E λ , we denote by w k = Π 0 v k , k = 0, 1, and w ± = 1 2 (Π + v 0 ± (−A) −1/2 Π + v 1 ). This is and the parallelogram law yields We also have, with where Q j is the matrix

34
The eigenvalues of Q j are sinh(X j ) X j ± 1 ≥ εe X j /2 on the set [2βλ 0 , +∞[, whereλ 0 is the first non-zero eigenvalue of −A, and ε only depends on 2βλ 0 . As a consequence, we obtain v 2 Secondly, we also have v 2 Combining the last two estimates together with (5.10) yields and hence the sought result when dividing by v 0 2 + v 1 2 1−δ .
The next step of the Lebeau-Robbiano method relies on a so-called "transmutation argument" to deduce from the observability of the elliptic system on E λ the observability of the heat equation on E λ , with a precise estimate on the cost in terms of λ and T (observation time). Here, we use an idea of Ervedoza and Zuazua [EZ11b,EZ11a] to simplify the original argument of Lebeau and Robbiano [LR95] (who used the moment method of Russell to pass from the elliptic system to the parabolic system, and was quite technically involved, see [Léa10] for a review of the method). , for all λ > 0, v 0 ∈ E λ .
Note that in the assumption of Lemma 5.9, sinh(s can equivalently be replaced by cosh(s √ −A). We need the following lemma, which is a slight variant on [EZ11b,EZ11a].
Lemma 5.10. Given S , T > 0, δ ∈ (0, 1), and α > S 2 1 (5.14) For the proof of Lemma 5.10, we follow [EZ11b, Section 3.1], where the authors go from the wave equation to the heat equation. Here, we use the method to go from an elliptic equation to heat equation. The only difference is that we take g 2k+1 = g (k) 1 where Ervedoza and Zuazua [EZ11b,EZ11a] take g 2k+1 = (−1) k g (k) From the low frequency observability estimate with precise cost, we may now deduce the full observability estimate. The original Lebeau-Robbiano strategy [LR95] does not provide with an optimal dependance on the blow-up of the constant as T → 0 + . The modified and simplified argument of [Mil10] does so, and we follow it here.
Lemma 5.11. Assume B : D(A) ⊂ H → K is an admissible observation for (e tA ). Assume for some a 0 , a, b > 0 we have e T A y ≤ a 0 e aλ+ b T Be tA y L 2 (0,T ;K) , for all y ∈ E λ , λ > 0, T > 0. (5.16) Then there is C, c > 0 such that we have e T A y ≤ Ce c T Be tA y L 2 (0,T ;K) , for all y ∈ H, T > 0.
A proof of this lemma (in much more generality) is included in the proof of [Mil10, Theorem 2.1], but we give it for the sake of readability. The key feature of the semigroup (e tA ) we shall use is that e tA y H ≤ e −λ 2 t y H , for all y ∈ E ⊥ λ , λ > 0, t > 0. (5.17) We also make use of the following particular case of [Mil10, Lemma 2.1].
Proof of Lemma 5.11. For y ∈ H, we decompose y = y λ + r λ with y λ ∈ E λ and r λ ∈ E ⊥ λ . Then, we estimate e T A y ≤ e T A y λ + e T A r λ . (5.18) Concerning the second term in (5.18), we only use (5.17) to write e T A r λ ≤ e −λ 2 T r λ ≤ e −λ 2 T y .
Lemma 5.12 implies f ((1 − q)T ) e T A y 2 ≤ Be tA y 2 L 2 (0,T ;K) , T ∈ (0, T ], y ∈ H, which is the sought result for t ∈ (0, T ]. The case T > T follows from the boundedness of the semigroup and the case T < T .

From interpolation inequality to the observability estimate for the heat equation
Let us now put the above context of the heat equation in the present abstract framework, and state the consequences of the above abstract setting. We have H = H 1 0 (M), A = ∆ D (the Dirichlet Laplacian) with D(A) = {u ∈ H 3 (M), u| ∂M = 0, ∆ g u| ∂M = 0}. We also have K = L 2 (Σ) × L 2 (Σ) as well as u → (u| Σ , ∂ ν u| Σ ).
Lemma 5.2 implies that B is an admissible observation for (e tA ) in the sense of (5.9). The first lemma is a consequence of the interpolation inequality of Theorem 5.5 and Lemma 5.8. Here, E λ is defined by (5.8) where φ j , λ j are an orthonormal basis of solutions to (−∆ g − λ 2 j )φ j = 0. This together with Lemma 5.9 this implies the following result. Then, Proof. Notice first that ∂M + = Σ (∂M ∩ M + ) and by elliptic regularity, we have u, v ∈ C ∞ (M + ). Moreover, if w ± ∈ C 2 (M ± ), then, in the distribution sense (with ∂ ν pointing towards M + ) Hence, (−∆ g + V)u e = (−∆ g + V)u o = 0 as distributions and by elliptic regularity, u e , u o ∈ C ∞ and hence have the desired properties.
We may now proceed to the proof of Proposition 1.8.