STABILITY ESTIMATE FOR THE SEMI-DISCRETE LINEARIZED BENJAMIN-BONA-MAHONY EQUATION∗,∗∗

In this work we study the semi-discrete linearized Benjamin-Bona-Mahony equation (BBM) which is a model for propagation of one-dimensional, unidirectional, small amplitude long waves in non-linear dispersive media. In particular, we derive a stability estimate which yields a unique continuation property. The proof is based on a Carleman estimate for a finite difference approximation of Laplace operator with boundary observation in which the large parameter is connected to the mesh size. Mathematics Subject Classification. 35B60, 35L05, 35Q35, 35R45, 65M06. Received November 26, 2020. Accepted August 29, 2021. Dedicated to Enrique Zuazua on the occasion of his 60th birthday


Introduction and results
In this work, we are interested in a linearized version of the Benjamin-Bona-Mahony equation (BBM) u t + u x + uu x − u xxt = 0, (1.1) proposed by T. Benjamin et al. [2] as a model for propagation of one-dimensional, unidirectional, small amplitude long waves in non-linear dispersive media. In the last years, several authors have widely studied dispersive equations in the context of controllability and inverse problems. Nevertheless, the BBM equation presents several particularities due to the structure of the operator. In particular, the infinitesimal generator of the semigroup is given by (I − ∂ 2 x ) −1 ∂ x , which is a compact operator, in opposition with the most common situation in PDE where the generator is an unbounded operator among others. We note some interesting results about the unique continuation property (UCP) for BBM for the continuous case. We refer to the reader to those works and their references for a more detailed discussion: L. Rosier and B.-Y. Zhang in [14] developed a UCP for (1.1) on a periodic domain. Moreover, in [6] P. L. da Silva and I. L. Freire give an alternative proof using geometrical arguments for the periodic case, and for the case when (1.1) is solved in R. In [16], X. Zhang and E. Zuazua considered a linearized BBM equation with space-dependent potential u t − u xxt = [α(x)u] x + β(x)u, (x, t) ∈ (0, 1) × (0, T ). (1.2) In that work, the authors established that the only solution of (1.2), such that u(0, t) = u(1, t) = 0, is the trivial one u ≡ 0 provided that both α and β do not vanish on some open subset of (0, 1). Furthermore, if α(x) = −1 and β(x) = 0 in (1.2), S. Micu proved in [12] a UCP with the additional boundary condition u x (1, t) = 0, and study controllabily results. On the other hand, in [15], M. Yamamoto established a UCP for BBM-like equation with time and space dependent potential where p ∈ L ∞ ((0, T ) × (0, 1)) and q ∈ L ∞ (0, T ; L 2 (0, 1)). It was shown that the solution of (1.3) shall vanish in (0, 1) × (0, T ), provided u(1, t) = ∂ x u(1, t) = 0 for all t ∈ (0, T ) and u(x, 0) = 0 for x ∈ (0, 1). The main tool to prove this result is a Carleman estimate for the Laplacian operator. Through a more refined version of this Carleman estimate, a stability estimate can be formulated for equation (1.3) (see Sect. 5). We note that the UCP's have been well studied in several continuous partial differential equations, but it is possible to see that the corresponding discrete case does not holds. For instance, if u is a harmonic function in a domain Ω and u = ∂ n u = 0 on Γ ⊂ ∂Ω then u ≡ 0 in Ω, it does not generally hold its discrete formulation. We refer to the counterexample due to O. Kavian, presented by E. Zuazua in [18]. However, for (1.4) it is possible to obtain a quantitative UCP under restriction over the mesh size. This result shall be discussed in more detail in Section 1.2.
In this paper, we are interested whether a unique continuation property, as in the work of Yamamoto [15], still holds for a semi-discrete approximation in space of (1.3). In this sense, for N ∈ N given, we set the space discretization parameter h := 1/(N + 1). We consider the pairs (x i , t) with t ∈ (0, T ), T > 0, and x i = ih, for i = 1, . . . , N . Thus, the space semi-discrete approximation of equation (1.3) by using the centered finite difference method with respect to the space variable is given by for i ∈ {1, 2, . . . , N } and t ∈ (0, T ), where u i (t) stands for u(x i , t).

Notation
We introduce the notation of meshes and operators that shall be used throughout this paper and necessary to state our results. We consider the following regular partition of the interval [0, 1] as M h := {x i | x i := ih, i = 0, 1, . . . , N + 1} , for N ∈ N and h := 1/(N + 1). For any sets of points W h ⊂ M h , we define the following dual meshes W h and W * h as  where We note that ifW h = W h , then for two consecutive points x i , x i+1 ∈ W h we have x i+1 − x i = h. Thus, any subset W h ⊂ M h that verifiesW h = W h is called regular mesh. Finally, we define the boundary of a regular mesh W h as ∂W h := W h \ W h . These sets defined above can be seen in Figures 1 and 2.
We define the average operator A h and the difference operator D h by To introduce the boundary conditions, we define the outward normal for (x h , t) ∈ ∂Q h as We indicate by ∂Q + (resp. ∂Q − ) the set of points such that n h (x h , t) = 1 (resp. n h (x h , t) = −1). We also introduce the trace operator for u h ∈ C(Q * h ) as Let us finally introduce the discrete integration on the boundary for u h ∈ C(∂W h ) as

Discrete Carleman estimates with boundary observation
For the discrete Carleman estimate, we consider the weight function of the form r(x, t) = e sϕ(x,t) for s ≥ 1, with ϕ(x, t) = e λψ(x,t) where ψ is a continuous function whose domain of definition Ω is contained in an enlarged smooth open and connected neighborhoodΩ. We also assume ψ ∈ C k Ω with k large enough such that it satisfies the following property (1.6) The assumption of the higher-order derivatives is needed to obtain the estimates on the weight function presented in Section 4, in contrast to the continuous case. We shall use the same notation for the sample of the continuous function on the discrete or semi-discrete sets. We now state a uniform Carleman estimate for the operator D 2 h with boundary observation. Although for the Carleman estimate just the condition (1.6) is needed, to achieve the UCP property for the system (1.4) we also consider the following assumption It is not difficult to find a function that verifies conditions (1.6) and (1.7), for instance the following function, Let ψ be a function verifying (1.6) and T > 0. For the parameter λ 0 ≥ 1 sufficiently large, there exist s 0 (λ 0 ) ≥ 1, h 0 > 0, ε 0 > 0 and C = C(ε 0 , s 0 , λ 0 ) independent of h > 0 such that Several recent works have been concerned with discrete and semi-discrete Carleman estimates for secondorder differential operators. The hyperbolic case has been developed for the one-dimensional case by L. Baudouin and S. Ervedoza [1], to study the stability of an inverse problem to recover a potential term in a semi-discrete wave equation. The elliptic case has been developed by F. Boyer et al. in [3], for the one-dimensional case, to establish a relaxed observability estimate for the associated semi-discrete parabolic equation. In [7], S. Ervedoza and F. de Gournay study the Laplacian operator in arbitrary dimension to prove the stability for the discrete Calderon problem, with limiting Carleman weight function. Semi-discrete Carleman estimates for parabolic operators have been established by F. Boyer and J. Le Rousseau in [4] for multidimensional Cartesian grids. Moreover, in [13], T. N. T. Nguyen studied in the one-dimensional setting a semi-discrete parabolic operator with discontinuous diffusion coefficient; both of them obtain relaxed controllability results for their respective systems.
In the aforementioned works, the discretization was based on a finite difference scheme. Also, the Carleman parameter cannot be arbitrarily large, which is related to the discretization step size, in contrast to the continuous setting. Let us finally mentioned that recently in [10], a fully discrete Carleman estimates for parabolic operator have been obtained by V. Hernández-Santamaría and P. González Casanova, where the spatial and time discrete step-size parameters are connected to the Carleman parameter.
We recall that T. Carleman introduced an estimate, known as the Carleman estimate, to prove a UCP for second-order elliptic partial differential equations in [5] when coefficients fail to be analytic. Nowadays, it has become an efficient tool to prove UCP, the study of controllability, observability, and stabilization for PDEs. We refer to the work of X. Fu et al. [8], and references therein, where the authors present a unified approach to Carleman estimates for second-order PDEs and their applications to control theory and inverse problems.
One of the main difficulties in the development of discrete or semi-discrete Carleman estimates is to compute multiple discrete operators such as D h and A h on the Carleman weight functions. For this reason, we establish Theorem 4.9 (see Sect. 4) to reduce some tedious computation, and it represents an extension of the results presented by F. Boyer et al. in [3] related to discrete estimate on the weight function.

Semi-discrete setting and main results
With the notation we have introduced we can rewrite the semi-discretization (1.4) as where p h , q h ∈ L ∞ h (Q h ). In (1.10) u h (t) provides an approximation of u(x h , t), u being the solution of the continuous equation (1.3). ∂ t u h stands for the first order differentiation with respect to t and the operators D h and D 2 h are the classical central finite-difference approximation of the space derivatives. We assume that there exists a constant M > 0 independent of h such that max{ As a consequence of Theorem 1.2, we have the following unique continuation property for semi-discrete BBM equation (1.10).
The methodology of the proof of Theorem 1.2 is similar in spirit to [15], where it have been obtained a UCP for equation (1.3). However, it cannot be followed straightly from the proof of the continuous case since the parameter s cannot be arbitrarily large. As we mentioned above, this parameter is related to the mesh size. Thus, a semi-discrete version of the Carleman estimate used in [15] is not enough. For this reason, we develop a more refined Carleman estimate, (5.2), and its semi-discrete counterpart, see Theorem 1.1.
The rest of this article is organized as follows. First, in Section 2, we introduce and prove some discrete calculus formulas for uniform meshes. Then, we establish a stability estimate for the semi-discrete numerical approximation based on a uniform spatial discretization of equation (1.3) in Section 3. The last three sections are devoted to the proof of the discrete Carleman estimate for a finite-difference approximation of Laplacian operator with boundary observation. In Section 4, we extend the estimations developed in Section 3 of [3] for arbitrary order, which reduce some computation in the proof of the Carleman estimate presented in Section 5. Some calculations are postponed and computed in Section 6.

Calculus formulas for uniform meshes
In this section, we state the elementary notions concerning discrete calculus formulas. First, we set some useful identities that shall be used in what follows. Then, we present the integration by parts for the difference and average operator. For the sake of presentation, we set the results in the discrete space variable framework instead of the semi-discrete setting.
The following Lemma gives us a calculus rule for finite difference operators. Its proof can be found in [3].
As a direct consequence of Lemma 2.1, we point out some identities to develop the rest of this paper.
In particular, for all u ∈ C(W h ), Note that as a consequence of the first identity in (2.1) of Lemma 2.1, it can be proved by induction the following identity.
Now, we state the following integration by parts presented for discrete space variable. We note that the same identities could be considered in the semi-discrete setting since the temporal variable does not play any significant role. (2.7) Combining (2.6) and (2.7) it follows that Similarly, averaging the equations (2.6) and (2.7) we obtain which completes the proof.
Remark 2.5. One can consider the previous result for the meshes W h , see for instance a similar result by S. Ervedoza and F. de Gournay in [7]. In that case we should change the definition of the boundary nodes of our mesh. As we are interested in the boundary that kind of integral-by-parts formulas does not fulfill our goal. In contrast with the integration-by-part formulas from F. Boyer et al. [3], we do not make any distinction on the difference and average operators, then we need to specify that difference in the respective meshes.

Stability estimate for space semi-discrete BMM equation
This section is devoted to proof Theorem 1.2. We follow as close as possible the ideas from its continuous formulation. For this reason we state a stability estimate for (1.3). The main tool for the proof is a Carleman estimate for Laplacian operator. For sake of exposition we postpone that proof, see Section 5. It is worth to mention that we refined the result presented in [15] (see Sect. 5.1).

The continuous case
We consider Q := (0, 1) × (0, T ), for T > 0, and we define the classical inner product Following the methodology from [15] we can obtain a stability estimate for (1.3). The proof is based on the Carleman estimate (5.2) and Lemma 6.4.2 from V. Isakov [11].
As a Corollary of Theorem 3.1, it follows the main result presented in [15].

Proof of Theorem 1.2
As we mentioned above the proof of Theorem 1.2 is based on the continuous setting strategy. Then, we write down a space semi-discrete version of Lemma 6.4.2 from [11], which is a Poincaré weighted inequality.
and integrating overM h we complete the proof.

Proof of Theorem 1.2
In this section, we shall give the proof of the Theorem 1.2.
Proof. Note that from (1.10) we have Then, applying the Carleman estimate (1.9) to v h = ∂ t u h we obtain On the other hand, we note that Now, we focus on the first term of the right-hand side above. Using (2.3) and a discrete integration by parts for the discrete average operator we get From Proposition 4.5 we have A h (e 2sϕ ) ≤ C λ e 2sϕ , we thus obtain (3.4)

Combining (3.2), (3.3) and (3.4) we get
We note that by choosing some s ≥ T 2/3 (1 + M 2 ) 1/3 + T 2 M , the last term on the right-hand side from (3.5) can be absorbs by its left-hand side. Thus, recalling the hypothesis on s from the Carleman estimates, by choosing Now, we need to connect the condition over the Carleman parameter s with the mesh size h. Defining it follows that for 0 < h ≤ min{h 0 , h 1 } we have sh ≤ ε 0 provided s ∈ (s 1 , ε 0 /h), which conclude the proof.
As a consequence, we obtain the UCP presented in Corollary 1.3 for semi-discrete BBM equation (1.10).

Some preliminary discrete calculus results
In this section, we establish some previous estimates for the Carleman weight function that shall be used in the next section to obtain the discrete Carleman estimate, see Theorem 1.1. Recall that our weight function is defined as e sϕ for s ≥ 1, with ϕ = e λψ , where ψ ∈ C k for k sufficiently large and λ ≥ 1. Our goal is to generalize the results presented previously in Section 3, obtained by F. Boyer et al. in [3], related to discrete operations performed on the Carleman weight functions, considering estimates and expansions for higher order discrete operators.
For easier comparison, we use the same notation by setting r = e sϕ and ρ = r −1 , these positive parameters s and h shall be large and small respectively and limited by the condition sh ≤ 1. The proofs are similar in spirit to those given in [3].
We denote by O λ (sh) the functions that verify O λ (sh) L ∞ (Q h ) ≤ C λ sh for some constant C λ depending on λ. By O(1) we denote bounded functions and by O λ (1) a bounded function once λ is fixed.
Proposition 4.1. Let us consider n ∈ N. Let f be a (n + 2)−times differentiable and g a twice differentiable functions on R. Then where R D n h and R A n h are given by Proof. The proof of this proposition follows from Taylor expansion First, we use (4.1) with i = 2 and y = (n − 2k)h 2 to obtain τ n−2k Then, it follows that Now, using n k=0 n k = 2 n and n k=0 n k k = n2 n−1 we write On the other hand, applying (4.1) for f with i = n + 2 and y = (n − 2k)h 2 , we have Thus, for the difference operator we get and the proof is complete.
Proof. It is enough to see that h . Now, we consider two fundamental estimates for our weight function. The proofs of these results can be found in [3]. We consider α = (α t , α x ) ∈ N 2 multi-indices.

Proof. From Corollary 4.2 we write
Then which yields the result. Lemma 4.6. Let α and β multi-index and n ∈ N. Provided sh ≤ 1, we have

By Lemma 4.3, it follows that
which concludes the proof of the first result.

By Lemma 4.7 we write
which is our claim.
Let us finally mention that the results of this section can be extended for time-dependent case. For instance, if we consider a weight function of the form r(x, t) = e sθ(t)ϕ(x) then the condition sh ≤ 1 must be replaced by sh(max [0,T ] θ(t)) ≤ 1 which implies that sθ(t)h ≤ 1.

Discrete Carleman estimate
In this section, we establish a discrete Carleman estimate with boundary observation for a finite difference approximation of the Laplacian operator in the one-dimensional setting. In order to do so, it is natural to look closer at the continuous version of such estimates. For this purpose, we follow the methodology of A. V. Fursikov and O. Y. Imanuvilov [9] to obtain a Carleman estimate for Laplacian operator in the continuous setting, which is similar to the estimate obtained by Yamamoto in [15]. The main difference with the one in [15] to our estimate is that we do not consider a density argument, and we thus obtain a Carleman estimate with boundary observation. Then, following the methodology in [3] we establish a discrete Carleman estimate.

The continuous case
The proof of the following Carleman estimate has two steps. First, we consider the conjugate operator defined by P ϕ u := e sϕ ∂ 2 x (e −sϕ u). In this case, our Carleman weight function is defined as e sϕ for s > 0 with ϕ = e λψ , where λ > 0, and satisfy Then, we split P ϕ into the operators P 1 and P 2 , and it is estimated the scalar product (P 1 u, P 2 u) L 2 (Q) .

Proof of the discrete Carleman estimate
Now, we establish a discrete Carleman estimate for the discrete operator D 2 h . Note that this is the discrete Laplacian in one-dimensional setting. There are Carleman estimates for this kind of operator (see F. Boyer et al. [3,4] and S. Ervedoza et al. [7]). The main difference respect to our estimate is the fact that we consider boundary observation, due to the choice of the weight function. Indeed, our Carleman weight function is defined as e sϕ for s ≥ 1, with ϕ = e λψ where ψ ∈ C k for k sufficiently large and λ ≥ 1. We also assume that We follow a classical scheme based on conjugating the original operator with a well chosen exponential function.

Proof Theorem 1.1
We make the change of variable u h = e sϕ v h . Our first task is to obtain an expression for P h,ϕ := e sϕ D 2 h (e −sϕ u h ) with the change of variable proposed. By using (2.5), we have We define the following coefficients α 1 := e sϕ A 2 h (e −sϕ ), α 2 := e sϕ D 2 h (e −sϕ ) and β 1 := e sϕ A h D h (e −sϕ ). On the other hand, we set We write Now, we shall estimate the scalar product For each term of (5.13), we obtain the following results.
Lemma 5.3. For sh ≤ 1, we have Lemma 5.4. For sh ≤ 1, we have where Lemma 5.5. For sh ≤ 1, we have where Lemma 5.6. For sh ≤ 1, we have and The proof of Lemmas 5.3-5.6 can be found in Section 6.
Combining the aforementioned Lemmas, for sh ≤ 1 there exist λ 1 ≥ 1 and ε small enough such that for λ ≥ λ 1 and 0 < sh ≤ min{ε 1 (λ), 1} = ε 1 (λ), there exists a constant C λ1,ε1 > 0 such that (5.14) Thus, from (5.11), (5.12) and (5.14) we get On the other hand, we have to deal with the boundary terms. To do this, we can estimate separately the right and left boundary observation. Indeed, let us denote by Y − i and Y + i the left and the right boundary observation of the term Y i , respectively. Once λ is fixed, for s large enough there exist positive constants C 0 and C 1 such that Therefore, if we fix λ = λ 1 , we can choose ε 0 and h 0 sufficiently small, with 0 < ε 0 ≤ ε 1 (λ 1 ), and s 0 ≥ 1 sufficiently large, such that for s ≥ s 0 , 0 < h ≤ h 0 , and sh ≤ ε 0 we obtain Finally, we return to the variable v h . To this end, we need the following Lemma.
Lemma 5.7. For sh ≤ 1, we have For a proof see Section 6. Combining (5.15) with Lemma 5.7, we can chooseε > 0 andh > 0 sufficiently small, with 0 <h ≤ h 0 , 0 <ε ≤ ε 0 , ands sufficiently large, such that for s ≥s, 0 < h ≤h, and sh ≤ε, we obtain where we have dropped the left boundary observation, and the proof is complete.

Proof of intermediate results
In this section, we shall prove some technical results used in the development of the discrete Carleman estimate. We consider sh ≤ 1 in the following Lemmas in order to ensure that every Lemma from Section 4 holds. Recall that our Carleman weight function defined as r(x) := e sϕ(x) for s ≥ 1, with ϕ(x) = e λψ(x) where ψ ∈ C k for k sufficiently large and λ ≥ 1. We denote ρ := r −1 and ψ verifies ∂ x ψ > 0 in Q h . The proof we develop in each Lemma is standard in the following sense. We begin rewritten the semi-discrete integral, if necessary, using some identity related to the discrete operators from Corollary 2.2. Then we apply a semidiscrete integration by parts from Proposition 2.4 to identify the leader terms of the Carleman estimate. Finally, thanks to Theorem 4.9, we can obtain the estimate claimed in each Lemma.

Proof of Lemma 5.3
Recalling the definition of C 1 and B 1 , and setting γ 11 := β 1 α 1 and I 11 := C 1 u h , B 1 u h Q h , we write From Corollary 2.2 the semi-discrete integral I 11 can be rewritten as Using Proposition 2.4, for I 11 we obtain

The proof is completed by showing that
which follows from Theorem 4.9 and Corollary 4.4.

Proof of Lemma 5.4
Set I 12 := C 1 u h , B 2 u h Q h . From the definition of the operators C 1 and B 2 , we have A semi-discrete integration by parts, Proposition 2.4, yields 12 .
Let us focus on I (a) 12 . We note that thanks to Lemma 2.1, I 12 can be rewritten as To estimate the term I (a2) 12 , due to Lemma 2.1, we write By using Proposition 4.1 we obtain the following estimates

Moreover, Lemma 4.8 leads to
The previous estimates enables us to write (6.1) as Therefore, I (a2) 12 can be estimated as On the other hand, by using (2.3), I (a1) 12 can be rewritten as A semi-discrete integration by parts with respect to the difference operator D h leads to By using (2.5), it follows that Now, applying Lemma 4.8 to α 1 := e sϕ A 2 h (e −sϕ ), we have Moreover, applying Proposition 4.1 to ∂ 2 x ϕ, we get Thus, (6.3) can be estimated as Similarly, we get Hence, for I (a1) 12 we obtain Finally, by using the Young's inequality, I 12 can be bounded as Therefore, collecting the estimates (6.2), (6.4) and (6.5) ,I 12 can be estimated as where X 2 and Y 2 are given by which is our claim.

Proof of Lemma 5.5
Setting γ 21 := α 2 β 1 and I 21 := C 2 u h , B 1 u h Q h . Let us compute By using Lemma 2.1 the above semi-discrete integral can be rewritten as A semi-discrete integration by parts with respect to the difference operator yields Then, by Proposition 2.4 we obtain Recalling that α 2 := e sϕ D 2 h (e −sϕ ) and β 1 := e sϕ A h D h (e −sϕ ) we have and D h (γ 21 ) = s 3 O λ (1), by virtue of Proposition 4.9 and Corollary 4.1. Hence, for I (a) 21 we obtain the following estimate On the other hand, I (b) 21 can be estimated as , due to Proposition 4.9. Thus, combining (6.6) with (6.7) and (6.8) the Lemma follows.

Proof of Lemma 5.6
Let I 22 := C 2 u h , B 2 u h Q h . By definition of C 2 and B 2 , let us estimate the semi-discrete integral To this end, by using (2.2), I 22 can be rewritten as Since sh ≤ 1, from Proposition 4.5 and Lemma 4.3 we have for α 2 := e sϕ D 2 h (e −sϕ ) the following estimate Furthermore, noting that ∂ 2 x ϕ = λ 2 (∂ x ψ) 2 ϕ + λϕ∂ 2 x ψ, with the previous estimate for α 2 we obtain Then, I 22 is estimated as Similarly, for I 22 , a semi-discrete integration by parts yields Let us estimate I (b2) 22 . Note that by using (6.9) and Young's inequality, I 22 can be bounded as We now use a semi-discrete integration by parts on the first integral above to obtain To obtain an estimate for I Indeed, to prove the estimate (6.12) we use Lemma 2.1 to write Then, thanks to Lemma 4.8, we obtain Moreover, using Proposition 4.1 we have and since ∂ 2 x ϕ = O λ (1), (6.12) follows. For the estimate (6.13), applying (2.5) it follows that Similarly, by using Lemma 4.8 and Proposition 4.1 we have These estimates and (6.15) establishes (6.13). The same methodology works for (6.14). We thus have, from (6.12)-(6.14), the following estimate for I (b1) 22 Therefore, combining (6.10) with (6.16) and (6.11) proves the estimate for I 22 .
6.5. Proof of Lemma 5.7 We begin proving the first inequality (5.16) of our Lemma. Recalling that v h = u h e −sϕ , thanks to Lemma 2.1 and Young's inequality, we have Let us first estimate J 2 . Using (2.3) and a discrete integration by part respect to the average operator we obtain Then, by virtue of Proposition 4.5, J 2 can be estimated as follows It remains to proves that which it follows from Proposition 4.5, and the proof for (5.16) is complete.
To prove the inequality (5.17), we note that

Repeated application of Lemma 2.1 and Young's inequality lead to
Then, using Proposition 4.5 we obtain which completes the proof.

Comments
The results presented in Section 4 are of independent interest in view of its potential applications on problems related to semi-discrete Carleman estimates. For instance, it could be used to answer the challenge proposed by C. Zheng in [17], that is, to obtain a semi-discrete global Carleman estimates for fourth-order Schrödinger equation and establish a semi-discrete counterpart of the main results presented in that paper. Even in the continuous setting, there are few papers about the stability of an inverse problem for higher-order equations, via Carleman estimates, due to tedious computation and the increased complexity. To our knowledge, there are no results about discrete or semi-discrete Carleman estimates for higher-order operators. Thus, Theorem 4.9 can be a useful tool to obtain results in that direction.
A possible extension of this paper could be to reformulate Theorem 1.2 for some families of non-uniform meshes. The Carleman estimate (1.11) is established for uniform mesh and could be adapted to some non-uniform meshes obtained as the smooth image of a uniform grid, following the methodology of [3].
Another interesting question is to consider the fully discrete case of our problem, particularly due to the term ∂ 2 x ∂ t , which mixes time and space. Perhaps a first attempt is just consider the time-discrete case u n+1 − u n ∆t − ∂ 2 x u n+1 − ∂ 2 x u n ∆t = p n+1 ∂ x u n+1 + q n+1 u n+1 , n = 0, 1, ..., which is a possible discretization in time.