We focus on the case of an observation of the flux in a part of the boundary that satisfies the Gamma of Lions conditions. In the control theory from the dependence of the exact controls for the waves with respect to the potentials;. In the applications that we have in mind and that will be developed in the following, it will be important to understand the dependence of the observational inequalities on the potentials.
The idea is to exploit Carleman's estimate of Theorem 1.1 to obtain controls whose dependence on the potential is weak. The proof of Theorem 1.3 is entirely variational and follows from the variational characterization of controllers. The idea now is to exploit Carleman's estimate of Theorem 1.2 to design an algorithm to reconstruct the potential from knowledge of the solution flux.
The first proofs of the stability of inverse problems for hyperbolic equations are based on uniqueness results obtained by local Carleman estimates (see e.g. [17, 22]). In particular, we give the proof of Theorem 1.3 on the dependence of the scheme with respect to the potential. P w−Rw|2dxdt, (2.7) the bulk of the proof then consists of bounding under the cross term.
Here and in the rest of the proof, Theorem 2.1 M >0 corresponds to a generic constant that depends at least on Ω and T but is independent of sand and λ.
Weighted Poincaré inequality
We now take s0 large enough so that the terms of the last line (coming from X1 and|Rw|2) are absorbed by the dominant term ins3λ3|w|2ϕ3 as soon as ass≥s0. Using (2.3) which by construction gives P w=esϕv, we can go back to the variable fin (2.13) and obtain that there exists some positive constantM such that for alls≥s0 andλ≥λ0,. In the sequel, we will fix λ=λ0 and use the fact that ϕ is then bounded from below by1 and from above by certain constants, depending on λ.
A Carleman estimate in time T large enough
Therefore v satisfies the required hypothesis sev(±T) =. 2.23) We will show that the last two terms of (2.23) can be absorbed in the left-hand side if the parameters and the time T are chosen sufficiently large. Integrating forτ between−T+ηandT−η, then is large, Es(T−η)≤M p. 2.29) Moreover, by combining it with the weighted Poincaré estimate of Lemma 2.4, we get To do so, one can introduce z(x, t) =˜ z(x,−t) and use the above estimates toz˜(we make the change of variable→ −t).
Note that, following the above proof carefully, the Carleman estimate (2.18) can actually be slightly improved.
A Carleman estimate with pointwise term in time − T
A Carleman estimate with pointwise term in time 0
We now conclude with the proof of Theorem 1.2, the main tool for studying the inverse problem in Section 4. Of course, since each term is odd or even, the integrals on (−T, T) are simply twice the integrals on (0 , T), thus the proof (1.9) is completed. In this section, our goal is to present the implications of the Carleman estimate of Theorem 1.1 with respect to the control properties of equation (1.10).
Setting
This is due to the fact that the weight function e2sϕ is bounded from above and below by positive constants, depending on us. Note that in (3.6) these equivalences of norms are proven uniformly with respect to tos≥s0(m) for potentials lying in L∞≤m(Ω×(−T, T)). In the following we will denote the space T with the norm k · kobs,s,p by (T,k · kobs,s,p).
But we draw the reader's attention to the fact that these · kobs,s rates are not uniformly equivalent with respect to tos >0. Note that, however, for all >0dhep∈L∞(Ω×(−T, T)), it is easily checked that, restricting the functions as necessary, there exist such C(s, p). 3.8) Finally, we also introduce the dual space L2(Ω)×H−1(Ω) in which we consider the family of norms.
Construction of a null-controlled trajectory
But this is exactly the dual formulation of equation (1.10) and integrations by parts yield, for allz∈ T,.
Dependence of the controls with respect to the potentials
In this section, we will propose an algorithm based on the Carleman estimator (1.9) and a data assimilation approach. Recall that the unknown is the potential Q=Q(x), which we want to obtain from the measurement of the normal derivative of the solution W[Q] from (1.15) on Γ0×(0, T). Note also that we are working under geometric assumptions and the function ϕ, which we will discuss below, always satisfies (1.6), so that Theorem 1.2 holds when the parameters are large enough.
As said in the introduction, in [2] one can find the proof that the additional information ∂νW[Q]onΓ0×(0, T) allows to identify Quniquely within the class of potentials inL∞≤m(Ω). We will then focus on the proof of Theorem 1.4, which is the main step in the proof of the convergence result of Theorem 1.5.
The general idea
Indeed, our goal is to prove that Algorithm 1, presented in Section 1.3, is convergent when large enough, as described in Theorem 1.5. Of course, it is very close to functionalKs,p, and we will therefore introduce the space T+=n. The properties of the space T+ and the family of normsk·kobs+,s,q are of course completely similar to those of T in (3.5), equipped with the family of normsk·kobs,s,p introduced in (3.2).
The continuity, strict convexity and constraint of the functional Js, p[µ, g] are straightforward and left to the reader. Of course, our goal is not only to prove that the functional Js,q[µ, g] has a minimum, but rather to study how the minimum of Js,q[µ, g] depends on the source termg. Indeed, zkin (4.1) is the minimum of the functionalJs,qk[µk, gk], whatever is >0, while in the algorithm Zk is the minimizer of the functionalJs,qk[µk,0], see (1.23).
As in Section 3.3, we will rely on the Euler-Lagrange equations satisfied by the minimum of the functions Js,q[µ, ga]andJs,q[µ, gb].
Convergence of Algorithm 1
To the best of our knowledge, the only stability result of the inverse problem proved by micro local analysis is a recent work [27] that requires GCC and convexity of the entire boundary. The control procedure proposed in Section 1.2 does not match the framework developed in [13], which proves that using the Hilbert Uniqueness Method (HUM) (slightly modified by introducing a smooth discontinuity function in time) to compute the controls if the data, to be controlled are smooth, then the corresponding control and the controlled path are smooth. Does the control procedure in Section 1.2 have smoothness properties similar to those of classical HUM control.
How the usual HUM control process depends on the potentials of the wave equation. Recently, in [4], we have proved discrete Carleman estimates for the discretized 1-d space semi-discrete wave equation using finite differences. This term, which corresponds somewhat to a kind of Tychonoff regularization of the Carleman estimates, is necessary because of the spurious waves created by the discretization process.
Based on these uniform Carleman estimates, we have been able to prove a convergence result for approximating a potential in the inverse problem given in Section 1.3, provided that a Tychonoff regularization term is added in the process. It would then be quite natural to try to adapt the algorithm developed here in the continuous case to the space-semi-discrete schemes and in numerical. The authors also wish to thank the Institut Henri Poincaré (Paris, France) for providing a very stimulating environment during the program "Control of Partial and Differential Equations and Applications" in autumn 2010.
Lipschitz stability in an inverse wave equation problem, 2010, http://hal.archives-ouvertes.fr/hal-00598876/fr/. Carleman global estimation on a network for the wave equation and application to an inverse problem. Analysis of the HUM control operator and exact controllability for semilinear waves in uniform time.
On the optimality of the observable inequalities for parabolic and hyperbolic systems with potentials. On the numerical implementation of the Hilbert uniqueness method for the exact boundary control of the wave equation. Carleman makes estimates for the non-stationary Lamé system and its application to an inverse problem.
