Carleman estimate for a one-dimensional system of m coupled parabolic PDEs with BV diffusion coefficients
© Ramoul; licensee Springer 2014
Received: 25 January 2014
Accepted: 29 July 2014
Published: 25 September 2014
This paper is devoted to deriving global Carleman estimate for a one-dimensional linear coupled parabolic system of m equations with bounded variations (BV) diffusion coefficients. This kind of estimate is a generalization of the scalar result (Le Rousseau in J. Differ. Equ. 233:417-447, 2007). The key ingredient is to derive a global Carleman estimate for piecewise- diffusion coefficients based on the construction of a suitable weight function. The Carleman estimate in the case of BV diffusion coefficients is then obtained using the approach of BV diffusion coefficients by piecewise-constant coefficients. This Carleman estimate is used to show the observability inequality which yields the controllability result.
MSC: 35K40, 26A45, 93B07.
1 Introduction and notations
In this paper we deal with one-dimensional m coupled parabolic equations with bounded variations (BV) diffusion coefficients.
Let be a one-dimensional bounded domain, and we assume that . Let us consider the following notations: , and .
The diffusion coefficients () are assumed to be of BV and satisfy the following.
The domain of is given by .
where for , and for all .
The main goal of this paper is to prove a global Carleman estimate for the operator with an interior observation region , where ω is a non-empty open subset of Ω and such that are of class on .
The Carleman estimate for piecewise regular diffusion coefficients is established by Doubova et al. in . In this work, the authors considered a scalar parabolic equation. They obtained observability inequality and controllability results by adding assumption on the monotonicity of the coefficient (i.e., the observability is supported in the region where the diffusion coefficient is the lowest). To obtain these results, the authors introduced a non-smooth weight function β, assuming that it satisfies the same transmission condition as the solution of a parabolic equation. An inverse problem for such a parabolic equation was studied in . In the same direction, we can also cite the work  of Bellassoued and Yamamoto which is devoted to determining a source term using the Carleman estimate established in . In 2007, a new Carleman estimate was established by Benabdallah et al. for the one-dimensional heat equation with a discontinuous diffusion coefficient. In this work the authors relaxed the monotonicity assumption on the diffusion coefficient by constructing a specific non-smooth weight function β. This function β satisfies suitable trace properties depending on the jumps of the derivatives of β at the singular points of the diffusion coefficient. In higher dimensions (), Le Rousseau and Robbiano in  showed that the monotonicity assumption on the diffusion coefficients can be relaxed and the observation region can be chosen independently of the jump’s sign of the diffusion coefficient. In the same way, we cite the work  about Carleman estimates in stratified media. In , Le Rousseau generalized the results obtained in  for the case of bounded variations diffusion coefficient (BV). In Le Rousseau’s paper, the author constructed a limit weight function as he approached BV coefficient by piecewise-constant coefficient. However, the relaxation of the monotonicity condition in the case of bounded variations diffusion coefficient in any dimension remains open.
For the first time, a Carleman-type estimate with one observation in parabolic systems was introduced by Ammar Khodja et al.,  where the authors used this estimate to establish observability inequality and deduce a controllability result by one control force. We also refer to ,  for this kind of works. In paper , Cristofol et al. obtained a new Carleman-type estimate with one observation acting on a subdomain ω of () for a reaction-diffusion system. They used this estimate for simultaneous identification of one parameter and initial conditions. We also cite the article , which represents an improvement of the work . It is about determining two coefficients by observation data of only one component in a nonlinear parabolic system. In the same direction, we can cite the works , .
If the observation region ω is replaced by , the Carleman estimate with observations for a system of m () coupled parabolic equations remains an open question.
In the same way, we cite the recent work  about an inverse problem for a one-dimensional coupled parabolic system (two equations) with discontinuous conductivities (assumed to be ). The paper  is devoted to proving the stability result using the Carleman estimate (with the observation of only one component) based on an adequate choice of weight function which is the same for each equation of a parabolic system. However, the authors needed additional assumptions on this Carleman weight function, and the method that was developed is completely different with respect to the approach obtained in our paper.
Roughly speaking, the aim of our paper is to extend the results obtained in  to the case of m coupled parabolic equations. One of the main difficulties in extending the scalar result comes from the fact that the weight function β has to be chosen the same for each equation and depends on the jumps of diffusion coefficients. Moreover, since the jump discontinuities may be located at different points for the diffusion coefficients (), this created an additional difficulty to find our weight function.
The major novelty of our work is to prove a global Carleman estimate (with m observations) in the case of BV diffusion coefficients () for the operator . In a first step, we derive a global Carleman estimate (with m observations) in the case of piecewise- diffusion coefficient. The main result, in this case, is Lemma 2.1, where we prove the existence of a suitable weight function for m coupled parabolic equations in the case of piecewise- coefficients. By comparison with , the idea in the proof of Lemma 2.1 lies in the fact that we have used adapted choices (more general) (see formulas (2.7) and (2.8)) for checking the trace property (2.3) in the case of m coupled parabolic equations. These choices are used later for constructing a function β (see formulas (3.2) and (3.3)) in the case of BV diffusion coefficients. The property (2.3) is needed to relax the condition of the monotonicity of the diffusion coefficients. In a second step, we follow the method developed in . Formulas (3.2) and (3.3) yield an explicit expression of an approached weight function that converges to a weight function β (see Lemma 3.2). The function allows us to establish a Carleman-type estimate (with m observations) associated to the operator with () piecewise constants that converge to the BV diffusion coefficients in -norm. At the end, we pass to the limit for each term in the Carleman estimate that holds for the operator as goes to zero. We then obtain the Carleman estimate for the operator with a relaxation of the monotonicity of BV diffusion coefficients .
To our knowledge, the weight Carleman function and its proof in our work has not been proposed in the literature review.
The article is organized as follows. In Section 2, we derive a Carleman estimate with m observations in the case of piecewise- diffusion coefficients. In Section 3, we prove a Carleman estimate with m observations in the case of BV diffusion coefficients. Finally, Section 4 is devoted to giving important comments and applications of our results on controllability for some parabolic systems.
2 Global Carleman estimate with ‘m observations’ in the case of piecewise-diffusion coefficients
In this section, we generalize the Carleman estimate obtained in  to a parabolic system. We prove here a global Carleman estimate in the case of piecewise- diffusion coefficients for a system of m coupled parabolic equations with an interior observation region , where ω is a non-empty open subset of Ω. In order to establish this estimate, we use similar arguments to those in  and  for constructing a suitable weight function in a subdomain of ℝ, which allows us to relax the monotonicity on the diffusion coefficients. The results obtained in this section are then used in the next section (the case of BV diffusion coefficients).
Let and . Let with .
We note : , , , , and .
The diffusion coefficients () are assumed here to be piecewise- such that () and satisfy Assumption 1.1.
If , the transmission conditions (TC) are then automatically satisfied.
We shall now prove the main result of this section. It concerns the construction of a suitable weight function.
In the case of one equation (), the proof of the existence of such a function is established in  and . However, in our case (m coupled equations), the main difficulty is to find β such that the trace property (2.3) is satisfied for all .
If (respectively, ), is equivalent to (respectively, ).
We assume here that the coefficients () cannot be smooth simultaneously at the same point (i.e., for and fixed ).
- 1.. In this case, we have . Then, for , we obtain
- 2.. In this case, we have and
and . We distinguish the following cases:
- (a)If with () and , then we have and with and we obtain
If with () and , then we have , which corresponds to the case .
If with () and . This case is reduced to the case .
- (b)If with () and , then we obtain , with , and
. In this case, we have . Then, for , we obtain .
- 2.. In this case, we have , and
and . We distinguish the following cases:
- (a)If with () and , then we have and with , thus
If with () and , we obtain . This case is reduced to the case .
If with () and . This case is reduced to the case .
If with () and , we have , with , and
(We have used .)
The case corresponds to the choice made in .
Carleman estimate (2.10) remains the same if we consider the operator instead of .
Using the previous proposition, we have the following theorem.
for any solutionof (1.1).
the last term on the right-hand side of (2.13) can be ‘absorbed’ by the terms in . This concludes the proof. □
Carleman estimate (2.12) remains valid if we consider the boundary observation (respectively ) instead of the interior observation ω. The result is obtained through a modified form of Lemma 2.1, namely:
3 Global Carleman estimate with ‘m observations’ in the case of BV diffusion coefficients
In this section, we generalize the Carleman estimate given in  to a parabolic system using the results obtained in the previous section. We show that we can prove the global Carleman estimate in the case of bounded variations (BV) diffusion coefficients for a system of m coupled parabolic equations with an interior observation region , where ω is a non-empty open subset of Ω. We follow the method developed in  and many notations and arguments of the previous paper will be reproduced here.
We consider system (1.1) with diffusion coefficients assumed here to be of BV such that are of class on and satisfy Assumption 1.1.
Our goal is to construct a limit weight function β (the same for each equation) using the approach of BV diffusion coefficients by piecewise-constant coefficients. This process allows us to derive a Carleman estimate for the operator .
Let . Without any loss of generality, we suppose that with . We denote the total variations of on and by and .
We consider the points () in the interval such that .
for some fixed . Observe that and , .
In a similar manner, we consider the points () in the interval such that .
for some fixed . Observe that and , .
and we design to be of class on .
In this section, we want to pass to the limit in Carleman estimate (2.11). We first need to control the behavior of the derivative of as ε goes to zero. This is the object of the following lemma.
(see , Lemma 3.2])
Let. We assume that the diffusion coefficientsand Assumption 1.1is satisfied, then there exist, andsuch that, for all, and.
Using Helly’s theorem (see ), the function (respectively ) converges everywhere to the function (respectively ) as ε goes to 0. Since the function (respectively ) is bounded in (respectively in ) uniformly with respect to ε, we deduce, by applying the dominated convergence theorem and Lemma 3.1, that the function (respectively ) converges everywhere to the function (respectively ).
converges everywhere to in .
converges to in .
From the above arguments, we obtain the following lemma.
(see , Lemma 3.3])
Let. We assume thatinis of classinand satisfies Assumption 1.1. Letbe piecewise-constant onand smooth on ω such that (3.1) is satisfied. Then there exists a functionthat satisfies the properties listed in Lemma 2.1for the associated coefficients. Furthermore, andare of classonand satisfy the above properties (1, 2, 3).
The results obtained in Lemma 3.2 imply that the constants and can now be chosen uniformly with respect to ε.
Under the same assumptions as in Lemma 3.2 and the properties of and defined as above, we obtain the following proposition.
(see , Proposition 3.4])
Let fixed. Then the constanton the right-hand side of Carleman estimate (2.11) for the operatorand the constantsandcan be chosen uniformly with respect to ε for.
The proof of Proposition (3.1) is established through the following lemmata.
where β is the function defined by (3.7).
with in , , .
The previous estimate holds through the Young and Gronwall inequalities. □
We recall that converges everywhere to β implies that and converge everywhere to and φ. Then, using Lemma 3.5, the Cauchy-Schwarz inequality and dominated convergence, the left-hand side of (3.14) converges to zero as ε goes as zero. We obtain the same result for the remaining terms in Carleman estimate (2.12).
In conclusion, using density arguments, we obtain the following theorem.
for any solutionof (1.1).
Carleman estimate (3.15) remains valid if we consider the boundary observation (respectively ) instead of the interior observation ω (see Remark 2.4). However, in this case, the assumption which corresponds to the fact that the coefficients are of class in is not needed to obtain (3.15).
4 Comments and applications
in the case ().
Choices (2.7) and (2.8) are taken in an optimal way in order to control the behavior of the function (see Lemma (3.1)). For example, choices (4.1) and (4.2) are not appropriate in the case of BV diffusion coefficients.
Using the results (Carleman estimate) obtained in the previous section, we deduce an observability inequality which yields null controllability. The proofs of such results can be adapted from the techniques used in  (also see the references therein). Consequently, we only highlight the main points.
where is the characteristic function of the non-empty set ω. The diffusion coefficients () are assumed to be BV such that are of class in and satisfy Assumption 1.1. We also assume that , , , , , and the controls . We have also for all and .
where and .
with and C a positive constant.
We then obtain the following result.
where β is the function defined through Lemma 2.1.
Let us consider the following assumption.
Using the results obtained in the previous section and proceeding as in , we obtain the following shifted Carleman estimate.
(see , Theorem 2.2])
for any solution (, ) of (1.1) ().
with and .
and , where .
The Carleman estimate obtained in paper  can be easily generalized in the case of BV diffusion coefficients by using similar arguments to those in the preceding sections.
with C a positive constant, depending on Ω, ω, , , T and .
The author would like to thank the editor and the anonymous referees for their valuable comments and helpful suggestions, which led to the improvement of the original manuscript.
The author also wishes to thank A Benabdallah, M González-Burgos, J Le Rousseau and N Boussetila for numerous discussions on the proofs in the paper.
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