B Mostafa

A Benslimane

E Boureau

M Molnar

M Saleh

Distributed Monitoring in 6LoWPAN based Internet of Things

Keywords: IoT, RPL, link monitoring, Vertex Cover

The emergence of the Internet of Things (IoT) is introducing more and more services and applications such as smart cities. For some services, the availability of elements and the connectivity between them are necessary. The robust functioning of a fragile and often dynamic system in IoT needs strong monitoring which is investigated in this paper. IPv6 Routing Protocol for Low Power and Lossy Networks (RPL) is the standardized routing protocol over IP-connected IoT networks. In tandem with RPL, we propose a new solution which aims to achieve distributed monitoring with minimal computational complexity. The main objective is to increase robustness in IoT via monitoring the links in the Destination Oriented Directed Acyclic Graphs (DODAG) constructed by RPL. Although RPL has a reactive repairing mechanism to tackle robustness in connectivity in case of node failures, we found that for real-time critical applications it is important to use proactive approaches for preventing faults and making recovery for connectivity faster. The problem can be modeled as a Vertex Cover Problem (VCP) on the DODAG. Such problem is a well known NP-hard optimization problem. The monitoring should be simple and energy aware. We demonstrate that the monitor placement in our case is only Fixed-Parameter Tractable, and also polynomial-time solvable, which is the best case.

I. INTRODUCTION

The 6LowPAN based Internet of Things (IoT) networks tend to experience unexpected communication problems during deployment, because resource-constrained embedded devices are unreliable by nature for a variety of reasons, such as uncertain radio connectivity and battery drain [START_REF] Kushalnagar | IPv6 over Low-Power Wireless Personal Area Networks (6LoWPANs): Overview, Assumptions, Problem Statement, and Goals[END_REF]. To that end, monitoring techniques for detecting, localizing and remedying network failures in IoT will definitely develop in significance. The primary objective of network monitoring in general is mapping the symptoms of the detected problems to possible root causes to take corrective measures [START_REF] Jurdak | Intelligence-Based Systems Engineering[END_REF].

Low power and Lossy Networks (LLN) experience several communication challenges such as energy and computational limitations of devices and extensive protocol overheads against memory. To handle such challenges, the Internet Engineering Task Force (IETF) has standardized the Routing Protocol for Low Power and Lossy Networks (RPL) as the routing protocol for IP-connected IoT [START_REF] Kintali | Computing Bounded Path Decompositions in Logspace[END_REF] [START_REF] Winter | RPL: IPv6 Routing Protocol for Low-Power and Lossy Networks[END_REF]. RPL is a self-healing routing and topology control protocol as it is able to respond to some node or link failures. It uses reactive local and global repair approaches for network recovery. However, the recovery time, which is the time required to establish a new route could be critical. If for any reason, the links connecting the node with its neighbors are all broken, as in the case of a sudden failure of multiple links, the node will not be able to regain connectivity quickly.

RPL favors the use of reactive repair approaches as they are more energy-efficient; to minimize the cost of monitoring links that are not being used [START_REF] Winter | RPL: IPv6 Routing Protocol for Low-Power and Lossy Networks[END_REF]. Furthermore, the neighbor unreachability detection (NUD) is not mandatory in RPL and active mechanisms for probing neighbors regularly do not exist in ContikiRPL implementation [START_REF] Bodlaender | Combinatorial Optimization on Graphs of Bounded Treewidth[END_REF] [START_REF] Gaddour | Survey RPL in a Nutshell: A Survey[END_REF].

Although a significant number of IoT applications are not time-sensitive, there is a whole class of real-time, missioncritical applications; where data must be processed and shared instantly and within strict reliability constraints. For instance, critical control and fault detection applications; where corrective actions must be taken with little delay. For critical, real-time IoT applications, instead of the embedded RPL reactive repair approaches, proactive monitoring mechanisms should be used for early detection and prevention of network failures. Node and link failures could then be detected beforehand, as a kind of preventive maintenance. This could greatly improve the robustness in connectivity, reliability and eventually Quality of Service (QoS) in the network, which will significantly increase the uptake of the technology by stakeholders. The added cost for link monitoring could be tolerated for critical-time IoT applications as it will help accelerate recovery, and decrease node unreachability times.

Consequently, we were motivated to investigate the problem of proactive link monitoring for IoT. The main objective is to increase robustness in IoT via monitoring the links in the Destination Oriented Directed Acyclic Graphs (DODAG) constructed by RPL. Distributed monitoring can be achieved by placing multiple monitoring nodes on the network. Those nodes are responsible for anticipating node/link failures by continuously tracking the status of the links in the DODAG and taking the corrective actions before failure happens. In this paper, we focus on calculating the minimum number of monitoring nodes required to track the entire set of links. We modeled the problem as a Vertex Cover Problem (VCP) on the DODAG. VCP is NP-hard, which implies that unless P = NP, efficient algorithms for solving it don't exist [START_REF] Dinur | On the hardness of approximating minimum vertex cover[END_REF]. However, due to the battery and computational constraints of embedded devices, the monitoring should be simple and energy-aware. We propose an algorithm to convert a DODAG into a nice-tree decomposition which makes solving the generally NP-hard VCP on this special graph achieved in polynomial-time.

The rest of the paper is organized as follows: section 2 is a brief overview of the RPL topology construction, repair mechanisms and a background of the concepts of vertex cover and tree decomposition problems. Section 3 presents the monitoring requirements, the proposed algorithm and proof of termination, followed by an analysis of the algorithm in Section 4. The conclusions and future research are presented in Section 5. Fig. 1. A sample RPL DODAG that has N nodes. The root is shaded, the numbers adjacent to the circles are the ranks, and the routes through siblings are dahsed.

(0) (9) (10) (7) (3) (5) (5)

II. BACKGROUND

A. RPL DODAG RPL is a versatile protocol that supports different modes of operation: many-to-one communication from the constrained nodes towards the root (typically the 6LoWPAN border router 6BR), one-to-many communication from the root to the constrained nodes (constrained in terms of battery and memory, etc.) and, one-to-one communication between the constrained nodes. The basic component of RPL is a Destination Oriented Directed Acyclic Graph (DODAG), a sample is shown in Fig. 1. It is a hierarchical organization, where each node has a node ID, one or more parents (except for the DODAG root), and a list of neighbors [START_REF] Raza | SVELTE: Real-time intrusion detection in the Internet of Things[END_REF].

Nodes also have a rank that determines their individual position in the hierarchy with respect to the DODAG root and relative to other nodes, which are the numbers adjacent to the nodes in Fig. 1. In RPL, the rank as well as the parent selection is calculated via an Objective Function (OF) that might use factors such as link quality, node energy and link reliability in its calculation [START_REF] Bodlaender | Combinatorial Optimization on Graphs of Bounded Treewidth[END_REF].

The DODAG construction depends on the Neighbor Discovery Protocol (NDP) and Upward route discovery protocol in 6LoWPAN. They allow a node to join a DODAG by discovering neighbors that are members of the DODAG and identifying a set of parents [START_REF] Winter | RPL: IPv6 Routing Protocol for Low-Power and Lossy Networks[END_REF]. The protocols build three logical sets of link-local nodes. Starting by the candidate neighbor set, which is a subset of the nodes that can be reached through link-local multicast. Then, the parent set, which is a restricted subset of the candidate neighbor set. Finally, the preferred parent set, a member (or members) of the parent set which is the preferred next hop in Upward routes. The exact policies for selecting neighbors is implementation dependent and driven by the OF.

B. DODAG Repair Mechanisms

Repair mechansims are of foremost significance for a routing protocol to update routes dynamically and adapt the network topology to potential failures. For that purpose, RPL supports two integral repair mechanisms, in particular local repair and global repair [START_REF] Gaddour | Survey RPL in a Nutshell: A Survey[END_REF].

When a node detects a network failure (e.g. a link between two nodes fails), it triggers local repair (see Fig. 2). It consists in searching for a backup path urgently without attempting to repair the entire DODAG. However, this alternate recovery path may not be the best path with respect to the defined OF.

The second mechanism is the global repair; it prompts the fundamental reconstruction of the entire network topology, but on the expense of additional control traffic in the network [START_REF] Kushalnagar | IPv6 over Low-Power Wireless Personal Area Networks (6LoWPANs): Overview, Assumptions, Problem Statement, and Goals[END_REF]. Nodes in the new DODAG version can choose a new position whose rank is not dependent on the previous one. The global repair is considered as an entire re-optimization of the routes. However, there are times when the nodes are unreachable during the DODAG rebuilding process. Therefore, relying solely on global repairs is neither efficient, in terms of the number of control messages, nor reliable especially for critical-time IoT applications where there is no tolerance for nodes unreachability.

C. Vertex Cover & Tree Decompositions

The Vertex Cover Problem (VCP) is a well known NPhard graph optimization problem. VCP is defined over an undirected graph G = (V, E) and searches for a set of vertices

S ⊆ V such that for each edge e ∈ E at least one of its endpoints belongs to S and |S| is as small as possible.

There are several general approaches for attacking NPhard problems, among them approximation algorithms, fixedparameter algorithms, and heuristics. VCP is one of the best studied problems concerning fixed-parameter tractability [START_REF] Moser | Exact Algorithms for Generalizations of Vertex Cover[END_REF]. Several techniques in parameterized complexity were successfully applied to VCP, as for instance data reduction, depth-bounded search trees, and dynamic programming [START_REF] Alber | Algorithm Theory -SWAT 2000: 7th Scandinavian Workshop on Algorithm Theory[END_REF].

Another approach for solving NP-hard graph optimization problems is the concept of tree decompositions for graphs, which was introduced by Robertson and Seymour [START_REF] Robertson | Graph minors. II. Algorithmic aspects of tree-width[END_REF] and plays an important role in algorithmic graph theory.

Tree decompositions were motivated by the observation that many NP-complete problems are easy to solve on trees, connected graphs without cycles [START_REF] Bodlaender | Combinatorial Optimization on Graphs of Bounded Treewidth[END_REF]. Trees are restricted structures when compared with general graphs; [START_REF] Robertson | Graph minors. II. Algorithmic aspects of tree-width[END_REF] addressed how hard problems can be solved for graphs that are "like" trees. Tree decompositions are the formal way to describe the "tree-likeness" of a given graph [START_REF] Moser | Exact Algorithms for Generalizations of Vertex Cover[END_REF]. See Fig. 3 for an example of tree decompositions. Fig. 3. Example of a graph G and its corresponding tree decomposition T [START_REF] Moser | Exact Algorithms for Generalizations of Vertex Cover[END_REF].

In the following definitions we give a brief description of tree decomposition, treewidth, and nice-tree decomposition. Fig. 3 illustrates an example of an optimal tree decomposition T. T is optimal in the sense that there is no tree decomposition for the given graph G such that every bag contains fewer than three vertices. Observe that the properties of tree decompositions as stated in Definition 1 hold [START_REF] Bodlaender | Combinatorial Optimization on Graphs of Bounded Treewidth[END_REF].

The concept of treewidth has been proven to be very useful in algorithmic graph theory. Several problems that are NPhard on general graphs are polynomial-time (some even linear-time) solvable on graphs with bounded treewidth. These problems include graph coloring and vertex cover [START_REF] Kintali | Computing Bounded Path Decompositions in Logspace[END_REF].

The usual approach of tree decomposition based algorithms is dynamic programming. The fact that the running time of dynamic programming algorithms on tree decompositions is commonly of the complexity O(c k * P(n)),

where k is the width of the tree decomposition, P(n) is a polynomial function of n, and n is the size of the instance, is useful for our problem. As these algorithms can be executed in a reasonable amount of time on graphs where k is relatively small (less than 20, for example) [START_REF] Carroll | Domain Specific Language for Dynamic Programming on Nice-tree Decompositions[END_REF]. If we restrict graph problems to graphs with small tree decompositions, we can solve many NP-hard problems by using dynamic programming. Such problems are called Fixed-Parameter Tractable (FPT) [START_REF] Kintali | Computing Bounded Path Decompositions in Logspace[END_REF] because they have algorithms that run in polynomial-time when the parameter, namely treewidth of the tree decomposition is fixed. Furthermore, algorithmic solutions are often easier when working with a nice-tree decomposition (cf. Definition 3) instead of a tree decomposition [START_REF] Carroll | Domain Specific Language for Dynamic Programming on Nice-tree Decompositions[END_REF].

Definition 1: Tree Decomposition [8]

Let G = (V, E) be a graph. A tree decomposition of G is a pair ({X i : i ∈ I}, T), where each X i ⊆ V is called a bag, and T is a tree with the elements of I as nodes. The following three properties must hold:

1. ⋃ ∈ = V, 2. for every edge e ∈ E there exists a bag X i with i ∈ I and e ⊆ X i , and 3. for all i, j, k ∈ I, if j lies on the path from i to k in T then X i ∩X k ⊆ X j . The third property is equivalent to the requirement that, for each v ∈ V, the nodes of all bags containing v induce a subtree of T.

Definition 3: Nice-Tree Decomposition [8]

A tree decomposition ({X i : i ∈ I}, T) is nice if:

• T is rooted at a designated node r ∈ I, called root node, • every vertex of T has at most two children (i.e.

the tree is a binary tree),

• if i is a leaf of T, | ( )| = 1 (I(i) is called a start bag), • if i has two children j and k, I(i) = I( j) = I(k) ( in this case I(i)

Definition 2: Treewidth [8]

The width of a tree decomposition is the maximum of | ( )| -1 over all v ∈ V T . Based on this information, converting a DODAG into a nice-tree decomposition with bounded treewidth will yield the advantage of solving many NP-hard graph optimization problems in polynomial-time. This approach can be used in our monitoring problem, which we addressed in this paper.

III. CONVERTING A DODAG INTO NICE-TREE DECOMPOSITION WITH UNITY TREEWIDTH

A. IoT Link Monitoring Requirements

For critical-time IoT applications in wireless networks, it is crucial to keep track of all the links in the network, especially the inactive ones which are not currently used for routing.

In order to meet the reliability requirements for critical applications, we were motivated to study the problem of network monitoring for IoT through monitoring the links in DODAGs. We propose a distributed monitoring model where there are several nodes carefully placed on the DODAG, which are responsible detecting and localizing network failures.

One of the main challenges for network monitoring is determining where to embed the monitoring nodes. Moreover, the monitoring computational cost, battery and memory requirements should be minimal in order to satisfy the low cost and energy constraints of IoT networks. Therefore, it is important to minimize the number of monitoring nodes placed on the graph, meanwhile balancing the monitoring load to maintain the status of all links and also maximize network lifetime. In other words, the objective is minimizing the number of monitoring nodes that "cover" the entire graph.

Taking the stated requirements into consideration, finding out the minimum number of monitoring nodes placed on the graph to keep track of all the links in the network can be modeled as the classic VCP. However, as mentioned in Section 2.3, VCP is NP-hard for general graphs and FPT when solved by dynamic programming on tree decompositions with bounded treewidth.

In view of this information, converting a DODAG into a nice-tree decomposition with bounded treewidth will yield the advantage of solving the generally NP-hard graph monitor placement problem in this special graph in polynomial-time. To the best of our knowledge, there is no work that tended to this problem in the literature.

B. Proposed Algorithm

A DODAG consists of a set of vertices v j : v j ∈ V, where j is a unique identifier for each vertex in D, and a set of edges E. Each vertex in the DODAG is connected via an edge e j ∈ E to one of three types of nodes: parents P(v j ) (preferred and alternative), children C(v j ) or siblings S(v j ).

The objective of the proposed algorithm (Algorithm 1) is to convert any DODAG D, into a nice-tree decomposition with unity treewidth, while holding the properties of nice-tree decompositions in Definition 3. Bounding the treewidth to the value of one will have the effect of reducing the complexity of solving the VCP on the DODAG to be polynomial-time, which is the main contribution of this paper. Step Step 2.6.1 Make leaf Ɩ a 'forget bag' via branching out one child X k , where X k = {v j , v q } and v q ∈ Step 2.6.2 Make X k an 'introduce bag' via branching out one child{v q } Step 2.6.3 L = L\ v q Step 2.6.

4 l= l + 1 END While Step 2.7 NL = NL\ v j END While END TABLE I. ALGORITHM 1 VARIABLES

Variable Description

D

The Destination Oriented Directed Acyclic Graph composed of a set of vertices v j ∈ V and a set of edges e j ∈ E

C(v j )

The entire set of children of vertex v j S(v j ).

The entire set of siblings of vertex v j c r A child of vertex v j s r A sibling of vertex v j X i A bag containing subset of V

I

The entire set of bags in the tree decomposition T A tree with the elements of I as nodes

NL

The set of Non-Leaf nodes in the DODAG no_of_required_ leaves

The number of required leaves for v j in the tree decomposition.

The algorithm is divided mainly into an initialization step (Step 1) and a main loop (Step 2). TABLE I summarizes the variables in Algorithm 1. Step 1.1 initializes a binary tree T and sets its root with a bag that only contains the DODAG root, which is considered as the destination of any node in D. In Step 1.2 we take all the Non-Leaf nodes in the DODAG, store them in a vector (NL), and then sort them ascendingly according to their ranks in D. The main part of the algorithm is implemented in Step 2, which iterates over the vector NL and constructs a binary tree for each of those vertices; by calling the recursive subroutine ConstructBinaryTree (see Algorithm 2).

Taking the first vertex in NL, steps 2.3 and 2.4 determine the required number of leaves (no_of_required_leaves) associated with it, which is the number of its children plus the number of siblings connected to it that are not already included in the tree decomposition.

The recursive subroutine ConstructBinaryTree is called at step 2.5, which takes as input a bag B and the required number of leaves and returns a binary tree t with the required number of leaves and all its bags are set equal to the input bag B. The returned tree t is then augmented with T at the right location; the location of the bag at current iteration.

Step 2.6, is an inner loop for all the leaves (the vertex's children or siblings that are not already included in T). In step 2.6.1, we make the leaf a 'forget bag' via branching out one child bag that contains two elements: the leaf and one of the vertex's children (or siblings), in list L. Then in step 2.6.2, we make this child bag an 'introduce bag' via branching out a child bag that contains only one element; i.e. the vertex's child or sibling. Finally, in step 2.6.3, we remove this vertex's child or sibling from the L list. After the algorithm exits this inner loop, the vertex is then removed from the vector NL at step 2.7; and the outer loop continues until there are no more vertices left in NL. Proof: In order to prove that Algorithm 1 terminates we need to show that the two nested loops, at steps 2 and 2.6 eventually terminate. At Step 2 the While loop starts with a finite subset of vertices, namely NL. The cardinality of NL is altered only at step 2.7, where it decreases by one. Since NL is finite and is strictly decreasing in cardinality with each iteration of the loop, the loop condition (NL≠ ∅) is eventually falsified and therefore the loop terminates. At step 2.6, the second loop iterates until l is greater than a finite integer quantity, the no_of_required_leaves. Note that l initially has the value of one, and the value of the no_of_required_leaves is the cardinality of the finite subset of vertices L. The quantity (no_of_required_leaves -l) is an integer that strictly decreases with each iteration of the loop. Therefore, eventually (no_of_required_leaves -l) is a negative quantity, thus (no_of_required_leaves > l ), and the loop terminates.

Lemma 2:

The construction of a binary tree (Algorithm 2) terminates.

Proof: Since Algorithm 2 is a recursive function, to prove that it terminates we need to show that its arguments get strictly smaller whenever there is a recursive call. The argument Z which is the required number of leaves is a finite integer that starts with a value greater than one, then strictly decreases until it reaches the value two, where the condition (Z > 2) is falsified and the recursive function terminates.

IV. ANALYSIS

A. Complexity Analysis

In any nice-tree decomposition produced by Algorithm 1, there are two types of bags in the nice-tree namely:

• single-element bags, which contain one vertex of the DODAG, and • two-elements bags, which contain two vertices that are connected via an edge in the DODAG.

The number of unique single-element bags is equal to the number of vertices in the DODAG. In addition, any twoelements bag will always be a unique bag; as the algorithm is designed to only map once an edge into a corresponding twoelements bag. Hence, we can conclude that the number of two-element bags is equal to the number of edges in the DODAG. Therefore, the number of unique bags in the nicetree decomposition is simply the summation of the number of vertices and the number of edges in the DODAG.

Single-element bags are either Non-Leaf single-element bags or leaf single-element bags. By experimental analysis, we identified the number of occurrences of any Non-Leaf singleelement bag as {2*(no_of_required_leaves) -1}. Moreover, the number of occurrences of any leaf single-element bag is determined by the number of alternative parents and siblings connected to its corresponding vertex, in the DODAG. Hence, we can conclude that the number of occurrences for both leaf and Non-Leaf single-element bags is of the order of O(v) at maximum, where v is the number of vertices. This leads to the conclusion that the total number of single-element bags is at maximum v 2 .

The complexity of the proposed algorithm is measured by the number of bags generated for each DODAG to be a nicetree decomposition. Therefore, we can conclude that the proposed algorithm has a complexity of O(m + v 2 ) , where m is the number of edges in the DODAG, which is polynomial in time.

Moreover, as mentioned before in Section 3, the running time of dynamic programming algorithms on tree decompositions is typically of the form O(c k * P(n)), where k is the treewidth and n is the size of the instance. Using the proposed algorithm for the construction of the nice-tree decomposition from DODAGs, we will have the treewidth to be only one. Consequently, our particular VCP turns out to be just polynomial in time.

B. Discussion

The initial objective was to monitor the unreliable links in a 6LowPAN network for IoT. Distributed monitoring is achieved via placing several monitoring nodes on the network, which track the status of the links in their neighborhood. We modeled the problem as the classic VCP, which finds out the minimum number of nodes that cover all links. For the purpose of reducing the complexity of solving such NP-hard problem, we proposed converting the DODAG into a nice-tree decomposition with unity treewidth. In order to verify that modeling the problem in such a way actually reaches the initial objective stated above, we need to prove that the DODAG maps the entire network, i.e. it contains a vertex and an edge for every node and link in the 6LoWPAN network.

As mentioned in Section 2 the DODAG construction in RPL is based on the NDP in 6LoWPAN which provides the candidate neighbor set for each node, but the exact policies for selecting neighbors and parents is implementation dependent and driven by the OF [START_REF] Winter | RPL: IPv6 Routing Protocol for Low-Power and Lossy Networks[END_REF].

Let the candidate neighbor set be N(v j ), and in Algorithm 1 we defined the set of parents, children and siblings of each vertex as P(v j ), C(v j ) and S(v j ), respectively. If we design the OF to construct the DODAG such that the DODAG maps exactly the entire network, i.e.:

P(vj) ∪ C(vj) ∪ S(vj) = N(vj)

(1) therefore, solving the VCP on the general graph G that maps the network is the same as solving VCP on the DODAG D, i.e.: VCP(D) = VCP(G) (2) By definition of tree decompositions (cf. Definition 1), each vertex and edge in D is present in T, therefore VCP(T) = VCP(D) (3) From ( 1), ( 2) and (3), we conclude that solving the VCP on the nice-tree achieves the objective of monitoring all the links of the network.

V. CONCLUSIONS & FUTURE WORK

In this paper, we investigated a strategy for increasing the robustness in connectivity of IoT for critical-time applications via a proactive monitoring of links in the DODAG constructed by RPL. We considered the Vertex Cover modeling of our problem, where the minimum number of monitoring nodes is required to cover the entire DODAG.

The VCP is a known NP-hard problem in general graphs but there exist Fixed Parameter Tractable dynamic programming algorithms for solving the VCP on tree decompositions. We developed a polynomial-time algorithm that converts the DODAG into a nice-tree decomposition with unity treewidth. This new strategy yields a significant reduction in the complexity of solving VCP on DODAGs to be only polynomial-time solvable. Meanwhile, it achieves the objective of monitoring all the links in a 6LowPAN network.

By definition of vertex cover, we now know the minimum number of required monitors to cover the entire network. However, we don't know the exact placement of the monitors. Therefore, it remains a challenge for future research to decide which nodes should be configured as monitors in order to minimize and balance the monitoring load, decrease energy consumption of the nodes and consequently maximize the lifetime of the network.

Fig. 2 .

 2 Fig.2. A DODAG after the local repairs. Borders of inactive nodes are dotted, repaired routes are dashed, the root is shaded, and the numbers adjacent to the circles are the ranks[START_REF] Korte | Dependable Networks and Services: 6th[END_REF].

  is called a join bag), and • if i has one child j, then either: o | ( )| = | ( )| -1 and I(i) ⊂ I( j), ( in this case I(i) is called a forget bag), or o | ( )| = | ( )| +1 and I(j) ⊂ I(i) (in this case I(i) is called an introduce bag).

Algorithm 2 :

 2 ConstructBinaryTree Input: Bag B and Z = required number of leaves Output: A binary tree with the required number of leaves and all its bags are equal to B Step 1: Create two branches; where each branch has a bag = B Step 2: If (Z > 2) then t = ConstructBinaryTree (B, Z -1) Augment the bag at the right branch with t End If END Algorithm 1: Convert DODAG into Nice-Tree with Treewidth 1 Input: DODAG D = ({v j : v j ∈ V }, E), Output: Nice Tree Decomposition ({X i : i ∈ I}, T) with unity Treewidth Let c r ∈ C(v j ) where C(v j ) is the set of children of vertex v j in DODAG D Let s r ∈ S(v j ) where S(v j ) is the set of siblings connected to vertex v j in DODAG D Let the set of Non-Leaf nodes in D be {NL: NL = ⋃ ∈ where ( ) + ( ) > 0}

Fig. 4 (

 4 Fig. 4 (a). DODAG D. Fig. 4 (b). Nice-tree decomposition of DODAG D with unity treewidth.

Fig. 4 ( 1 :

 41 a) shows a DODAG D and Fig. 4(b) shows the output of Algorithm 1, which is the nice-tree T of D with unity treewidth. C. Proof of Termination Lemma The conversion of a DODAG (Algorithm 1) into a Nice-Tree terminates.

  1 Initialization Step 1.1 Initialize an empty tree T then set its root to a bag that only includes the DODAG root Step 1.2 Sort ascendingly NL according to the ranks of v j Step 2 While NL≠ ∅ do Step 2.1 Select v j from NL {its top vertex} Step 2.2 Search T for the first bag X i = {v j } using breadth first search Step 2.3 Set L = C(v j ) ∪ S(v j ) where s r ∈ NL Step 2.4 Let no_of_ required_leaves = | | Step 2.5 If (no_of_ required_leaves > 1) then t = ConstructBinaryTree ( X i , no_of_ required_leaves ) At X i augment T with t End If Step 2.6 While l ≤ the no_of_ required_leaves


Assia Benabdallah

Michel Cristofol
email: michel.cristofol@cmi.univ-mrs.fr

Patricia Gaitan
email: patricia.gaitan@cmi.univ-mrs.fr

Masahiro Yamamoto

Inverse problem for a parabolic system with two components by measurements of one component

We consider a 2×2 system of parabolic equations with first and zeroth coupling and establish a Carleman estimate by extra data of only one component without data of initial values. Then we apply the Carleman estimate to inverse problems of determining some or all of the coefficients by observations in an arbitrary subdomain over a time interval of only one component and data of two components at a fixed positive time θ over the whole spatial domain. The main results are Lipschitz stability estimates for the inverse problems. For the Lipschitz stability, we have to assume some non-degeneracy condition at θ for the two components and for it, we can approximately control the two components of the 2 × 2 system by inputs to only one component. Such approximate controllability is proved also by our new Carleman estimate.

Finally we establish a Carleman estimate for a 3×3 system for parabolic equations with coupling of zeroth-order terms by one component to show the corresponding approximate controllability with a control to one component.

Introduction and notations

This article is devoted to the question of the identification of coefficients for a reaction diffusion convection system of two equations in a bounded domain, with the main particularity that we observe only one component of the system. Let Ω ⊂ R n be a bounded connected open set with C 2 -boundary ∂Ω, and we set x = (x 1 , ...,

x n ) ∈ R n , ∂ j = ∂ ∂x j , 1 ≤ j ≤ n, ∂ t = ∂ ∂t , ∇ = (∂ 1 , ..., ∂ n ), ∆ = n j=1 ∂ 2 j .
For any fixed T > 0, we set Ω T = Ω × (0, T ), Σ T = ∂Ω × (0, T ) and we consider the following 2 × 2 reaction-diffusion-convection system :

                           ∂ t U = ∆U + aU + bV + A • ∇U + B • ∇V + f in Ω T , ∂ t V = ∆V + cU + dV + C • ∇U + D • ∇V + g in Ω T , U = h 1 , V = h 2 on Σ T , U (•, 0) = U 0 , V (•, 0) = V 0 in Ω, (1.1)
where a, b, c, d are scalar functions and A, B, C, D vectorial fields both defined on Ω. The boundary condition h i as well as f, g shall be kept fixed. If we change the reaction coefficients b, c into b, c, we let ( U , V ) be the solution of (1.1) associated to b, c and ( U 0 , V 0 ) for the initial condition. Let ω ⊂ Ω be a non-empty subdomain and T > 0. We assume that we can measure both U | ω×(0,T ) and (U, V )| Ω×{θ} .

at a time θ ∈ (0, T ).

We set ω T = ω × (0, T ). For m ∈ N, 1 ≤ p ≤ ∞, by W m,p (Ω) and L p (0, T ; X) we denote the classical Sobolev space with the norm • W m,p (Ω) , and the space of X-valued p-Bochner integrable functions respectively (e.g., [START_REF] Adams | Sobolev Spaces[END_REF]). As usual we write W 0,p (Ω) = L p (Ω) and H m (Ω) = W m,2 (Ω) for m ∈ N. We define a Banach space

W m, m 2 2 (Ω T ) = {u : Ω × (0, T ) → R; ∂ α x ∂ α n+1 t u ∈ L 2 (Ω T ), for |α| + 2α n+1 ≤ m}, with the norm u W m, m 2 2 (Ω T ) = |α|+2α n+1 ≤m ∂ α x ∂ α n+1 t u L 2 (Ω T ) .
Here α = (α 1 , . . . , α n ) is a multi-index,

|α| = α 1 + • • • + α n , ∂ α x = ∂ α 1 1 • • • ∂ αn n
, and the differentiation is to be understood in the weak sense. Let M be an arbitrary positive constant. We denote by ν the outward unit normal to Ω and by B X (0, r) the closed ball of a metric space X centered on 0 of radius r.

We pose the following assumptions.

(f ) | U (•, θ)|, | V (•, θ)| > δ 0 on Ω T with some constant δ 0 > 0, (g) U C(Ω T ) , V C(Ω T ) ≤ M , (h) U C 3 (ω T ) , V C 3 (ω T ) ≤ M .
If the functions and the coefficients appearing in (1.1) satisfy sufficient smoothness and compatibility conditions, then Assumption 1.1 (g) and (h) are satisfied. By Ladyzenskaja, Solonnikov and Ural'ceva [START_REF] Ladyzenskaja | Linear and Quasilinear Equations of Parabolic Type[END_REF] for example, we can describe such conditions, but we are interested mainly in the inverse problem and we will not exploit these conditions.

Our first main result is the stability in determining the reaction coefficients b, c : Theorem 1.2 Let θ ∈ (0, T ) be fixed. We suppose that Assumption 1.1 is satisfied and that

(U, V )(•, θ) = ( U , V )(•, θ) in Ω.
Then there exists a constant κ > 0 such that

b -b L 2 (Ω) + c -c L 2 (Ω) ≤ κ ∂ t (U -U ) W 2,1 2 (ω T ) + U -U W 2,1
The key ingredient to these stability results is a global Carleman estimate for system (1.1).

Since the pioneer work of Bukhgeim-Klibanov [START_REF] Bukhgeim | Uniqueness in the large of a class of multidimensional inverse problems[END_REF], Carleman estimates have been successfully used for the following problems:

(i) the uniqueness and the stability in determining coefficients: Especially for parabolic equations, see Benabdallah, Dermenjian and Le Rousseau [START_REF] Benabdallah | Carleman estimates for the onedimensional heat equation with a discontinuous coefficient and applications to controllability[END_REF], Benabdallah, Gaitan and Le Rousseau [START_REF] Benabdallah | Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation[END_REF],

Imanuvilov and Yamamoto [START_REF] Yu | Lipschitz stability in inverse parabolic problems by the Carleman estimate[END_REF], [START_REF] Yu | Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations[END_REF], Imanuvilov, Puel and Yamamoto [START_REF] Yu | Carleman estimates for parabolic equations with nonhomogeneous boundary conditions[END_REF], Isakov [START_REF] Isakov | Inverse Problems for Partial Differential Equations[END_REF], Klibanov [START_REF] Klibanov | Inverse problems and Carleman estimates Inverse Problems[END_REF], [START_REF] Klibanov | Global uniqueness of a multidimensionnal inverse problem for a non linear parabolic equation by a Carleman estimate[END_REF] Klibanov and Timonov [START_REF] Klibanov | Carleman Estimates for Coefficient Inverse Problems and Numerical Applications[END_REF], Yuan and Yamamoto [START_REF] Yuan | Lipshitz stability in the determination of the principal parts of a parabolic equation by boundary measurements[END_REF] and the references therein. For hyperbolic problems, among many works, we restrict ourselves to a few works such as Imanuvilov and Yamamoto [START_REF] Yu | Global Lipschitz stability in an inverse hyperbolic problem by interior observations[END_REF], Isakov [START_REF] Isakov | A nonhyperbolic Cauchy problem for 2 b 2 c and its applications to elasticity theory[END_REF], [START_REF] Isakov | Inverse Problems for Partial Differential Equations[END_REF], Klibanov [START_REF] Klibanov | Inverse problems and Carleman estimates Inverse Problems[END_REF], Klibanov and Timonov [START_REF] Klibanov | Carleman Estimates for Coefficient Inverse Problems and Numerical Applications[END_REF] and see the references also in Isakov [START_REF] Isakov | Inverse Problems for Partial Differential Equations[END_REF] and Klibanov and Timonov [START_REF] Klibanov | Carleman Estimates for Coefficient Inverse Problems and Numerical Applications[END_REF].

(ii) observability inequalities and related estimates: see Fursikov and Imanuvilov [START_REF] Fursikov | Controllability of Evolution Equations[END_REF], Imanuvilov [START_REF] Yu | Controllability of parabolic equations[END_REF], Isakov [START_REF] Isakov | A nonhyperbolic Cauchy problem for 2 b 2 c and its applications to elasticity theory[END_REF], [START_REF] Isakov | Inverse Problems for Partial Differential Equations[END_REF], Kazemi and Klibanov [START_REF] Kazemi | Stability estimates for ill-posed Cauchy problem involving hyperbolic equations and inequalities[END_REF], Klibanov and Malinsky [START_REF] Klibanov | Newton-Kantorovich method for 3-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time dependent data[END_REF]. Furthermore the exact controllability of linear systems is equivalent to the observability of the corresponding adjoint system and we can refer to [START_REF] Fursikov | Controllability of Evolution Equations[END_REF], [START_REF] Yu | Controllability of parabolic equations[END_REF]. Imanuvilov and Yamamoto [START_REF] Yu | Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations[END_REF] discuss the global exact zero controllability for a semilinear parabolic equation. Also see Ammar-Khodja, Benabdallah and Dupaix [START_REF] Ammar-Khodja | Null controllability of some reaction-diffusion systems with one control force[END_REF], and Ammar-Khodja, Benabdallah, Dupaix and Kostine [START_REF] Ammar-Khodja | Controllability to the trajectories of phase-field models by one control force[END_REF], [START_REF] Ammar-Khodja | Null controllability of some systems of parabolic type by one control force[END_REF], González-Burgos and Pérez-Garc ía [START_REF] González-Burgos | Controllability results for some nonlinear coupled parabolic systems by one control force[END_REF] for semilinear parabolic systems.

Apart from the last previous works quoted, the existing Carleman estimates require observations of all the components when we will discuss inverse problems for a system such as (1.1). It is very desirable to establish the stability for inverse problems for a 2×2 parabolic system by means of only one component, because for a reaction-diffusion system it may be frequently difficult to observe the both components. There are not many papers devoted to such inverse problems for 2 × 2 parabolic systems, and we can refer, for instance, to Cristofol, Gaitan and Ramoul [START_REF] Cristofol | Inverse problems for a 2X2 reaction-diffusion system using a Carleman estimate with one observation[END_REF].

The article is organized as follows. In Section 2 we derive a new Carleman estimate for system (1.1).

In Section 3 we prove the stability result. In Section 4 we will remove Assumption 1.1 (f) on positivity of Ũ , Ṽ at a time θ > 0. Section 5 is devoted to some comments and open problems. The appendices provide technical proofs of lemmata stated in Sections 2 and 4. We want to point that the Carleman estimate proved in Section 2 implies a new approximate controllability result for a 2 × 2 reaction-diffusion-convection system with one localized control. As it will be seen in Section 5, this result can be extended to a 3 × 3 reaction-diffusion system.

2 Carleman estimate 2.1 A Carleman estimate for a 2 × 2 system by extra data of one compoment

Let (a ij ) 1≤i,j≤2 ∈ L ∞ (Ω T ) and (A ij ) 1≤i,j≤2 ∈ L ∞ (Ω T ) n . Let u 0 , v 0 ∈ L 2 (Ω) and f, g ∈ L 2 (Ω T ).
Consider the following reaction-diffusion system with convection terms :

                           ∂ t u = ∆u + a 11 u + a 12 v + A 11 • ∇u + A 12 • ∇v + f in Ω T , ∂ t v = ∆v + a 21 u + a 22 v + A 21 • ∇u + A 22 • ∇v + g in Ω T , u = v = 0 on Σ T , u(•, 0) = u 0 , v(•, 0) = v 0 in Ω. (2.3)
Uniqueness existence and stability results in solving an initial value-boundary value problem (2.3) can be proved by the semigroup theory for example (e.g., [START_REF] Ladyzenskaja | Linear and Quasilinear Equations of Parabolic Type[END_REF], Pazy [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF], Tanabe [START_REF] Tanabe | Equations of Evolution[END_REF]). In particular it admits a unique solution (u,

v) ∈ C([0, T ]; L 2 (Ω)) 2 ∩ L 2 (0, T ; H 1 0 (Ω)) 2 .
Our main interest is to derive a Carleman estimate of (u, v) solution of (2.3) by solely observing u in ω × (0, T ). We make the following main assumptions :

Assumption 2.1 (a) Let ω ⊂ Ω with ∂ω ∩ ∂Ω = γ and |γ| = 0. (b) |A 12 (x, t) • ν(x)| = 0, (x, t) ∈ γ T , with γ T = γ × (0, T ), (c) A 12 C 2 (ω T ) n , a 12 C 2 (ω T ) , A 11 C 1 (ω T ) n ≤ M
, where M > 0 is an arbitrarily fixed constant.

In the sequel κ will denote a generic constant and their values may change from a line to others.

The dependence of κ on s will be specified.

In this section, we prove:

Theorem 2.
≤ κ 1 (s, τ )( u 2 W 2,1 2 (ω T ) + f 2 L 2 (ω T ) ) + κ Ω T (sρ) τ e -2sηω (|f | 2 + |g| 2 )
for all s ≥ s 0 and any solution (u, v) to (2.3). Here we set

η ω (x, t) = α ω (x) t(T -t) , ρ(t) = 1 t(T -t) . (2.4)
This is a Carleman estimate for a 2 × 2 system with extra data in ω T of only one component. In [START_REF] Ammar-Khodja | Null controllability of some reaction-diffusion systems with one control force[END_REF] and [START_REF] Cristofol | Inverse problems for a 2X2 reaction-diffusion system using a Carleman estimate with one observation[END_REF], it is assumed that A 11 = A 12 = 0. In that case, the proof can be completed by directly substituting v by means of u in ω T . By the first-order coupling, we extra need Assumption 2.1 (a)

and (b).

Proof of Theorem 2.2 First we prove Lemma 2.3 Let ω ⊂ Ω be a subdomain and ∂ω ∩ ∂Ω = γ. We consider n j=1 p j (x, t)∂ j u(x, t) + q(x, t)u(x, t) = f (x, t), x ∈ ω ⊂ Ω, 0 < t < T.

(2.5)

Here p j , q ∈ L ∞ (0, T ; C 1 (Ω)) for 1 ≤ j ≤ n. We set p = (p 1 , ..., p n ) and let ν(x) = (ν 1 (x), ..., ν n (x))

be the unit outward normal vector to ∂ω at x. We assume that

|p(x, t) • ν(x)| = 0, x ∈ γ, 0 ≤ t ≤ T. (2.6)
Let u = u(x, t) satisfy (2.5) and u| γ×(0,T ) = 0. Then there exist a subdomain ω ′ ⊂ ω and a constant κ > 0, which is dependent on p and q and independent of f , such that

u L 2 (ω ′ T ) ≤ κ f L 2 (ω ′ T ) .
Proof of Lemma 2.3. We set x = (x 1 , ..., x n ) = (x ′ , x n ) and y = (y 1 , ..., y n ) = (y ′ , y n ). Without loss of generality, we can assume that

ω = {(x ′ , x n ); h(x ′ ) < x n < h 1 (x ′ ), |x ′ | < ρ} and γ = {(x ′ , x n ); x n = h 1 (x ′ ), |x ′ | < ρ}.
Here ρ > 0 is sufficiently small and h, h

1 ∈ C 2 ({|x ′ | ≤ ρ}) satisfy h = h 1 on {|x ′ | = ρ}. We change independent variables y ′ = x ′ and y n = x n -h(x ′ ). Then ω is transformed to ω = {(y ′ , y n ); 0 < y n < (h 1 -h)(x ′ ), |y ′ | < ρ}.
Set ũ(y, t) = u(x, t), p(y, t) = p(x, t), q(y, t) = q(x, t), f (y, t) = f (x, t), Γ1 = {(y ′ , 0); |y ′ | < ρ} and

Γ2 = {(y ′ , y n ); y n = (h 1 -h)(y ′ ), |y ′ | < ρ}. Then ∂ ω = Γ1 ∪ Γ2 , ( P ũ)(y, t) = n-
y n = (h 1 -h)(y ′ ), |y ′ | < ρ, 0 < t < T. (2.8) Moerover ν(x) is parallel to (∂ 1 h(x ′ ), ...., ∂ n-1 h(x ′ ), -1) on {(x ′ , x n ); x n = h(x ′ ), |x ′ | < ρ}.
Therefore, in terms of (2.6), without loss of generality, we can assume that there exists a constant δ > 0 such that r(y ′ , 0, t) > 2δ for |y ′ | < ρ and 0 < t < T . We choose ρ > 0 sufficiently small, so that r(y, t) > δ, y ∈ ω, 0 < t < T.

(2.9)

Let ν(y) = (ν 1 (y), ..., νn (y)) be the unit outward normal vector to ∂ ω at y. Then ν(y) is parallel to (0, ..., 0, -1) for y ∈ Γ1 and to

-∂(h 1 -h) ∂y 1 (y ′ ), ..., -∂(h 1 -h) ∂y n-1 (y ′ ), 1 for y ∈ Γ2 .
Hence, by choosing h 1 , h such that h 1h C 1 ({|y ′ |≤ρ}) is sufficiently small if necessary, by (2.9) we have

Γ 1 ⊂ y ∈ ∂ ω; n-1 j=1 pj (y, t)ν j (y) + r(y, t)ν n (y) ≤ 0 (2.10) and ũ(•, t) = 0 on Γ2 , Γ2 ⊂    y ∈ ∂ ω; n-1 j=1 pj (y, t)ν j (y) + r(y, t)ν n (y) > 0    . (2.11)
For the proof of the lemma, it suffices to prove a Carleman estimate for (2.5), whose proof is similar for example to Lemma 3.2 in [START_REF] Yu | Carleman estimates for the non-stationary Lamé system and the application to an inverse problem[END_REF]. We set

P0 ũ = P u -qũ = n-1 j=1 pj (y, t) ∂ ũ ∂y j (y, t) + r(y, t) ∂ ũ ∂y n (y, t), w = w(•, t) = ũ(•, t)e syn and Qw = e syn
P0 (e -syn w). Then Qw = P0 wsr(y, t)w.

We arbitrarily fix t ∈ [0, T ]. Hence by integration by parts and (2.8) -(2.11) we obtain

ω | P0 ũ| 2 e 2syn dy = ω |Qw| 2 dy = P0 w 2 L 2 (ω) + s 2 r(•, t)w 2 L 2 (ω) -2s ω   n-1 j=1 pj ∂w ∂y j (y, t) + r ∂w ∂y n   rwdy ≥ s 2 ω r2 w 2 dy -s ω   n-1 j=1 pj r ∂w 2 ∂y j + r2 ∂w 2 ∂y n   dy ≥ s 2 δ ω w 2 dy + s ω   n-1 j=1 ∂(p j r) ∂y j + ∂r 2 ∂y n   w 2 dy -s Γ1 + Γ2 r   n-1 j=1 pj νj + rν n   w 2 dS ≥ s 2 ω δ - κ 1 s w 2 dy.
Henceforth κ j > 0 depends on max 1≤j≤n p j C 1 (Ω T ) and ω. Hence we have

s 2 ω |ũ| 2 e 2syn dy ≤ κ 2 ω | P0 ũ| 2 e 2syn dy
for all large s > 0. Since

ω | P0 ũ| 2 e 2syn dy ≤ 2 ω | P ũ| 2 e 2syn dy + 2 ω |q ũ| 2 e 2syn dy ≤ 2 ω | f | 2 e 2syn dy + 2 q 2 C(Ω T ) ω |ũ| 2 e 2syn dy, by choosing s large such that s 2 κ 2 -2 q 2 C(Ω T ) ≥ s 2 2 , we have s 2 ω |ũ| 2 e 2syn dy ≤ κ 3 ω | f | 2 e 2syn dy
for all large s > 0. Since 1 ≤ e 2syn ≤ e 2sκ 4 for y ∈ ω where

κ 4 = h 1 -h C({|y ′ |≤ρ})
, for all large s > 0, we fix s > 0 large and we have ũ(

•, t) L 2 (ω) ≤ κ 5 f (•, t) L 2 (ω)
. By integrating over t ∈ (0, T ), the proof of Lemma 2.3 is completed.

By (2.3), we have

A 14 • ∇v + a 12 v = ∂ t u -∆u + a 11 u + A 13 .∇u + f in ω T and v = 0 on ∂Ω × (0, T ).
In terms of Assumption 2.1 (b), we apply Lemma 2.3 so that we can choose a subdomain

ω ′ ⊂ Ω such that v L 2 (ω ′ T ) ≤ κ u W 2,1 2 (ω ′ T ) + f L 2 (ω ′ T ) .
(2.12)

By [START_REF] Fursikov | Controllability of Evolution Equations[END_REF] and [START_REF] Yu | Controllability of parabolic equations[END_REF], for ω ′ , there exist β ω ′ ∈ C 2 (Ω) with β ω ′ > 0 on Ω and two positive constants s 0 and κ, which depend on T, Ω, ω ′ , τ and L ∞ norms of a ij , A ij , such that for all s ≥ s 0 , there exist positive constants κ 1 (s, τ ) and κ such that

Ω T (sρ) τ -1 e -2s η ω ′ (|∂ t u| 2 + |∆u| 2 + (sρ) 2 |∇u| 2 + (sρ) 4 |u| 2 ) ≤ κ Ω T (sρ) τ e -2s η ω ′ |a 12 v + A 12 • ∇v + f | 2 + κ ω ′ T (sρ) τ +3 e -2s η ω ′ |u| 2
and

Ω T (sρ) τ -1 e -2s η ω ′ (|∂ t v| 2 + |∆v| 2 + (sρ) 2 |∇v| 2 + (sρ) 4 |v| 2 ) ≤ κ Ω T (sρ) τ e -2s η ω ′ |a 21 u + A 21 • ∇u + g| 2 + κ ω ′ T (sρ) τ +3 e -2s η ω ′ |v| 2
for all large s > 0. Here and henceforth we set

η ω ′ (x, t) = β ω ′ (x) t(T -t) .
Adding them and choosing s > 0 sufficiently large to absorb the terms of u, v, ∇u, ∇v on the right hand side into the left hand side.

Hence

Ω T (sρ) τ -1 e -2s η ω ′ (|∂ t u| 2 + |∂ t v| 2 + |∆u| 2 + |∆v| 2 + (sρ) 2 |∇u| 2 + (sρ) 2 |∇v| 2 + (sρ) 4 |u| 2 + (sρ) 4 |v| 2 ) ≤ κ Ω T (sρ) τ e -2s η ω ′ (|f | 2 + |g| 2 ) + κ ω ′ T (sρ) τ +3 e -2s η ω ′ (|u| 2 + |v| 2 )
for all large s > 0. Moreover we have |(sρ

) τ +3 e -2s η ω ′ | ≤ κ 2 (s, τ ) on Ω T by β ω ′ > 0 on Ω. Hence ω ′ T (sρ) τ +3 e -2s η ω ′ (|u| 2 + |v| 2 ) ≤ κ 2 (s, τ )( u 2 L 2 (ω ′ T ) + v 2 L 2 (ω ′ T ) ). Apply Lemma 2.3, set α ω = β ω ′ and note by ω ′ ⊂ ω that u 2 W 2,1 2 (ω ′ T ) ≤ u 2 W 2,1
2 (ω T ) . Then the proof of Theorem 2.2 is completed.

Proof of Theorem 1.2

Let us recall that (U, V ) satisfie (1.1) and ( Ũ , Ṽ ) system (1.1) with b, c, U 0 , V 0 replaced by b, c, Ũ0 , Ṽ0 respectively.

We set

u = U -U , v = V -V .
Then (u, v) satisfies

∂ t u = ∆u + au + bv + A • ∇u + B • ∇v + (b -b) Ṽ , ∂ t v = ∆v + cu + dv + C • ∇u + D • ∇v + (c -c) Ũ in Ω T , u = v = 0 on Σ T and u(•, θ) = v(•, θ) = 0 in Ω.
By Assumption 1.1, we can assume that | U (x, t)|, | V (x, t)| = 0 for all (x, t) ∈ Ω T by taking T > 0 sufficiently small if necessary. Moreover we can assume that θ = T 2 . Because we take small δ > 0 such that 0 ≤ θδ < θ < θ + δ ≤ T and we can replace ω × (0, T ) by ω × (θδ, θ + δ). Shifting t by t -(θδ), we can set θ = δ and T = 2δ.

Setting

u = u V , v = v U , f = b -b, g = c -c,
we have

∂ t u = ∆ u + a 11 u + a 12 v + A 13 • ∇ u + A 14 • ∇ v + f in Ω T , (3.13)
∂ t v = ∆ v + a 21 u + a 22 v + A 23 • ∇ u + A 24 • ∇ v + g in Ω T , (3.14)
where

a 11 = a - ∂ t V V + ∆ V V + A • ∇ V V , a 12 = b U V + B • ∇ U V , A 13 = A + 2∇ V V , A 14 (x, t) = B U V (x, t) ≡ B(x)W (x, t), a 21 = c V U + C ∇ V U , a 22 = d - ∂ t U U + ∆ U U + D • ∇ U U and A 23 = C V U , A 24 = D + 2∇ U U . Let y = ∂ t u, z = ∂ t v.
Since b, c, b, c are independent of t, we obtain

∂ t y = ∆y + a 11 y + a 12 z + A 13 • ∇y + A 14 • ∇z +(∂ t a 11 ) u + (∂ t a 12 ) v + (∂ t A 13 ) • ∇ u + (∂ t A 14 ) • ∇ v, (3.15)
∂ t z = ∆z + a 21 y + a 22 z + A 23 • ∇y + A 24 • ∇z +(∂ t a 21 ) u + (∂ t a 22 ) v + (∂ t A 23 ) • ∇ u + (∂ t A 24 ) • ∇ v (3.16) y = z = 0 on Σ T .

First

Step. In terms of y, we estimate an L 2 -norm of z in a subdomain of Ω. Since ũ(x, t) = t θ y(x, ξ)dξ and ṽ(x, t) = t θ z(x, ξ)dξ by ũ(•, θ) = ṽ(•, θ) = 0, we rewrite (3.15) as

B(x) • ∇z(x, t) + b 1 (x)z(x, t) + W 1 (x, t)B(x) • t θ ∇z(x, ξ)dξ + b 2 (x, t) t θ z(x, ξ)dξ = 1 W (x, t) ∂ t y(x, t) -∆y(x, t) -a 11 y(x, t) -A 13 • ∇y -(∂ t a 11 ) t θ y(x, ξ)dξ -(∂ t A 13 ) • t θ ∇y(x, ξ)dξ ≡ Q(y)(x, t) x ∈ ω, 0 < t < T. (3.17)
Here we set

b 1 (x, t) = a 12 (x, t) W (x, t) , b 2 (x, t) = ∂ t a 12 (x, t) W (x, t) , W 1 (x, t) = ∂ t W (x, t) W (x, t) .
We will estimate z in a subdomain ω ′ of ω by means of (3.17), and the argument is similar to Lemma 2.3 but we need a special weight function for treating the integral terms t θ ∇z(x, ξ)dξ and t θ z(x, ξ)dξ. First we show Lemma 3.1 Let T = 2θ and let φ ∈ C 1 [0, T 2 ] and let us assume that there exists a constant κ 0 > 0

such that d φ dt (t) ≤ -κ 0 for t ∈ [0, T 2 ]. Then T 0 t θ g(ξ)dξ 2 e 2s φ((t-θ) 2 ) dt ≤ 1 4sκ 0 T 0 |g(t)| 2 e 2s φ((t-θ) 2 ) dt.
The proof is given by Klibanov and Timonov p.78, [START_REF] Klibanov | Carleman Estimates for Coefficient Inverse Problems and Numerical Applications[END_REF].

Henceforth we choose φ(t) = -t and we set ϕ 1 (t) = φ((tθ) 2 ) = -(tθ) 2 . Then the conclusion of Lemma 3.1 holds true. We set

w(x, t) = z(x, t) + W 1 (x, t) t θ z(x, ξ)dξ, x ∈ Ω, 0 < t < T. (3.18)
Then direct calculations yield

B(x) • ∇w(x, t) = Q(y)(x, t) -b 1 z -b 2 t θ z(x, ξ)dξ + (B • ∇W 1 ) t θ z(x, ξ)dξ in ω T . (3.19)
Henceforth κ j > 0 denote generic constants which are dependent on M, δ 0 in Assumption 1.1 and independent of s > 0. In terms of Assumption 1.1 (d), we can apply Lemma 2.3 to obtain x) dx for all large s > 0. Here and henceforth we set ϕ 0 (x) = x nγ(x ′ ).

s 2 ω ′ |w(x, t)| 2 e 2sϕ 0 (x) dx ≤ κ 1 ω ′ |Q(y)(x, t)| 2 e 2sϕ 0 (x) dx + κ 1 ω ′ |z(x, t)| 2 e 2sϕ 0 (x) dx + κ 1 ω ′ t θ z(x, ξ)dξ 2 e 2sϕ 0 (
Hence by Lemma 3.1, we have

s 2 T 0 ω ′ |w(x, t)| 2 e 2s(ϕ 0 (x)+ϕ 1 (t)) dxdt ≤ κ 1 T 0 ω ′ |Q(y)(x, t)| 2 e 2s(ϕ 0 (x)+ϕ 1 (t)) dxdt + κ 1 T 0 ω ′ |z(x, t)| 2 e 2s(ϕ 0 (x)+ϕ 1 (t)) dxdt + κ 1 ω ′ T 0 t θ z(x, ξ)dξ 2 e 2sϕ 1 (t) dt e 2sϕ 0 (x) dx ≤ κ 1 ω ′ T |Q(y)(x, t)| 2 e 2s(ϕ 0 (x)+ϕ 1 (t)) dxdt + κ 1 ω ′ T |z(x, t)| 2 e 2s(ϕ 0 (x)+ϕ 1 (t)) dxdt + κ 1 s ω ′ T |z(x, t)| 2 e 2s(ϕ 0 (x)+ϕ 1 (t)) dxdt.
Consequently

s 2 ω ′ T |w(x, t)| 2 e 2s(ϕ 0 (x)+ϕ 1 (t)) dxdt ≤ κ 2 e 2sκ 3 y 2 W 2,1 2 (ω ′ T ) +κ 2 ω ′ T |z(x, t)| 2 e 2s(ϕ 0 (x)+ϕ 1 (t)) dxdt (3.20)
for all large s > 0. On the other hand, (3.18) and Lemma 3.1 yield

ω ′ T |z(x, t)| 2 e 2s(ϕ 0 (x)+ϕ 1 (t)) dxdt = ω ′ T w(x, t) -W 1 (x, t) t θ z(x, t)dξ 2 e 2s(ϕ 0 (x)+ϕ 1 (t)) dxdt ≤ κ 4 ω ′ T |w(x, t)| 2 e 2s(ϕ 0 (x)+ϕ 1 (t)) dxdt + κ 4 ω ′ T t θ z(x, t)dξ 2 e 2s(ϕ 0 (x)+ϕ 1 (t)) dxdt ≤ κ 5 ω ′ T |w(x, t)| 2 e 2s(ϕ 0 (x)+ϕ 1 (t)) dxdt + κ 5 s ω ′ T |z(x, t)| 2 e 2s(ϕ 0 (x)+ϕ 1 (t))
dxdt for all large s > 0. Hence choosing s > 0 sufficiently large, we have

ω ′ T |z(x, t)| 2 e 2s(ϕ 0 (x)+ϕ 1 (t)) dxdt ≤ κ 6 ω ′ T |w(x, t)| 2 e 2s(ϕ 0 (x)+ϕ 1 (t)) dxdt (3.21)
for all large s > 0. Substituting (3.21) into (3.20) and fixing s > 0 sufficiently large, we obtain

w L 2 (ω ′ T ) ≤ κ 7 e κ 7 s y W 2,1 2 (ω ′ T ) .
Hence by (3.21) we have

z L 2 (ω ′ T ) ≤ κ 8 e κ 7 s y W 2,1 2 (ω ′ T ) .
(3.22)

Second

Step. We will estimate ∇z L 2 (ω 1 ×(δ,T -δ)) where ω 1 ⊂ ω and δ > 0. For it, we use the interior regularity estimate for a heat equation (3.16) in z. Let us recall that ρ(t) = 1 t(T -t) . Setting z(x, t) = e -ρ(t) z(x, t), we rewrite (3.16) as

∂ t z(x, t) = ∆z(x, t) -ρ ′ (t)z(x, t) + a 22 z + A 24 • ∇z +(∂ t a 22 ) t θ e ρ(ξ)-ρ(t) z(x, ξ)dξ + (∂ t A 24 ) • t θ e ρ(ξ)-ρ(t) ∇z(x, ξ)dξ +e -ρ(t) a 21 y + A 23 • ∇y + (∂ t a 21 ) t θ y(x, ξ)dξ + (∂ t A 23 ) • t θ ∇y(x, ξ)dξ . (3.23) We choose subdomains ω 1 , ω 2 of C ∞ class such that ω 1 ⊂ ω 1 ⊂ ω 2 ⊂ ω 2 ⊂ ω ′ and choose χ ∈ C 1 (ω ′ ), ≥ 0 such that χ(x) =          1, x ∈ ω 1 , 0, x ∈ ω ′ \ ω 2 .
Moreover we can take χ satisfying

|∇χ(x)| 2 χ(x) ≤ κ 9 x ∈ ω ′ (3.24)
(e.g., p.414 in Lions [START_REF] Lions | Contrôlabilité Exacte Perturbations et Stabilisation de Systèmes Distribués[END_REF]). Multiplying (3.23) with χz and integrating over ω ′ × (0, T ), we have

1 2 T 0 ω ′ χ(x)∂ t (z 2 )dxdt = - T 0 ω ′ χ|∇z| 2 dxdt - T 0 ω ′ ∇χ • z∇zdxdt - T 0 ω ′ χρ ′ (t)e -2ρ(t) |z| 2 dxdt + T 0 ω ′ (a 22 |z| 2 χ + A 24 • ∇zχz)dxdt + T 0 ω ′ (∂ t a 22 )χz t θ e ρ(ξ)-ρ(t) z(x, ξ)dξ dxdt + T 0 ω ′ (∂ t A 24 ) • χz t θ e ρ(ξ)-ρ(t) ∇z(x, ξ)dξ dxdt + T 0 ω ′ e -ρ(t) χz a 21 y + A 23 ∇y + (∂ t a 21 ) t θ y(x, ξ)dξ + (∂ t A 23 ) • t θ ∇y(x, ξ)dξ dxdt.
By the Cauchy-Schwarz inequality and (3.24), we have

|∇χ • z∇z| = ∇χ √ χ z • √ χ∇z ≤ 1 8 χ|∇z| 2 + 2|∇χ| 2 χ |z| 2
and

|A 24 • χz∇z| = | √ χ∇z • A 24 √ χz| ≤ 1 8 χ|∇z| 2 + 2|A 24 | 2 χ|z| 2 . Hence, since z(•, 0) = z(•, T ) = 0, sup 0≤t≤T |ρ ′ (t)e -2ρ(t) | < ∞
and ρ(ξ)ρ(t) ≤ 0 if ξ is between θ and t, we have

T 0 ω ′ χ|∇z| 2 dxdt ≤ 1 4 T 0 ω ′ χ|∇z| 2 dxdt + κ 10 T 0 ω ′ (|z| 2 + |z| 2 )dxdt + κ 10 T 0 ω ′ |z| t θ z(x, ξ)dξ dxdt + κ 10 T 0 ω ′ χ|z| t θ ∇z(x, ξ)dξ dxdt + κ 10 T 0 ω ′ |z|(|y| + |∇y|) + |z| t θ y(x, ξ)dξ + t θ ∇y(x, ξ)dξ dxdt.
Moreover the Cauchy-Schwarz inequality yields

T 0 ω ′ χ|z(x, t)| t θ ∇z(x, ξ)dξ dxdt ≤ T 0 ω ′ √ χ|z(x, t)| T 0 χ(x)|∇z(x, ξ)|dξ dxdt ≤ T 0 ω ′   1 8T 2 T 0 √ χ|∇z(x, ξ)|dξ 2 + 2T 2 χ|z(x, ξ)| 2   dxdt ≤ 2T 2 T 0 ω ′ |z(x, t)| 2 dxdt + T 0 ω ′ 1 8T T 0 χ|∇z(x, ξ)| 2 dξdxdt ≤ 2T 2 T 0 ω ′ |z(x, t)| 2 dxdt + 1 8 T 0 ω ′ χ|∇z(x, t)| 2 dxdt. Hence 5 8 T 0 ω ′ χ|∇z(x, t)| 2 dxdt ≤ κ 11 T 0 ω ′ (|z| 2 + |z| 2 )dxdt + κ 11 T 0 ω ′ (|y| 2 + |∇y| 2 )dxdt.
Let δ > 0 be fixed sufficiently small. Then |∇z(x,

t)| ≥ κ 12 (δ)|∇z(x, t)| for δ ≤ t ≤ T -δ. Since χ = 1 in ω 1 , we have T -δ δ ω 1 |∇z| 2 dxdt ≤ κ 13 (δ)( z 2 L 2 (ω ′ T ) + y 2 L 2 (0,T ;H 1 (ω ′ )) ).
By means of (3.22), we obtain

z L 2 (δ,T -δ;H 1 (ω 1 )) ≤ κ 14 (δ) y W 2,1 2 (ω ′ T ) .
(3.25)

Third

Step. We apply Theorem 2.2 to (3.15) and (3.16) for ω ′ ⊂ Ω and (δ, Tδ). We set

η(x, t) = α ω ′ (x) (t -δ)(T -δ -t) .
Using also (3.25), we obtain that there exist two positive constants s 0 and κ such that for all s ≥ s 0 , one has

T -δ δ Ω (sρ) -1 e -2sη (|∂ t y| 2 +|∂ t z| 2 +|∆y| 2 +|∆z| 2 +(sρ) 2 |∇y| 2 +(sρ) 2 |∇z| 2 +(sρ) 4 |y| 2 +(sρ) 4 |z| 2 )dxdt ≤ κ 15 T -δ δ Ω |(∂ t a 11 ) u + (∂ t a 12 ) v + (∂ t A 13 ) • ∇ u + (∂ t A 14 ) • ∇ v| 2 e -2sη dxdt +κ 15 T -δ δ Ω |(∂ t a 21 ) u + (∂ t a 22 ) v + (∂ t A 23 ) • ∇ u + (∂ t A 24 ) • ∇ v| 2 )e -2sη dxdt +κ 16 (s) y 2 W 2,1 2 (ω ′ ×(δ,T -δ)) +κ 16 (s) (∂ t a 11 ) t θ y(x, ξ)dξ + (∂ t a 12 ) t θ z(x, ξ)dξ + (∂ t A 13 ) • t θ ∇y(x, ξ)dξ +(∂ t A 14 ) • t θ ∇z(x, ξ)dξ 2 L 2 (ω ′ ×(δ,T -δ)) +κ 16 (s) (∂ t a 21 ) t θ y(x, ξ)dξ + (∂ t a 22 ) t θ z(x, ξ)dξ + (∂ t A 23 ) • t θ ∇y(x, ξ)dξ +(∂ t A 24 ) • t θ ∇z(x, ξ)dξ 2 L 2 (ω ′ ×(δ,T -δ)) ≤ κ 15 T -δ δ Ω |(∂ t a 11 ) u + (∂ t a 12 ) v + (∂ t A 13 ) • ∇ u + (∂ t A 14 ) • ∇ v| 2 e -2sη dxdt +κ 15 T -δ δ Ω |(∂ t a 21 ) u + (∂ t a 22 ) v + (∂ t A 23 ) • ∇ u + (∂ t A 24 ) • ∇ v| 2 )e -2sη dxdt +κ 16 (s) y 2 W 2,1 2 (ω T ) . (3.26)
for all large s > 0. In order to improve inequality (3.26), we use the following lemma. ( [START_REF] Klibanov | Global uniqueness of a multidimensionnal inverse problem for a non linear parabolic equation by a Carleman estimate[END_REF] , Lemma 3.1.1 in [START_REF] Klibanov | Carleman Estimates for Coefficient Inverse Problems and Numerical Applications[END_REF]).

Lemma 3.2 Let θ = T

2 . There exists a positive constant κ 17 such that

T -δ δ Ω t θ q(x, ξ)dξ 2 e -2sη dxdt ≤ κ 17 s T -δ δ Ω |q(x, t)| 2 e -2sη dxdt
Since u(x, t) = t θ y(x, ξ)dξ and v(x, t) = t θ z(x, ξ)dξ, by a direct application of this lemma, the first integral on the right hand side of (3.26) can be absorbed into the left hand side. Hence

T -δ δ Ω 1 sρ (|∂ t y| 2 + |∂ t z| 2 + |∆y| 2 + |∆z| 2 ) + sρ(|∇y| 2 + |∇z| 2 ) + s 3 ρ 3 (|y| 2 + |z| 2 ) e -2sη dxdt ≤ κ 16 (s)( ∂ t (U -U ) 2 W 2,1 2 (ω T ) + U -U 2 W 2,1 2 (ω T ) )
for all large s > 0. We choose t 0 > 0 sufficiently small such that

δ < t 0 < θ < T -t 0 < T -δ, so that T -t 0 t 0 Ω 1 sρ (|∂ t y| 2 + |∂ t z| 2 + |∆y| 2 + |∆z| 2 ) + sρ(|∇y| 2 + |∇z| 2 ) + s 3 ρ 3 (|y| 2 + |z| 2 ) e -2sη dxdt ≤ κ 16 (s)( ∂ t (U -U ) 2 W 2,1 2 (ω T ) + U -U 2 W 2,1 2 (ω T ) ). Since 1 ρ e -2sη , ρe -2sη ≥ κ 0 (t 0 , s) on Ω × [t 0 , T -t 0 ],
we fix s > 0 sufficiently large, so that

u 2 H 1 (t 0 ,T -t 0 ;H 2 (Ω)) + u 2 H 2 (t 0 ,T -t 0 ;L 2 (Ω)) + v 2 H 1 (t 0 ,T -t 0 ;H 2 (Ω)) + v 2 H 2 (t 0 ,T -t 0 ;L 2 (Ω)) ≤ κ 16 (s)( ∂ t (U -U ) 2 W 2,1 2 (ω T ) + U -U 2 W 2,1 2 (ω T ) ).
By the trace theorem, we have

∂ t u(•, θ) 2 L 2 (Ω) + ∂ t v(•, θ) 2 L 2 (Ω) + u(•, θ) 2 H 2 (Ω) + v(•, θ) 2 H 2 (Ω) ≤ κ 16 (s)( ∂ t (U -U ) 2 W 2,1 2 (ω T ) + U -U 2 W 2,1 2 (ω T ) ).
Since f and g satisfy (3.13) and (3.14

) at t = θ, we see that b -b 2 L 2 (Ω) + c -c 2 L 2 (Ω) ≤ κ( ∂ t (U -U ) 2 W 2,1 2 (ω T ) + U -U 2 W 2,1 2 (ω T ) ).
Thus the proof of Theorem 1.2 is completed.

Removing the positivity assumption

For the stability in our inverse problem, the non-vanishing condition Assumption 1.1 (f) is crucial and does not hold automatically. We are going to prove that one can realize this assumption by a suitable control.

Let m ∈ N be fixed such that

m 4 > n. (4.27)
We assume that

a, b, c, d, A, B, C, D ∈ W 2m-2,∞ (Ω). (4.28)
We set

L(u, v) = L(a, b, c, d, A, B, C, D)(u, v) =      L 1 (u, v) L 2 (u, v)      = -      ∆u + au + bv + A • ∇u + B • ∇v ∆v + cu + dv + C • ∇v + D • ∇v      (4.29)
and

D(L) = H 2 (Ω) ∩ H 1 0 (Ω) 2 . For h ∈ L 2 (ω T ), let ( U , V ) := ( U ( Ũ0 , Ṽ0 , h)(•, •), V ( Ũ0 , Ṽ0 , h)(•, •)) satisfy ∂ t ( U , V ) = -L(a, b, c, d, A, B, C, D)( U , V ) + (χ ω h, 0) in Ω T , ( U , V ) = (0, 0) on Σ T , ( Ũ , Ṽ )(•, 0) = ( Ũ0 , Ṽ0 ) in Ω. ( 4
= c in ω 1 . Let (U, V )(•, θ) = ( U , V )(•, θ). Then there exists h ∈ L 2 (ω T )
depending on a, b, c, d, A, B, C, D, U 0 , V 0 and ω, such that there exists a constant κ > 0 such that

b -b L 2 (Ω) + c -c L 2 (Ω) ≤ κ( ∂ t (U -U ) W 2,1 2 (ω T ) + U -U W 2,1 2 (ω T ) ) (4.31)
for arbitrary b, c, U, V satisfying Assumption 1.1 (a), (e), (h).

The rest of this section is devoted to the proof of Theorem 4.1.

First

Step. First we prove

h ∈ L 2 (ω T ) such that | Ũ (•, θ)|, | Ṽ (•, θ)| = 0 on Ω \ ω 1 . (4.32)
In this step, we will give the proof of Lemma 4.2, which is based on the approximate controllability and our Carleman estimate Theorem 2.2.

Taking M > 0 for a, b, c, d, A, B, C, D, and setting U 1 = e -M t U and V 1 = e -M t V , we have

∂ t U 1 = ∆U 1 + (a -M )U 1 + bV 1 + A • ∇U 1 + B • ∇V 1 + e -M t χ ω h and ∂ t V 1 = ∆V 1 + cU 1 + (d -M )V 1 + C • ∇U 1 + D • ∇V 1 .
Consequently, by choosing M > 0 sufficiently large, the integration by parts yields

((L + M I)(u, v), (u, v)) (L 2 (Ω)) 2 ≥ κ 1 (u, v) 2 (H 1 (Ω)) 2 , (u, v) ∈ D(L).
Again the elliptic regularity yields

(u, v) (H 3 (Ω)) 2 ≤ κ 1 (∆u, ∆v) (H 1 (Ω)) 2 + κ 1 (u, v) (L 2 (Ω)) 2 ≤ κ 1 (-L -Q)(u, v) (H 1 (Ω)) 2 + κ 1 (u, v) (L 2 (Ω)) 2 ≤ κ 1 L(u, v) (H 1 (Ω)) 2 + κ 1 (u, v) (H 2 (Ω)) 2 . (4.35)
On the other hand, we have L(u, v) ∈ D(L) and apply (4.33) to L(u, v) to have

L(u, v) (H 1 (Ω)) 2 ≤ κ 1 L 2 (u, v) (L 2 (Ω)) 2 .
Applying this and (4.34) to (4.35), we obtain

(u, v) (H 3 (Ω)) 2 ≤ κ 4 L 2 (u, v) (L 2 (Ω)) 2 .
Repeating these arguments, we can complete the proof of Lemma 4.3.

Moreover by [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF] and [START_REF] Tanabe | Equations of Evolution[END_REF] for example, we see:

Lemma 4.4
The operator -L generates an analytic semigroup in (L 2 (Ω)) 2 .

There are no general result on the approximate controllabilty for parabolic systems with controls of a restricted number of components and see e.g., [START_REF] Ammar-Khodja | Null controllability of some reaction-diffusion systems with one control force[END_REF] and [START_REF] Leiva | Controllability of a system of parabolic equations with non-diagonal diffusion matrix[END_REF] as related works. For controllability for systems, see [START_REF] Ammar-Khodja | Null controllability of some reaction-diffusion systems with one control force[END_REF] - [START_REF] Ammar-Khodja | Null controllability of some systems of parabolic type by one control force[END_REF], [START_REF] González-Burgos | Controllability of some coupled parabolic systems by one control force[END_REF] - [START_REF] González-Burgos | Controllability results for cascade systems of m coupled parabolic PDEs by one control force[END_REF]. Next we will prove the approximate controllability with control χ ω h to only one component.

Lemma 4.5 For any ε > 0, ( Ũ0 , Ṽ0 ) ∈ (L 2 (Ω)) 2 , ( Ũ1 , Ṽ1 ) ∈ (L 2 (Ω)) 2 , and any t 0 ∈ (0, δ), there

exists h ε ∈ L 2 (ω T ) such that Ũ ( Ũ0 , Ṽ0 , h)(•, t 0 ) -Ũ1 L 2 (Ω) + Ṽ ( Ũ0 , Ṽ0 , h)(•, t 0 ) -Ṽ1 L 2 (Ω) < ε.
Proof. Consider the following reaction-diffusion-convection system :

∂ t u = ∆u + au + bv -∇ • (Au) -∇ • (Bv) in Ω T , ∂ t v = ∆v + cu + dv -∇ • (Cu) -∇ • (Dv) in Ω T , u = v = 0 on Σ T . (4.36)
The approximate controllability is equivalent to the uniqueness: Let u, v satisfy (4.36). Then u = 0 in ω T implies u = v = 0 in Ω T (e.g., Zabczyk [START_REF] Zabczyk | Mathematical Control Theory: An Introduction[END_REF]). This uniqueness follows from Theorem 2.2 by replacing the coefficients in (2.3) suitably and verifying Assumption 1.1 (d).

Now we will complete

Proof of Lemma 4.2 The proof is be done in three steps. Henceforth for fixed ( U 0 , V 0 ), by

( U , V )(h) we denote ( U , V )( Ũ0 , Ṽ0 , h).
Existence of a control in L 2 (ω T )

Let us arbitrarily fix ( Ũ1 , Ṽ1 )

∈ (H 2m+2 0 (Ω)) 2 satisfying | Ũ1 |, | Ṽ1 | = 0 on Ω \ ω 1 .
Then for any ε > 0 and any T 1 ∈ (0, θ), there exists

h ε ∈ L 2 (ω T 1 ) such that ( U , V )(h ε )(•, T 1 ) -( Ũ1 , Ṽ1 ) (L 2 (Ω)) 2 ≤ ε. (4.37)
A more regular control

By the density of C ∞ (ω T 1 ) in L 2 (ω T 1 ), for any δ > 0, there exists h ε,δ ∈ C ∞ (ω T 1 ) such that h ε -h ε,δ L 2 (ω T 1 ) ≤ δ. (4.38) Therefore ( U , V )(h ε )(•, T 1 ) -( U , V )(h ε,δ )(•, T 1 ) (L 2 (Ω)) 2 = T 1 0 e -(T 1 -s)L χ ω (h ε -h ε,δ )(s)ds (L 2 (Ω)) 2 ≤ κ 5 δ.

Use of the time regularizing effect

By (4.37) and (4.38), we obtain

( U , V )(h ε,δ )(•, T 1 ) -( U 1 , V 1 ) (L 2 (Ω)) 2 ≤ ε + κ 5 δ. (4.39)
Since -L generates an analytic semigroup in (L 2 (Ω)) 2 , by e.g., [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF], [START_REF] Tanabe | Equations of Evolution[END_REF], we see that

e -(θ-T 1 )L ( U , V )(h ε,δ )(•, T 1 ) ∈ D(L m )
and

L m [e -(θ-T 1 )L ( U , V )(h ε,δ )(•, T 1 ) -e -(θ-T 1 )L ( Ũ1 , Ṽ1 )] (L 2 (Ω)) 2 ≤ κ 6 (θ -T 1 ) -m ( U , V )(h ε,δ )(•, T 1 ) -( Ũ1 , Ṽ1 ) (L 2 (Ω)) 2 ≤ κ 6 (θ -T 1 ) -m (ε + κ 5 δ).
Extending h ε,δ (•, t) = 0 for t > T 1 , we have

e -(θ-T 1 )L ( U , V )(h ε,δ )(•, T 1 ) = ( U , V )(h ε,δ )(•, θ),
and so

L m [( Ũ , Ṽ )(h ε,δ )(•, θ) -e -(θ-T 1 )L ( Ũ1 , Ṽ1 )] (L 2 (Ω)) 2 ≤ κ 6 (θ -T 1 ) -m (ε + κ 5 δ). (4.40)
Moreover as ( Ũ1 , Ṽ1 ) ∈ D(L m+1 ), we have

L m [e -(θ-T 1 )L ( Ũ1 , Ṽ1 ) -( Ũ1 , Ṽ1 )] (L 2 (Ω)) 2 ≤ κ 6 (e -(θ-T 1 )L -I)L m ( Ũ1 , Ṽ1 ) (L 2 (Ω)) 2 ≤ κ 6 θ-T 1 0 d dη (e -ηL )L m ( Ũ1 , Ṽ1 )dη (L 2 (Ω)) 2 ≤ κ 6 θ-T 1 0 e -ηL L m+1 ( Ũ1 , Ṽ1 )dη (L 2 (Ω)) 2 ≤ κ 6 (θ -T 1 ) L m+1 ( Ũ1 , Ṽ1 ) (L 2 (Ω)) 2 .
In terms of (4.40), we obtain

L m [( Ũ , Ṽ )(h ε,δ )(•, θ) -( Ũ1 , Ṽ1 )] (L 2 (Ω)) 2 ≤ κ 6 (θ -T 1 ) -m (ε + κ 5 δ) +κ 6 (θ -T 1 ) L m+1 ( Ũ1 , Ṽ1 ) (L 2 (Ω)) 2 . (4.41)
For any ε 1 > 0 and ( Ũ1 , Ṽ1 ) ∈ D(L m+1 ), we choose T 1 ∈ (0, θ) such that

κ 6 (θ -T 1 ) L m+1 ( Ũ1 , Ṽ1 ) (L 2 (Ω)) 2 < ε 1 3 .
Then, with this T 1 , we choose ε > 0 such that

κ 6 (θ -T 1 ) -m ε < ε 1 3 .
Finally with this h ε , we choose δ > 0 sufficiently small such that

κ 6 δ ≤ ε 1 3 .
Therefore (4.41) yields Second Step We will complete the proof of Theorem 4.1. Let h ∈ L 2 (ω T ) be chosen in Lemma 4.2. We set

L m [( Ũ , Ṽ )(h ε,δ )(•, θ) -( Ũ1 , Ṽ1 )] L 2 (Ω) 2 ≤ ε 1 . ( 4
u = U -U , v = V -V .
Then (u, v) satisfies

∂ t u = ∆u + au + bv + A • ∇u + B • ∇v + f Ṽ , ∂ t v = ∆v + cu + dv + C • ∇u + D • ∇v + g Ũ in Ω T , u = v = 0 on Σ T , (4.43)
where

f = b -b, g = c -c.
We consider the time derivative of system (4.43). Setting y = ∂ t u and z = ∂ t v, we obtain

∂ t y = ∆y + a(x)y + b(x)z + A • ∇y + B • ∇z + f ∂ t V in Ω T , ∂ t z = ∆z + c(x)y + d(x)z + C • ∇y + D • ∇z + g∂ t U in Ω T , y = z = 0 on Σ T . (4.44)
Applying the Carleman estimate Theorem 2.2 to system (4.43) and using f = 0 in ω 1 , we have Furthermore, for large s > 0, we can prove that

Ω T (sρ) -1 e -2sηω
Ω T |f (x)| 2 e -2sηω (x,t) dxdt ≤ o(1) Ω |f (x)| 2 e -2sηω(x,θ) dx as s → ∞. (4.46)
In fact, we can prove similarly to [START_REF] Yu | Lipschitz stability in inverse parabolic problems by the Carleman estimate[END_REF]. Recall that T = 2θ. Setting ℓ(t) = t(Tt), by (2.4) we have ∂(-ηω) ∂t (x, θ) = 0, x ∈ Ω and

∂ 2 (-η ω ) ∂t 2 (x, t) = -α ω (x) 2ℓ ′ (t) 2 -ℓ(t)ℓ ′′ (t) ℓ(t) 3 and ∂ 3 (-η ω ) ∂t 3 (x, t) = -α ω (x) 6ℓ ′ (t)(ℓ(t)ℓ ′′ (t) -ℓ ′ (t) 2 ) ℓ(t) 4 , (x, t) ∈ Ω T . ∂ 2 (-η ω ) ∂t 2 (x, t) ≤ - κ 8 ℓ(t) 3 , (x, t) ∈ Ω T
with a positive constant κ 8 and

∂ 3 (-η ω ) ∂t 3 (x, t) ≥ 0, 0 ≤ t ≤ θ, x ∈ Ω, ∂ 3 (-η ω ) ∂t 3 (x, t) ≤ 0, θ ≤ t ≤ T, x ∈ Ω.
Consequently by the mean value theorem, we can take t 1 such that t 1 is between t and θ and

-η ω (x, t) = -η ω (x, θ) + 1 2 ∂ 2 (-η ω ) ∂t 2 (x, t)(t -θ) 2 + 1 6 ∂ 3 (-η ω ) ∂t 3 (x, t 1 )(t -θ) 3 ≤ -η ω (x, θ) - κ 8 2t 3 (T -t) 3 (t -θ) 2 , (x, t) ∈ Ω T .
Hence, noting that κ 8 > 0 and -1 t(T -t) ≤ -4 T 2 , we obtain T 0 e -2sηω(x,t) dt ≤ e -2sηω (x,θ)

T 0 exp - sκ 8 t 3 (T -t) 3 (t -θ) 2 dt ≤ e -2sηω(x,θ) T 0 exp - sκ 9 T 2 (t -θ) 2 dt.
The Lebesgue theorem yields .

We have

Ω (|y(x, θ)| 2 + |z(x, θ)| 2 )e -2sηω(x,θ) dx = Ω ∂ ∂t θ 0 (|y(x, t)| 2 + |z(x, t)| 2 )e -2sηω (x,t) dtdx = Ω θ 0 {2sα ω (x)ρ(t) 2 (T -2t)(|y(x, t)| 2 + |z(x, t)| 2 ) + 2(y∂ t y + z∂ t z)}e -2sηω (x,t) dtdx ≤ κ 10 Ω T {(sρ) 3 (|y(x, t)| 2 + |z(x, t)| 2 ) + (sρ) -1 (|∂ t y(x, t)| 2 + |∂ t z(x, t)| 2 )}e -2sηω (x,t) dtdx.
At the last inequality, we used

|y∂ t y| = |(sρ) -1 2 ∂ t y(sρ) 1 2 y| ≤ 1 2 (sρ) -1 |∂ t y| 2 + 1 2 (sρ)|y| 2 ≤ 1 2 (sρ) -1 |∂ t y| 2 + κ ′ 10 (sρ) 3 |y| 2 .
Hence, by (4.45) and (4.46), noting that f = g = 0 in ω 1 , we have

Ω (|y(x, θ)| 2 + |z(x, θ)| 2 )e -2sηω (x,θ) dx ≤ κ 7 (s) y 2 W 2,1 2 (ω T ) + o(1) Ω\ω 1 (|f (x)| 2 + |g(x)| 2 )e -2sηω (x,θ) dx (4.47)
for all large s > 0.

On the other hand, since u(•, θ) = v(•, θ) = 0, we have y(x, θ) = f (x) Ṽ (x, θ) and z(x, θ) = g(x) Ũ (x, θ) for x ∈ Ω. Therefore, by (4.32) and (4.47) we obtain

κ 11 Ω (|f (x)| 2 + |g(x)| 2 )e -2sηω (x,θ) dx ≤ κ 7 (s) y 2 W 2,1 2 (ω T ) + o(1) Ω\ω 1 (|f (x)| 2 + |g(x)| 2 )e -2sηω (x,θ) dx as s -→ ∞.
Taking s > 0 large and fixing, we absorb the second term on the right hand side into the left hand side and the proof of Theorem 4.1 is completed.

5 Some generalization and comments

Identification of all the coefficients

Indeed we can determine all the coefficients of (1.1). For it, we need repeats of measurements by choosing suitable interior controls. We choose m ∈ N such that (Ω)) 2 , there exists

m > n 4 + 1 2 . We recall that ( Ũ , Ṽ ) = ( Ũ (h)(•, •), Ṽ (h)(•, •)) satisfies (4.29) and that (U, V ) = (U (h)(•, •), V (h)(•, •)) satisfies (4.
h ∈ L 2 (ω T ) such that ( Ũ (h), Ṽ (h))(•, θ) -( Ũ1 , Ṽ1 ) (C 1 (Ω)) 2 ≤ ε. ( 5
., h 2n+2 ∈ L 2 (ω T ) such that det                            U (h 1 ) V (h 1 ) 0 0 ∇ U (h 1 ) ∇ V (h 1 ) 0 0 0 0 U (h 1 ) V (h 1 ) 0 0 ∇ U (h 1 ) ∇ V (h 1 ) U (h 2 ) V (h 2 ) 0 0 ∇ U (h 2 ) ∇ V (h 2 ) 0 0 0 0 U (h 2 ) V (h 2 ) 0 0 ∇ U (h 2 ) ∇ V (h 2 ) . . . . . . . . . . . . . . . . . . . . . . . . U (h 2n+2 ) V (h 2n+2 ) 0 0 ∇ U (h 2n+2 ) ∇ V (h 2n+2 ) 0 0 0 0 U (h 2n+2 ) V (h 2n+2 ) 0 0 ∇ U(h 2n+2 ) ∇ V (h 2n+2 )                            = 0 x ∈ Ω \ ω 1 , t = θ (5.49)
and we choose a constant κ > 0 depending on M, m, γ, s, Ω, ω, T and h 1 , ..., h 2n+2 such that

a -a L 2 (Ω) + b -b L 2 (Ω) + c -c L 2 (Ω) + d -d L 2 (Ω) + A -A (L 2 (Ω)) n + B -B (L 2 (Ω)) n + C -C (L 2 (Ω)) n + D -D (L 2 (Ω)) n ≤ κ 2n+2 j=1 ( ∂ t (U (h j ) -Ũ (h j )) W 2,1 2 (ω T ) + U (h j ) -Ũ (h j ) W 2,1 2 (ω T ) )
for all (a, b, c, d, A, B, C, D) satisfying Assumption 1.1.

Example for Theorem 5.1:

Let n = 1 and let p 1 , p 2 , q 1 , q 2 , q 3 be constants such that p 1 q 2p 2 q 1 = 0 and p 3 (x 1 ), q 4 (x 1 ) satisfy (∂ 1 p 3 )(x 1 ) = 0 and ∂ 1 q 4 (x 1 ) = 0 for x 1 ∈ Ω \ ω 1 , and let q 3 be an arbitrarily smooth function.

Then for x = x 1 ∈ Ω \ ω 1 , we can verify that det

                           p 1 q 1 0 0 ∂ 1 p 1 ∂ 1 q 1 0 0 0 0 p 1 q 1 0 0 ∂ 1 p 1 ∂ 1 q 1 p 2 q 2 0 0 ∂ 1 p 2 ∂ 1 q 2 0 0 0 0 p 2 q 2 0 0 ∂ 1 p 2 ∂ 1 q 2 . . . . . . . . . . . . . . . . . . . . . . . . p 4 q 4 0 0 ∂ 1 p 4 ∂ 1 q 4 0 0 (a) (a ij ) i,j=1,3 ∈ W 2,∞ (Ω T ), a ij W 2,∞ (Ω T ) ≤ M . (b) ω of class C 2 , ∂ω ∩ ∂Ω = γ and |γ| = 0. (c) (∇a 12 -a 12 a 13 ∇a 13 ) • ν = 0 on γ × (0, T ). (d) a 12 , a 13 ∈ W 3,∞ (ω T ), a 12 W 3,∞ (ω T ) , a 13 W 3,∞ (ω T ) ≤ M .
(e) a 13 = 0 on Ω T .

We show a Carleman estimate with extra data of one component.

Theorem 5.3 Under Assumption 5.2, there exist α ω ∈ C 2 (Ω) with α ω > 0 on Ω and a constant s 0 > 0 which depends on T, M, Ω, ω, τ and the L ∞ (Ω)-norms of a ij , 1 ≤ i, j ≤ 3 such that we can choose positive constants κ 1 (s) and κ satisfying: and G = a 12 g + a 13 h.

Ω T (sρ) -1 e -2sηω
By [START_REF] Fursikov | Controllability of Evolution Equations[END_REF], [START_REF] Yu | Controllability of parabolic equations[END_REF] and the proof of Theorem 2.2, we see that there exist a subdomain ω ′ ⊂ ω and for all s ≥ s 0 , where we set ηω ′ (x, t) =

β ω ′ ∈ C 2 (
β ω ′ (x)
t(T -t) . Here (5.52) is obtained by applying the Carleman estimate in [START_REF] Fursikov | Controllability of Evolution Equations[END_REF] or [START_REF] Yu | Controllability of parabolic equations[END_REF] to the first and the second equations in (5.51), while (5.53) is seen by applying Theorem 2.2 to the second and the third equations in (5.51) and noting Assumption 5.2 (c). We further notice that the weight function ηω ′ can be taken the same, which can be seen from the proof of Theorem 2.

≤ κ 2 Ω T (z 2 + u 2 + v 2 + |∇v| 2 )e -2sη ω ′ dxdt + κ 2 Ω T (f 2 + g 2 + h 2 )e -2sη ω ′ dxdt +κ 3 (s)( z 2 W 2,1 2 (ω ′ T ) + u 2 L 2 (ω ′ T ) + g 2 L 2 (ω ′ T ) + h 2

Assumption 1 . 1

 11 (a) a, b, b, c, c, d ∈ B L ∞ (Ω) (0, M ), (b) A, B, C, D ∈ B L ∞ (Ω) n (0, M ), (c) ω ⊂ Ω satisfies ∂ω ∩ ∂Ω = γ and |γ| = 0, and ω is of class C 2 , (d) |B(x) • ν(x)| = 0, x ∈ γ, (e) B ∈ C 2 (ω) n , A ∈ C 1 (ω) n and b ∈ C 2 (ω),

. 30 )Theorem 4 . 1

 3041 By (U, V ) we denote the solution to (4.30) with b, c replacing b, c. Our main result in this section is the following : Suppose Assumption 1.1 except for (f ). Let ω 1 be a neighbourhood of ∂Ω such that ω ⊂ ω 1 and let b = b and c

Lemma 4 . 2

 42 Let Assumption 1.1 except for (f ) hold and let b = b, c = c in ω 1 . Then there exists

  .42) In terms of Lemma 4.3 and (4.27), by choosing ε > 0 sufficiently small for inf x∈Ω\ω 1 | Ũ1 (x)| and inf x∈Ω\ω 1 | Ṽ1 (x)|, the proof of Lemma 4.2 is completed.

  29) where ã, b, c, d, Ã, B, C, D are replaced by a, b, c, d, A, B, C, D respectively. Then, with m >

Theorem 5 . 1

 51 Let ω, a, a, b, b, c, c, d, d, A, A, B, B, C, C, D, D satisfy Assumption 1.1 and a, b, c, d, A, B, C, D ∈ W 2m,∞ (Ω). Let ω 1 be a neighbourhood of ∂Ω such that ω ⊂ ω 1 and let the coefficients (a, b, c, d, A, B, C, D)and ( a, b, c, d, A, B, C, D) coincide in ω 1 . Then there exist h 1 , h 2 , ..

L 2

 2 s > 0. We can absorb the first terms on the right hand side into the left hand side by choosing s > 0 large, and we use z = ∂ t u -∆ua 11 uf by the first equation in (5.51), so that the proof of Theorem 5.3 is completed.The approximate controllability is a direct consequence of Theorem 5.3. That is, we consider∂ t u = ∆u + a 11 (x)u + a 21 (x)v + a 31 (x)w + χ ω f in Ω T , ∂ t v = ∆v + a 12 (x)u + a 22 (x)v + a 32 (x)w in Ω T , ∂ t w = ∆w + a 13 (x)u + a 23 (x)v + a 33 (x)w in Ω T , u = v = w = 0 on Σ T , u(•, 0) = u 0 , v(•, 0) = v 0 , w(•, 0) = w 0 in Ω.(5.55)

2

  Let τ ≥ 1 and ω ⊂ Ω be a subdomain such that ω ⊂ Ω. Under Assumption 2.1, there exist α ω ∈ C 2 (Ω) with α ω > 0 on Ω and two positive constants s 0 and κ which depend on T, M, Ω, ω, τ and the L ∞ -norms of a ij , A ij , such that there exist positive constants κ 1 (s, τ ) and κ -2sηω (|∂ t u| 2 + |∂ t v| 2 + |∆u| 2 + |∆v| 2 + (sρ) 2 |∇u| 2 + (sρ) 2 |∇v| 2 + (sρ) 4 |u| 2 + (sρ) 4 |v| 2 )

	such that the following Carleman estimate holds
	(sρ) τ -1 e
	Ω T

  (|∂ t y| 2 + |∂ t z| 2 + |∆y| 2 + |∆z| 2 +(sρ) 2 |∇y| 2 + (sρ) 2 |∇z| 2 + (sρ) 4 |y| 2 + (sρ) 4 |z| 2 )dxdt

	≤ κ 7 (s) y 2 W 2,1 2 (ω T ) + κ	Ω T	e -2sηω (|f ∂ t V | 2 + |g∂ t U | 2 )dxdt.	(4.45)

  (|∂ t u| 2 + |∂ t v| 2 + |∂ t w| 2 + |∆u| 2 + |∆v| 2 + |∆w| 2 +(sρ) 2 |∇u| 2 + (sρ) 2 |∇v| 2 + (sρ) 2 |∇w| 2 + (sρ) 4 u 2 + (sρ) 4 v 2 + (sρ) 4 w 2 )dxdtProof Setting z = a 12 v + a 13 w, we rewrite (5.50) as∂ t u = ∆u + a 11 u + z + f in Ω T , ∂ t z = ∆z + A • ∇z + az + eu + B • ∇v + bv + G in Ω T , ∂ t v = ∆v + a 21 u + dv + cz + g in Ω T ,33 + a 12 a 23 + ∂ t a 13 -∆a 13 a 13 , b = a 12 a 22 + a 13 a 32 + ∂ t a 12 -∆a 12 + 2∇a 13 • ∇ a 12 a 13 -a 12 a 13 (a 12 a 23 + a 13 a 33 + ∂ t a 13 -∆a 13 ), c = a 23 a 13 , d = a 22 -a 12 a 23 a 13 , e = a 21 a 12 + a 31 a 13

	where	≤ κ 1 (s)( u 2 W 4,2 2 (ω T ) + f 2 W 2,1 A = -2 ∇a 13 a 13 , B = -2∇a 12 + 2 on Σ T , a 12 a 13 2 (ω u = v = z = 0 a = 2 |∇a 13 | 2 13 a 2 + a	∇a 13 ,	(5.51)

T ) + g 2 L 2 (ω T ) + h 2 L 2 (ω T ) )

+κ

Ω T (|f 2 | + |g| 2 + |h| 2 )e -2sηω dxdt

for all s ≥ s 0 and (u, v, w) satisfying (5.50). Here we set

η ω (x, t) = α ω (x) t(Tt)

.

  Ω) with β ω ′ > 0 on Ω such thatΩ T (sρ) -1 e -2sη ω ′ (|∂ t u| 2 + |∂ t z| 2 + |∆u| 2 + |∆z| 2 +(sρ) 2 |∇u| 2 + (sρ) 2 |∇z| 2 + (sρ) 4 u 2 + (sρ) 4 z 2 )dxdt + |eu| 2 )e -2sη ω ′ dxdt + κ 2 Ω T (f 2 + G 2 )e -2sη ω ′ dxdt |B • ∇b + bv| 2 e -2sη ω ′ dxdt + κ 2 -1 e -2sη ω ′ (|∂ t z| 2 + |∂ t v| 2 + |∆z| 2 + |∆v| 2 +(sρ) 2 |∇z| 2 + (sρ) 2 |∇v| 2 + (sρ) 4 z 2 + (sρ) 4 v 2 )dxdt ≤ κ 3 (s)( z 2 G| 2 + |a 21 u + g| 2 )e -2sη ω ′ dxdt(5.53)

	and	≤ κ 2 (z 2 +κ 2 Ω T Ω T (sρ) W 2,1 2 (ω ′ T ) + eu + G 2 L 2 (ω ′ T ) ) + κ 2	Ω T	ω ′ T (|eu + (sρ) -1 (u 2 + z 2 )e -2sη ω ′ dxdt	(5.52)
		Ω T

  [START_REF] Ammar-Khodja | Null controllability of some reaction-diffusion systems with one control force[END_REF]. By(5.52) and(5.53), in terms of Assumption 5.2 (a), (d) and (e), we haveΩ T (sρ) -1 e -2sη ω ′ (|∂ t u| 2 + |∂ t z| 2 + |∂ t v| 2 + |∆u| 2 + |∆z| 2 + |∆v| 2 +(sρ) 2 |∇u| 2 + (sρ) 2 |∇z| 2 + (sρ) 2 |∇v| 2 + (sρ) 4 u 2 + (sρ) 4 z 2 + (sρ) 4 v 2 )dxdt

(ω T ) (1.2)

Acknowledgements. Most part of this paper was written during the stays of the fourth named author in 2007 at Université de Provence and he thanks the invitation.

The fourth author was partially supported by l'Agence Nationale de la Recherche under grant ANR JC07 183284.

Therefore with fear of confusion, we may denote a -M and d -M by a and d respectively. Then (u, v) (H 1 (Ω)) 2 ≤ κ 1 L(u, v) (L 2 (Ω)) 2 , (u, v) ∈ D(L).

(4.33)

Here and henceforth κ j > 0 denote generic constants which depend on Ω, M ,

We can prove Lemma 4.3 Under assumption (4.28), there exists a constant κ 2 > 0 such that

Proof of Lemma 4.3 The proof is done by the classical regularity property for the Dirichlet problem for the Poisson equation (e.g., Theorem 8.13 in Gilbarg and Trudinger [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]) and given here for completeness.

We recall (4.29) and we set

. By the elliptic regularity (e.g., Theorem 8.13 in [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]) in the Dirichlet problem for ∆u = f , we have

and

Hence by (4.33), we have

Therefore in (5.48), we can choose ( Ũ1 , Ṽ1 ) = (p j , q j ), 1 ≤ j ≤ 4 to construct h 1 , h 2 , h 3 , h 4 satisfying (5.49).

Carleman estimate for a 3×3 reaction-diffusion system with one observation

We consider now a 3 × 3 reaction-diffusion system

(5.50)

We will assume Assumption 5.2

Here we assume that all the coefficients are independent of t.

Then

Theorem 5.4 Under Assumption 5.2, for all ε > 0, T > 0, (u 0 , v 0 , w 0 ) ∈ (L 2 (Ω)) 3 and (u 1 , v 1 , w 1 ) ∈ (L 2 (Ω)) 3 , there exists f ∈ L 2 (ω T ) such that the corresponding solution of (5.55) satisfies

Similarly to section 4, we can apply the Carleman estimate of Theorem 5.3 for determining the nine coefficients a ij , 1 ≤ i, j ≤ 3 by suitably repeated observations of only one component u and we will here omit further details.