Observability and control of parabolic equations on networks with loops

Network theory can be useful for studying complex systems such as those that arise, for example, in physical sciences, engineering, economics and sociology. In this paper, we prove the observability of parabolic equations on networks with loops. By using a novel Carleman inequality, we find that the observability of the entire network can be achieved under certain hypothesis about the position of the observation domain. The main difficulty we tackle, due to the existence of loops, is to avoid entering into a circular fallacy, notably in the construction of the auxiliary function for the Carleman inequality. The difficulty is overcome with a careful treatment of the boundary terms on the junctions. Finally, we use the observability to prove the null controllability of the network and to obtain the Lipschitz stability for an inverse problem consisting on retrieving a stationary potential in the parabolic equation from measurements on the observation domain.


Presentation of the problem and state of the art
Network theory can be useful for studying complex systems such as those that arise, for example, in physical sciences, engineering, economics and sociology. These systems can be modeled as networks, also known as metric graphs, and their elements and interactions or links are identified respectively by vertices and edges. During the last decades, the use of networks has been helpful and effective, among others, in the study of pipes, neural systems (the brain can be thought of as a network of neurons), the flow of traffic on roads, the global economy and the human circulatory system (see, for example, [9,Chapter 9], [11,22,43,55]).
In this work, we consider the propagation of heat on a network with loops. We seek to control these networks by acting in its interior with a source term, and to estimate the solutions with an observation domain located in the interior of the network. Indeed, the main purpose of this research is to extend the results of [38] to networks with • The first results on the study of the null controllability of the heat equation date back to the 1970s, when the controllability is derived from that of the wave equation. In [57], boundary controllability is proved for parabolic PDEs using some observability estimates of hyperbolic equations. Then, in [53,54] a null controllability result is given for the heat equation using the transmutation method and proving more accurate bounds of the controllability cost. The transmutation method relates the null controllability of the heat equation to the exact controllability of the wave equation in a direct way, different from the indirect complex and harmonic analysis method used in [57]. Similar methods allow to obtain more precise results, as: [23], where the estimates on the cost of controllability are improved assuming the geometric control condition, [16], where a lower bound of the reachable set of the one-dimensional heat equation is obtained and, [17], where better bounds were obtained for the 1D heat equation. There are other results such as [60], where the work of [53] is improved by giving new estimates on the norm of the operator associating the minimal norm control to any initial state driving the system to zero. • In [47] in 1995, the control to zero of the heat equation is constructed thanks to some spectral inequalities proved there by using Carleman inequalities and duality arguments. Based on this method, several studies have been carried out: -The null controllability is studied when the control domain is a measurable set with positive measure in [2,62,65]. -New observability inequalities are presented in [3], where bounded Lipschitz locally star-shaped domains are studied, and with these inequalities the controllability is obtained. -Appropriate observability inequalities are proved and null controllability of parabolic equations with fourth or higher order derivatives is obtained in [24,25].
• A new method is provided in 1996 in [35], where an observability inequality is proved using global Carleman parabolic inequalities and energy methods. It extends the result of null controllability to parabolic equations whose second order coefficient depends on space and time variables. This outcome also allowed other researchers to later obtain controllability results for more general parabolic linear and non-linear equations as in [31,32,35]. Additional examples are given when the diffusion coefficient is not continuous in [7,20,46]. Moreover, it is well-known that this method can be generalized to cubes and to any Cartesian product of C 2 domains (see, for instance, [36]). More recently, the heat equation in pseudo-cylindrical domains has been studied in [4]. • Finally, the flatness approach introduced in [34] provides the possibility to prove the null controllability of the heat equation in a bounded cylinder in [51], giving an explicit and very regular control. The flatness approach parametrizes the solution of the equation and the control by the derivatives of a flat output. This technique is first introduced for parabolic one-dimensional equations in [44] for motion planning.
The paper is organized as follows. In Sect. 1.2 we recall basic definitions from network theory and its functional framework. In Sect. 1.3, we introduce the control problem based on a parabolic system that models the dynamics of the flux in a network with loops, we next set up the required hypotheses for the observation domain in order to control those networks, and finally state the main results regarding the null controllability and the existence of a feedback control. In Sect. 1.4, we complete the introduction by presenting a result that gives us a Lipschitz stability to obtain the solution of an inverse problem related to our parabolic system. Then, in Sect. 2 we prove an observability inequality for our system. In particular, in Sect. 2.1 we construct explicitly the necessary auxiliary function for the Carleman inequality, proved later in Sect. 2.2. Lastly, in Sect. 3 we present applications of the previous Carleman inequality to the null and feedback controllability (see Sect. 3.1) and to an inverse problem (see Sect. 3.2), and we conclude with some open problems related to Control and Network research areas (see Sect. 3.3).

Basic definitions
We first define some concepts related to graph theory that we use in this work. Let G = (V, E) be a graph.
• An edge e ∈ E that is incident to the vertices v andṽ ∈ V is expressed as e = vṽ, where v andṽ are the ends of e. The set of ends of e is denoted by V(e).
Similarly, for every vertex v ∈ V, E(v) denotes the set of edges incident to v.
The degree of a vertex v ∈ V, denoted by d(v), is |E(v)|. • V 0 = {v : |E(v)| ≥ 2} denotes the set of inner vertices, and V ∂ = V\V 0 denotes the set of boundary vertices of the graph.
If all the vertices are distinct, it is called a path, and if all the vertices are distinct except v 0 = v n it is called a cycle.
• A graph is connected if there is a walk joining each pair of vertices.
We define a network as a tuple G = (V, E, I), where V is a finite set of vertices, E ⊂ V × V is the set of edges, and I is the identification of each edge e = vṽ and its ends v andṽ with a closed interval [0, L e ] and the ends 0 and L e respectively. Formally, the identification can be viewed as a function from E to X × V × V, where X is the set of compact intervals. Notably, where v is the end of e identified with 0, andṽ is the other end of e, which we identify with L e . This definition is also referred in the literature as metric graph. With the identification I, for every edge e ∈ E, we define the following numbers for the vertex v identified with 0 and for the vertexṽ identified with L e : n e (v) = −1 and n e (ṽ) = 1.
This allows to define the operator ∂ n e (v) y = n e (v)∂ x y e (v), which can be shorten to ∂ n e y when the vertex is clear. Usually, we make a small abuse of notation and do not write the identification I explicitly when we denote the network G.
In this paper we work in the functional spaces W k, p pw (E), which denotes the set of functions that belong to W k, p (e) for all e ∈ E, k ∈ N and 1 ≤ p ≤ ∞, and

The controllability result
The problem that we study here is the dynamics of the flux and the control, which can be modelled by the following parabolic system: (1.1) In this model y denotes the flux of the heat on the entire network. Throughout this paper, we denote the restriction of a function to an edge e by adding the superscript e. In addition, a and μ are positive coefficients and b and c are coefficients which characterize the properties of the pipes of the network (roughness or properties of the heat flux, for example). Moreover, γ is a real coefficient measuring the flux of the heat on junctions, and ω ⊂ E is the control domain. Here, when writing ω ⊂ E we make a small abuse of notation to mean ω ⊂ ∪ e∈E e. In addition, by coefficients we mean functions which model the properties of the systems like the heat diffusivity and, unless stated otherwise, depend on the time and spatial variables. It is trivial to prove in system (1.1) the usual energy estimations in L 2 (0, T ; H 1 (E))∩ C 0 ([0, T ]; L 2 (E)) and regularity result in L 2 (0, T ; H 2 pw (E)) ∩ H 1 (0, T ; L 2 (E)) for parabolic equations. This can be done by multiplying the first equation of the system by y and y t and integrating it in (0, T ) × E (see [21] for a particular case).
In order to solve the controllability and inverse problems with respect to the parabolic system (1.1), we study the observability properties of the adjoint system, which is given by: System (1.2) might not be observable unless the control domain intersects a sufficient number of edges. In particular, in order to avoid some of those non-observable cases, we assume that the control domain intersects a sufficient number of edges: Moreover, we suppose that there exists a function u : {e ∈ E : e ∩ ω = ∅} → V 0 such that: Roughly speaking, the state of the equation in the edge e is controlled by ω if e ∩ ω = ∅, and by u(e) otherwise, which is controlled by the rest of the adjacent edges. Identifying the right hypothesis, in the sense that allows us to prove the results without being too restrictive, is not trivial and is one of the contributions of our paper. Indeed, the main breakthrough with respect to the previous work, and notably [38], is to make sure that we do not enter a circular reasoning fallacy. This is done with Hypothesis 1, as the proof of the controllability follows in a fluid way. . In addition, we consider ω = e 1 ∪e 2 ∪e 3 (see Fig. 1). It is clear that ω does not intersects the central cycle, but there exists a function that satisfies Items 2 and 3 of Hypothesis 1, for instance u(e 4 ) = v 5 , u(e 5 ) = v 6 and u(e 6 ) = v 4 . The parabolic system (1.1) may not be approximately controllable on such a network. For instance, if a = μ = 1, γ = 0 and b = c = 0, then the network is not approximately controllable. In fact, for those parameters the system (1.2) does not satisfy the unique continuation principle, given that there is an eigenfunction of the Laplacian null on e 1 , e 2 and e 3 ; for example: ω = e 7 ∪ e 8 ∪ e 9 (see Fig. 2). In that network, we cannot find any injective function u because |V 0 | = |{v 7 , v 8 , v 9 }| = 3 and because |{e 1 , e 2 , e 3 , e 4 , e 5 , e 6 }| = 6. The system (1.1) may not be approximately controllable on such a network. For instance, if a = μ = 1, γ = 0 and b = c = 0, then the network is not approximately controllable. In fact, for those parameters the system (1.2) does not satisfy the unique continuation principle, given that there is an eigenfunction of the Laplacian null on e 7 , e 8 and e 9 ; for example:
and the solution of (1.1) satisfies y(T, ·) = 0. Theorem 1.4 is proved in Sect. 3.1 by duality. Next, we focus on feedback control, also known as closed-loop control, in order to prove the stabilization properties for the simplified system of (1.1) when b = c = 0.
Many applications of feedback type controls can be found in industry and engineering: water level controller, air conditioner, adaptive measurement in quantum systems or servo voltage stabilizer (see [1,18,61,66]). The study and construction of feedback controls is a well-established research topic (see for instance, [13,48,58] and the survey [14]) and, in this work, we obtain a feedback control from Theorem 1.4.
In the works [58,59] we see the relation between the mild solution of Riccati equation and the null controllability of a system. The purpose of introducing Riccati equations into the study of control theory (see [42] and [63,Chapter 2]) is to design feedback controls for linear quadratic control problems. We apply Theorem 1.4 to obtain the existence of feedback controls for a simplified version of system (1.1), based mainly on the theory of Riccati equations and it is formulated in Theorem 1.5. In order to state our result we need first to define the following concepts: • A and B are bounded operators defined as: is the Banach space of all symmetric and positive operators acting • For each T > 0 and g ∈ L 2 ((0, • A * and B * denote the adjoint operators of the above mentioned corresponding operators A and B.
• A function P is called a mild solution to the Riccati equation: if for each δ ∈ (0, T ) the function P satisfies: and z 0 ∈ L 2 (E), and the equality lim 3.1 by using auxiliary results from [58].

Application to the resolution of inverse problems
Carleman estimates can also be used to obtain results in the field of inverse problems, which is an additional objective of our paper. In fact, the link between Carleman inequalities and their applications is well known. Some important references regarding this topic include [39,40], and detailed surveys are included in [6,64].
In this paper, we seek to generalize the results of [38] to systems with loops. With that purpose, let us consider the system: for μ a piecewise constant function, γ a real parameter, y 0 the initial state and p the static potential. Moreover, we denote by y[ p, y 0 ] the solution of (1.4).
Our objective is to recover the potential p by making measurements on the flux of the heat at a time t 0 > 0 and also on the observation domain ω but throughout the whole time interval (0, T ). In particular, we prove the following result: ) and such that for some t 0 ∈ (0, T ) the following estimate holds: (1.5) we have: The proof of Theorem 1.6 can be found in Sect. 3.2. In fact, this result is an easy consequence of the Carleman inequality proved in Sect. 2.2.

The observability problem
In this section we prove the observability inequality for system (1.2). With that purpose, in Sect. 2.1 we construct an auxiliary function of Fursikov-Imanuvilov type, and in Sect. 2.2, using appropriate weights, we obtain the observability of system (1.2) with a Carleman inequality.

Construction of the auxiliary function
In this section we construct an auxiliary function that is required to define the Fursikov-Imanuvilov weights in Sect. 2.2. Throughout this section we consider an open subdomainω ⊂ ω such thatω ⊂ ω and such that, for all e ∈ E,ω ∩ e = ∅ if and only if ω ∩ e = ∅.
The construction of the auxiliary function is one of the main contributions of the paper. We need to make sure that for all edge e, if e ∩ω = ∅, the maximum of η e is achieved inω and if e ∩ω = ∅, the maximum of η e is achieved on u(e), being its derivative small near u(e). As the "smallness" depends on the coefficients of the system, we get a family of auxiliary functions whose derivatives near u(e) are as small as needed, and such that they are uniformly bounded in W 2,∞ pw (E).
2. For all edges e such that e ∩ω = ∅, then: If an edge e that we identify with [0, L e ] satisfies e ∩ω = ∅, then: The proof of the existence of such function is based on an induction on the number of edges of G, and is one of the contributions of the paper. In order to prove Proposition 2.1, we first need to study the case of an edge assuming we have some restrictions on the boundary. This is done with Lemmas 2.2 and 2.3, whose proofs are standard (see [35]), but which we prove for the sake of completeness. We first study the construction of the auxiliary function for edges that have no intersection withω.
Then, there is a function η e ∈ C 2 (e) such that: Proof. For the case p = 0 it suffices to consider: Indeed, • As η e is increasing on [0, L e ] and as δ ≤ 1: Similarly, for p = L e we have the auxiliary function: Indeed, − x), so, on [0, L e ] we have that: • As η e is increasing on [0, L e ] and as δ ≤ 1: Next, in the following Lemma, we study the construction of the auxiliary function for edges which intersectω. Proof. Let us fix an interval ( p 1 , p 2 ) ⊂ω ∩ e and let us consider χ a C ∞ function such that χ(x) = 0 for all x ≤ 0 and χ(x) = 1 for all x ≥ 1. Then, it suffices to consider: It is easy to prove that η e satisfies the required properties. In particular, the second one is satisfied because ( p 1 , p 2 Remark 2.4. Of course, Lemma 2.3 may be applied in a context of fewer constraints. In that case, we proceed as follows: if we are given the value that η e takes in L e , that is, R 2 , we assume that R 1 = 0, and if we are given the value that η e takes in 0, that is, R 1 , we assume that R 2 = 0. This is relevant for proving Proposition 2.1.
Proof of Proposition 2.1. We prove the existence by induction on the number of edges on graphs that satisfy Hypothesis 1 and are connected. If they are not connected, it suffices to apply the result to each of the connected components.
The base case, a connected graph with one edge, is trivial, as a metric graph with one edge is just a segment, and the intersection withω is non-trivial (since V 0 = ∅, ω intersects the segment). Consequently, the existence on the base case follows from Lemma 2.3 applied with R 1 = R 2 = 0.
Next, we consider a network G = (V, E) with an indexing function u and suppose that the result is proved for any graph G = (V , E ) such that |E | ≤ |E| − 1 and which satisfies Hypothesis 1. We recall that each edge is identified with a segment [0, L e ] and that the identification of the edges is done in accordance with Remark 1.3; that is, whenω ∩ e = ∅, the value L e is identified with the vertex u(e). Also, in the following, if we have an indexing function u defined in all e ∈ E such thatω ∩ e = ∅ and E ⊂ E, u |E denotes the indexing function u restricted to all e ∈ E which satisfies thatω ∩ e = ∅. This is a small abuse of notation but which makes the proof more readable.
In order to prove the inductive case, we consider a division of cases based on the structure of G, whose union covers all possible graphs:  With that, we define η as follows:  Fig. 4 for an example). We may assume, by changing the indexes, that e n ∩ω = ∅. We define the function η 1 in e 1 as follows: . Finally, we define η n in e n by using Lemma 2.3 with R 1 = η n−1 (v n ) and if v n is identified with L e n . Thus, the function that satisfies the conclusion of Proposition 2.1 is the following: As the degree of v 1 is at least 2, we have v 1 ∈ V 0 (G 2 ). Thus, both G and G satisfy the inductive hypothesis with the restriction of u, so we can define an auxiliary function η 1 and η 2 in each graph, respectively. With these functions we can define a function η in G which is globally continuous: ii. If d(v 1 ) = 3 (see Fig.7 for an example), then we consider the following two graphs: for v * and additional vertex that we define, which we joint to v 1 by the new edge v 1 v * . Clearly, G satisfies Hypothesis 1 with u |{e 1 ,...,e n } , so we With that, we define η as follows: can define there a function η 1 . In addition, G satisfies Hypothesis 1 with ω replaced byω ∪ {v 1 v * } and with indexing function u |(E∪{v 1 v * })\{e 1 ,...,e n } . Thus, as G has less edges than G , we can define a function η 2 in G . With these two functions, we can define the auxiliary function for G: thanks to the inductive hypothesis. Then, with Lemma 2.2 we prolong it to e 0 with p = 0 and R = η(v −1 ) and to e 1 considering p = L e and R = η(v 2 ). Finally, we prolong it to e 1 using Lemma 2.3 with R 1 = η(v 0 ) and if v −m = v n we are in Case 4a (here d(v n ) > 2 as G is connected and have a vertex of degree at least 3), and otherwise in Case 4b.

A new Carleman inequality
The auxiliary function constructed in the previous section allows us to define the usual Fursikov-Imanuvilov weights: where η is defined in Proposition 2.1 forω an open domain compactly included in ω such that e ∩ω = ∅ if and only if e ∩ ω = ∅ for all e ∈ E, and for δ > 0 a sufficiently small parameter to be defined later on in the proof of Proposition 2.5 for absorbing boundary terms. Bearing this in mind, we state and prove the next Carleman inequality: and g ∈ L 2 (Q). Then, there is C > 0 depending on G, ω, a, and μ such that for all ϕ T ∈ L 2 (E), λ ≥ C and s ≥ C(T + T 2 ) the following inequality is satisfied:

2)
for α and ξ the weights defined in (2.1), and ϕ the solution of: In the proof we denote by o (J (s, λ)) a function depending on s, λ such that for all > 0 there is C > 0 depending on G, a, μ and ω such that if λ ≥ C and s ≥ C(T +T 2 ), then |o(J (s, λ))| < J (s, λ). Throughout the proof, we denote Q := (0, T ) × E and Q ω := (0, T ) × ω, and the constants c, C > 0 depend on G, a, μ and ω and their value may be different from line to line.
We now proceed as in [5]. The main differences are on how to deal with the boundary terms at junctions and that the observation domain is in the interior. As usual, we denote by (L i ψ) j the j-th term in the expression of L i ψ given above, for i ∈ {1, 2, 3} and j ∈ {1, 2, 3}. From (2.4), we get: (2.7) As usual, we estimate the scalar product to obtain the Carleman inequality, which is done in steps 1 and 2, and we then conclude in step 3 by using (2.7).
Step 1: Estimates in the interior. In this step we perform integrations by parts in the spirit of [5,35], but keeping track of the boundary terms appearing at the junctions. This part of the proof is standard and is presented here for the sake of completeness.
To begin with, we compute: (2.8) Secondly, integration by parts (with respect to the time variable) yields, for λ ≥ C and s ≥ C(T + T 2 ), In addition, Moreover, we can prove that: (2.12) Step 2: Estimation of the boundary terms. In this part of the proof we estimate the boundary terms I 1 , I 2 , I 3 and I 4 . The exterior vertices, if any, can be treated as in [35], since those terms appear when studying the heat equation in a segment with Dirichlet boundary conditions. However, the boundary terms at junctions require new more precise computations: Step 2.1: Estimation of I 1 and I 4 . To begin with, let us deal with the boundary term I 1 and I 4 , in (2.8) and (2.12) respectively. By using the Dirichlet boundary conditions of ψ on V ∂ we have that: (2.13) Let us now, estimate the terms in V 0 . For each interior node v ∈ V 0 as the function ξ and ψ are continuous at junctions we get that: Indeed, if v ∈ Range(u), ∂ n e (v) η e = −1 for all e ∈ E(v) and (2.14) is straightforward. Otherwise, we have to use that ∂ nẽ(v) ηẽ(v) = δ for the edgeẽ = u −1 (v) and ∂ n e (v) η e (v) = −1 for all e ∈ E(v)\ẽ, so we obtain (2.14) by choosing δ small enough just depending on a e , μ e and on the number of edges adjacent to each junction (see Item 2 of Hypothesis 1). Moreover, since combining (2.13)-(2.15), we obtain that: Step 2.2: Estimation of I 2 . To continue with, let us study the boundary term I 2 given in (2.10) for each v ∈ V, i.e.

−sλ
If v ∈ Range(u), then ∂ n e (v) η e = −1 for all e ∈ E(v) by Proposition 2.1. Consequently, we have the estimate: (2.17) Note that this includes all v ∈ V ∂ , as Range(u) ⊂ V 0 . Otherwise, let us denotẽ e = u −1 (v). From Proposition 2.1 we have that ∂ nẽ ηẽ(v) = δ and ∂ n e η e (v) = −1 for all e ∈ E(v)\{ẽ}. Thus, we have that: So, we have to absorb the boundary term of the edgeẽ. For that, we differentiate ψ = ϕe −sα and use Cauchy-Schwarz inequality to get: (2.19) We can absorb I 5 by (2.14) if δ is sufficiently small. As for the second one, using the continuity of ξ , α and ψ on the junctions, the condition on the junctions (2.3) 4 , the facts that ∂ x α = −λ∂ x ηξ , e −sα ∂ n e ϕ e = ∂ n e ψ e +s∂ n e αψ e , and that (x 1 +· · ·+ x n ) 2 ≤ n(x 2 1 + · · · + x 2 n ) for all n ∈ N and x 1 , . . . , x n ∈ R (we shall use this for n = |E(v)|): Taking δ small enough and using the continuity of ξ and ψ on junctions, we can absorb the second and third term on the right-hand side of (2.20) by (2.14). As for the first term in the right-hand side of (2.20), by taking δ small enough it can be absorbed by the first term in the right hand side of (2.18).
As an easy consequence of Proposition 2.5, we have the following result: Corollary 2.6. (Observability of the heat equation on networks with loops) Let G = (V, E) be a network satisfying Hypothesis 1, a, μ ∈ W 1,∞ ((0, T ); Then, there exists C > 0 such that for all ϕ T ∈ L 2 (E) we have the inequality: for ϕ the solution of (1.2).

Applications of the Carleman inequality and open problems
In this section we show some applications of the Carleman inequality proved in Proposition 2.5. Notably, in Sect. 3.1 we show the controllability of (1.1) and prove Theorems 1.4 and 1.5, in Sect. 3.2 we estimate the potential and prove Theorem 1.6 and, finally, in Sect. 3.3 we present some problems that remain open.

The controllability problem
We start this section giving the proof of Theorem 1.4.
Proof of Theorem 1.4. Theorem 1.4 follows from Corollary 2.6 and the Hilbert Uniqueness Method (see [35,49,50]), which assures that the null controllability is equivalent to prove an observability inequality for the adjoint equation (in our case (1.2)).
We now prove the result that provides a feedback control for the simplified case of system (1.1).

The inverse problem
Let us now prove Theorem 1.6. For that, we follow the steps in [38], which uses the technique of obtaining the inverse problem result from a Carleman inequality dating back to paper [39] in 1998: This implies that ∂ t z is a solution of: μ e ∂ n e (∂ t z e ) = γ (∂ t z), on (0, T ) × V 0 , ∂ t z(0, ·) = (q − p)y 0 , in E.
Next, we consider that, because of (1.5), Thus, by considering s ≥ C(T + T 2 ) and λ ≥ C large enough and by estimating the weight α we obtain (1.6) from (3.5).

Open problems
We highlight the following problems that remain open and may be considered for future work: • Bang-bang controls and semi-linear heat equation. One of the main applications of the Carleman parabolic inequalities are bang-bang controls and, with that, the controllability of the semi-linear heat equation. Indeed, in parabolic equations whose domains are smooth manifolds it is known that if the nonlinearity grows smoothly, then the system is controllable to trajectories (see, for example, [19,26,28,32,33] and, more recently, [27,45]). An open problem is whether or not our results may be applied to obtain the controllability of the following equation: μ e ∂ n e y e = γ y, on (0, T ) × V 0 , y(0, ·) = y 0 , in E, for G ∈ C 1 satisfying the same growth hypothesis as in [19]. Indeed, we should be careful as regularity results on networks are not as powerful as in segments, as regular solutions belong to L 2 (0, T ; H 2 pw (E)) instead of L 2 (0, T ; H 2 (E)). • Minimum number of edges on which the control acts. Another open problem is the characterization of the minimum number of edges where the control domain has to be positioned so that the system (1.1) is controllable. The construction of algorithms for that purpose also remains an open problem. • Non-controllable cases. The counterexamples in Remarks 1.1 and 1.2 show that system (1.1) is not controllable for arbitrary coefficients, but it suggests that such problems only arise for some critical coefficients when the quantity of controls is small. Since Carleman inequalities do not depend on the coefficients, it is likely that it is not the right tool for approaching this problem. A possible approach is to use spectral-oriented results, like those in [8,10]. Additionally, a possible solution is to use the characterization of controllability obtained in [41]. • Less regular coefficients. To obtain controllability results with less regular coefficients, for instance, when they all belong to L ∞ . This may be done, for instance, using the techniques presented in [29,30].