E Panteley
email: elena.panteley@cnrs.fr.

A Loría

Strict Lyapunov functions for dynamic consensus in linear systems interconnected over directed graphs

Maitreyee Dutta Elena Panteley Antonio Loría Srikant Sukumar

Abstract-We study dynamic consensus for general networked (homogeneous) linear autonomous systems, that is, it is only assumed that they are stabilizable. Dynamic consensus pertains to a general form of consensus in which, as a result of the systems' interactions, they exhibit a rich collective dynamic behavior. This generalizes the classical consensus paradigm in which case all systems stabilize to a common equilibrium point. Our main statements apply to systems interconnected over generic directed connected graphs and, most significantly, the proofs are constructive. Indeed, even though our controllers are reminiscent of others previously used in the literature, to the best of our knowledge, we provide for the first time in the literature strict Lyapunov functions for fully distributed consensus over generic directed graphs.

Index Terms-Multiagent system, directed graphs, Lyapunov stability, linear systems.

I. INTRODUCTION

I N the study of the collective behavior of multiagent networked systems, a common problem studied in the literature pertains to the case in which all the systems stabilize at a common equilibrium point -see [START_REF] Ren | Distributed consensus in multivehicle cooperative control[END_REF]. The collective behavior, however, may be much richer than converging to an equilibrium. In general, it depends on the nature of the systems dynamics-they may be, e.g., linear [START_REF] Chowdhury | Synchronization networks of identical linear systems with reduced information[END_REF]- [START_REF] Scardovi | Synchronization in networks of identical linear systems[END_REF] or nonlinear [START_REF] Panteley | Synchronization and dynamic consensus of heterogeneous networked systems[END_REF], [START_REF] Li | Consensus of multi-agent systems with general linear and lipschitz nonlinear dynamics using distributed adaptive protocols[END_REF]-, on whether the networked systems are homogeneous [START_REF] Chowdhury | Synchronization networks of identical linear systems with reduced information[END_REF], [START_REF] Scardovi | Synchronization in networks of identical linear systems[END_REF] or heterogeneous [START_REF] Panteley | Synchronization and dynamic consensus of heterogeneous networked systems[END_REF], [START_REF] Lv | Distributed pi control for consensus of heterogeneous multiagent systems over directed graphs[END_REF], [START_REF] Xiao | Adaptive consensus in leader-following networks of heterogeneous linear systems[END_REF], on the nature of the interconnections graph-whether it is undirected [START_REF] Li | Consensus of multi-agent systems with general linear and lipschitz nonlinear dynamics using distributed adaptive protocols[END_REF] or directed [START_REF] Chowdhury | Synchronization networks of identical linear systems with reduced information[END_REF], [START_REF] Scardovi | Synchronization in networks of identical linear systems[END_REF]-, etc.

In this letter, we study the collective behavior of multiagent linear systems interconnected over directed graphs via a distributed consensus algorithm. The class of systems that we consider is fairly general since the only standing assumption regarding the systems' model is that it is stabilizable -cf. [START_REF] Yang | Decentralized event-triggered consensus for linear multi-agent systems under general directed graphs[END_REF], [START_REF] Li | Consensus of multi-agent systems with general linear and lipschitz nonlinear dynamics using distributed adaptive protocols[END_REF], and [START_REF] Li | Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs[END_REF]. This class covers stable systems, as in [START_REF] Chowdhury | Synchronization networks of identical linear systems with reduced information[END_REF] and [START_REF] Lv | Distributed pi control for consensus of heterogeneous multiagent systems over directed graphs[END_REF], neutrally stable systems, as in [START_REF] Scardovi | Synchronization in networks of identical linear systems[END_REF] and [START_REF] Xiao | Adaptive consensus in leader-following networks of heterogeneous linear systems[END_REF], but also unstable systems or oscillators. Therefore, the collective behavior of such systems is not bound to stabilizing over a common equilibrium point, but is much richer.

Following [START_REF] Panteley | Synchronization and dynamic consensus of heterogeneous networked systems[END_REF], we study dynamic consensus. This pertains to the case in which, due to the systems' interaction, an emerging dynamics is generated, which, roughly, corresponds to a weighted average dynamics of the interconnected systems. Then, we say that dynamic consensus is achieved if the motion of each system in the network matches asymptotically that of the emergent dynamics. For instance, a network of oscillators may behave collectively as a weighted averaged oscillator. Other works in which synchronization is considered with respect to a dynamical system, as opposed to an equilibrium point, include [START_REF] Chowdhury | Synchronization networks of identical linear systems with reduced information[END_REF] and [START_REF] Scardovi | Synchronization in networks of identical linear systems[END_REF]. The systems studied in [START_REF] Yang | Decentralized event-triggered consensus for linear multi-agent systems under general directed graphs[END_REF], [START_REF] Li | Consensus of multi-agent systems with general linear and lipschitz nonlinear dynamics using distributed adaptive protocols[END_REF], [START_REF] Li | Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs[END_REF] and [START_REF] Xiao | Adaptive consensus in leader-following networks of heterogeneous linear systems[END_REF], too, lead to reach collective behaviors since they are assumed to be merely stabilizable in [START_REF] Li | Consensus of multi-agent systems with general linear and lipschitz nonlinear dynamics using distributed adaptive protocols[END_REF], [START_REF] Li | Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs[END_REF] and [START_REF] Yang | Decentralized event-triggered consensus for linear multi-agent systems under general directed graphs[END_REF], and neutrally stable in [START_REF] Xiao | Adaptive consensus in leader-following networks of heterogeneous linear systems[END_REF].

We analyze two controllers that are reminiscent of consensus algorithms previously reported, in which the control gain is defined upon the solution to a linear matrix inequality -cf. [START_REF] Yang | Decentralized event-triggered consensus for linear multi-agent systems under general directed graphs[END_REF], [START_REF] Li | Consensus of multi-agent systems with general linear and lipschitz nonlinear dynamics using distributed adaptive protocols[END_REF], [START_REF] Li | Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs[END_REF], and [START_REF] Xiao | Adaptive consensus in leader-following networks of heterogeneous linear systems[END_REF]. The first controller, as the one proposed in [START_REF] Chen | Event-triggered coordination of multiagent systems via a Lyapunov-based approach for leaderless consensus[END_REF], relies on a constant coupling strength, so dynamic consensus is reached exponentially fast (we provide an explicit bound), but knowledge of the maximum eigenvalue of a matrix that solves a Lyapunov equation involving the Laplacian matrix is required. For the second controller this requirement is removed, so, as in [START_REF] Li | Consensus of multi-agent systems with general linear and lipschitz nonlinear dynamics using distributed adaptive protocols[END_REF] and [START_REF] Li | Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs[END_REF], it is fully distributed. We show that by rendering the coupling strength time-varying and monotonically increasing, the aforementioned information is not required. To some extent, our control law recalls that proposed in [START_REF] Li | Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs[END_REF], but even if our controller relies on the same principle, it differs from that in [START_REF] Li | Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs[END_REF] in that in the latter the coupling gain is computed dynamically in function of the norm of the network's state, akin to high-gain adaptive control.

In regards to the network's topology, as for instance in [START_REF] Yang | Decentralized event-triggered consensus for linear multi-agent systems under general directed graphs[END_REF], we consider networks interconnected over generic connected directed graphs, that is, containing a rooted directed spanning tree. In particular, the graph may consist in a spanning tree, as in [START_REF] Maghenem | Lyapunov functions for persistentlyexcited cascaded time-varying systems: application in consensus analysis[END_REF], it may consist in a directed graph contain a root node, as in [START_REF] Chowdhury | Synchronization networks of identical linear systems with reduced information[END_REF], [START_REF] Chen | Event-triggered coordination of multiagent systems via a Lyapunov-based approach for leaderless consensus[END_REF] and [START_REF] Zhang | Consensus control of linear systems with optimal performance on directed topologies[END_REF], or in rooted graphs with a specific leader, as in [START_REF] Xiao | Adaptive consensus in leader-following networks of heterogeneous linear systems[END_REF], [START_REF] Li | Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs[END_REF], [START_REF] Lv | Distributed adaptive output feedback consensus protocols for linear systems on directed graphs with a leader of bounded input[END_REF], and possibly bidirectional links among the followers, as in [START_REF] Xiao | Adaptive consensus in leader-following networks of heterogeneous linear systems[END_REF], to mention a few. Similar results to those obtained here are presented in [START_REF] Li | Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs[END_REF], but they are restricted to a leader-follower configuration, so the directed graph is assumed to have a directed spanning tree with a root node as a leader. See also [START_REF] Lv | Distributed adaptive output feedback consensus protocols for linear systems on directed graphs with a leader of bounded input[END_REF]. On the other hand, in [START_REF] Zhang | Consensus control of linear systems with optimal performance on directed topologies[END_REF] the leaderless consensus problem is addressed for directed-graph networked systems, but under the assumption that the (static) graph is balanced. Thus, in none of these references generic directed graphs are considered.

From the viewpoint of analysis, in many cases consensus among linear systems may be assessed by relying on linear algebra and graph theory. However, these methods fail for nonlinear systems -cf. [START_REF] Panteley | Synchronization and dynamic consensus of heterogeneous networked systems[END_REF], [START_REF] Li | Consensus of multi-agent systems with general linear and lipschitz nonlinear dynamics using distributed adaptive protocols[END_REF]. In this case, the consensus analysis (resp. design) problem may be broached as one of stability analysis (resp. stabilization), relying on the construction of a Lyapunov function. There are many articles in which Lyapunov functions are proposed, for multiagent systems interconnected over undirected graphs [START_REF] Li | Consensus of multi-agent systems with general linear and lipschitz nonlinear dynamics using distributed adaptive protocols[END_REF] as well as over directed graphs [START_REF] Xiao | Adaptive consensus in leader-following networks of heterogeneous linear systems[END_REF]- [START_REF] Chen | Event-triggered coordination of multiagent systems via a Lyapunov-based approach for leaderless consensus[END_REF], [START_REF] Zhang | Consensus control of linear systems with optimal performance on directed topologies[END_REF]- [START_REF] Panteley | Strict Lyapunov functions for consensus under directed connected graphs[END_REF]. Actually, our proofs follow a similar rationale as that in [START_REF] Li | Consensus of multi-agent systems with general linear and lipschitz nonlinear dynamics using distributed adaptive protocols[END_REF], but they certainly do not constitute a straightforward extension.

Furthermore, we stress that except for [START_REF] Panteley | Strict Lyapunov functions for consensus under directed connected graphs[END_REF], which is restricted to systems of the form ẍi = u i , in none of these references a strict Lyapunov function is proposed. Furthermore, we stress that except for [START_REF] Panteley | Strict Lyapunov functions for consensus under directed connected graphs[END_REF], which is restricted to systems of the form ẍi = u i , in none of these references a strict Lyapunov function (i.e., having a negative-definite derivative) is proposed. The analysis methods used most commonly in the literature relies on tools such as La Salle's invariance principle or Barbalat's lemma. The former does not apply to networks with time-varying topology and the latter does not lead to uniform asymptotic stability. Only the latter, however, guarantees robustness with respect to bounded additive disturbances. Disposing of strict Lyapunov functions is an important step to consider more general scenarrii, such as that of linear heterogeneous systems or with added nonlinearities, but they may also be useful in the study of perturbed networked systems or of robustness with respect to neglected dynamics.

Thus, to the best of our knowledge, although several aspects studied in this letter have been addressed in the literature, separately, never have they been considered simultaneously. Beyond the generality of the network topology as well as in the class of systems that we consider, our primary contribution resides in the construction of strict Lyapunov functions for multiagent systems in such a general scenario. Indeed, we are unaware of strict Lyapunov functions for generic linear systems interconnected over arbitrary connected directed graphs. The construction is based on that for second-order integrators, proposed in [START_REF] Panteley | Strict Lyapunov functions for consensus under directed connected graphs[END_REF], and as a byproduct of our main results, we provide explicit exponential stability bounds on the synchronization error trajectories. We show explicitly how the speed of convergence depends on the maximum eigenvalue of the matrix that solves an algebraic Riccati equation with unity weighting matrix.

The remainder of this Letter is organized as follows. In the next section we present the problem statement and its solution.

In Section III we present our main results, in Section IV we provide illustrative numerical examples, before wrapping up the paper with some concluding remarks in Section V.

II. PROBLEM FORMULATION AND ITS SOLUTION

We consider N multiagent linear systems with identical dynamics,

ẋi = Ax i + Bu i , i ∈ {1, 2, • • • , N }, (1)
where x i ∈ R n , u i ∈ R p , and A ∈ R n×n , B ∈ R n×p , and N ∈ N. For the systems (1) we address the consensus problem, i.e., to guarantee that lim t→∞ |x i (t) -x j (t)| = 0 ∀i = j, under the following Standing Assumption: 1) the agents communicate over a directed connected graph, 2) the pair (A, B) is stabilizable. That is, it is assumed that the systems communicate through reliable unidirectional channels of intensity a ij ≥ 0. That is, if there exists an unidirectional edge ε ji interconnecting the node j to the node i, we have a ij > 0 and if no information flows from the jth to the ith node a ij = 0. In general, for any pair (i, j), a ij = a ji . As is customary, the graph topology may be modeled using the Laplacian matrix L = [l ij ] ∈ R N ×N , where

l ii = N j=1,j =i a ij , l ij = -a ij , i = j. (2)
The first part of the Standing assumption is that the directedinterconnection graph contains a rooted spanning tree. The latter is a necessary condition for consensus in such networks [START_REF] Ren | Consensus seeking in multiagent systems under dynamically changing interaction topologies[END_REF] and it means that there exists a subgraph containing a node called root which has no incoming edge and from which all nodes may be reached. It seems important to stress that we do not assume that the graph contains a root node with no incoming edges, as in a leader-follower configuration [START_REF] Xiao | Adaptive consensus in leader-following networks of heterogeneous linear systems[END_REF], [START_REF] Li | Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs[END_REF], [START_REF] Lv | Distributed adaptive output feedback consensus protocols for linear systems on directed graphs with a leader of bounded input[END_REF]. Under the assumption that the graph is connected we have the following.

Lemma 1: [START_REF] Ren | Consensus seeking in multiagent systems under dynamically changing interaction topologies[END_REF], [START_REF] Ren | Distributed consensus in multi-vehicle cooperative control[END_REF] If the directed graph G has a rooted spanning tree then its associated Laplacian matrix has a simple zero eigenvalue with 1 N := 1, • • • , 1 as its eigenvector of dimension N and all of the remaining eigenvalues lie in the open right half plane.

Another important characteristic of the Laplacian corresponding to a directed graph is that it is not symmetric. This is significant because it adds considerable difficulty to the task of constructing strict Lyapunov functions to study consensus. Yet, the following statement holds.

Lemma 2: [START_REF] Panteley | Strict Lyapunov functions for consensus under directed connected graphs[END_REF] Let G be a directed graph of order N and L ∈ R N ×N be the associated non-symmetric Laplacian matrix. Then, the following statements are equivalent:

1) the graph G has a spanning tree, 2) for any matrix

Q L ∈ R N ×N , Q L = Q L > 0
and for α > 0, there exists matrix P = P > 0 such that

P L + L P = Q L -α[P 1 N v l + v l 1 N P ], (3)
where v l is the left eigenvector associated to the single zero eigenvalue of L. Under the standing assumptions above, we propose two solutions to the consensus problem. Both involve the use of a control input of the form

u i = -c(•)F N j=1 a ij (x i -x j ), (4)
where c(•) > 0 is a coupling weight that may be constant or a time-varying function c : R ≥0 → R >0 and F ∈ R p×n is the feedback matrix defined as F = B M , where M is solution to the Riccati equation

M A + A M -M BB M = -Q 0 , (5)
for any given Q 0 = Q 0 > 0 -cf. [START_REF] Yang | Decentralized event-triggered consensus for linear multi-agent systems under general directed graphs[END_REF], [START_REF] Li | Consensus of multi-agent systems with general linear and lipschitz nonlinear dynamics using distributed adaptive protocols[END_REF], [START_REF] Xiao | Adaptive consensus in leader-following networks of heterogeneous linear systems[END_REF]- [START_REF] Chen | Event-triggered coordination of multiagent systems via a Lyapunov-based approach for leaderless consensus[END_REF]. The existence of such matrix M is guaranteed by the assumption that the pair (A, B) is stabilizable -cf. [START_REF] Ledyaev | On analytical solutions of matrix Riccati equations[END_REF], [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. Indeed, in this case, for any symmetric positive definite matrices Q and R ∈ R n×n , there exists a positive definite matrix M = M ∈ R n×n that satisfies the matrix algebraic Riccati equation

M A + A M -M BR -1 B M = -Q.
Thus, the existence of the control parameters c and F is guaranteed by Lemmata 1-2 and the standing assumption.

III. CONSENSUS ANALYSIS: MAIN RESULTS

Our main statements, which are presented in this section farther below, establish consensus of the systems (1) under the control law (4) and, more significantly, provide a strict Lyapunov function for the closed-loop system. In the first statement it is assumed that the coupling strength c is a constant majorating the largest eigenvalue of P in (3) -cf. [START_REF] Yang | Decentralized event-triggered consensus for linear multi-agent systems under general directed graphs[END_REF], [START_REF] Xiao | Adaptive consensus in leader-following networks of heterogeneous linear systems[END_REF], [START_REF] Chen | Event-triggered coordination of multiagent systems via a Lyapunov-based approach for leaderless consensus[END_REF], [START_REF] Zhang | Consensus control of linear systems with optimal performance on directed topologies[END_REF]. Hence, the computation of the appropriate gain c relies on the solution P to Eq. ( 3) and, indirectly, on the knowledge of L. The second statement relaxes this dependence by introducing a time-varying, strictly increasing, gain t → c(t).

A. The networked system's equations

The rest of the paper is devoted to the analysis of the consensus manifold {x i = x j } for all i, j ≤ N for the systems (1) under the control law (4). To that end, we write the closedloop equations in the compact matrix form

ẋ = (I N ⊗ A)x + (I N ⊗ B)u, (6)
where '⊗' denotes the Kronecker product, I N denotes the identity matrix of dimension R N × R N , and

u := [u 1 • • • u N ] is given by u = -(L ⊗ cF )x, (7)
so, replacing [START_REF] Lv | Distributed pi control for consensus of heterogeneous multiagent systems over directed graphs[END_REF] in [START_REF] Li | Consensus of multi-agent systems with general linear and lipschitz nonlinear dynamics using distributed adaptive protocols[END_REF], we obtain

ẋ = [(I N ⊗ A) -(L ⊗ cBF )]x. (8)
Then, we recall that, in contrast to systems that have asymptotically stable equilibria, in general, for the multiagent system (8) consensus, if it takes place, is dynamic. That is, on the consensus manifold, on one hand a collective dynamic behavior arises and, on the other, the synchronization errors with respect to such behavior, converge to zero. To analyze the dynamic consensus we follow the framework laid in [START_REF] Panteley | Synchronization and dynamic consensus of heterogeneous networked systems[END_REF].

To that end, we start by recalling a suitable convertible transformation that maps the space of the states x into two orthogonal spaces, one containing the states of the "averaged" states x m and one containing the synchronization errors e. More precisely, we have

x m e = T x, T := v l ⊗ I n (I N -1 N v l ) ⊗ I n . (9)
It is important to remark that the matrix T is invertible and it exists under the mild assumption that the Laplacian has a simple eigenvalue equal to zero and all others have positive real parts. That is, after Lemma 2 under the assumption that the graph is directed and connected. We remark that the state x m = (v l ⊗I n )x may be regarded as a weighted average of the individual systems' states and the synchronization errors are defined relative to it, i.e.,

e =      e 1 e 2
. . .

e N      =       x 1 - N j=1 v lj x j x 2 - N j=1 v lj x j . . . x N - N j=1 v lj x j       =      x 1 -x m x 2 -x m . . . x N -x m      . (10)
Therefore the collective behavior of the networked system is completely described by the dynamics of x m and of e. These are computed by differentiating on both sides of the first equality in (9) and using [START_REF] Xiao | Adaptive consensus in leader-following networks of heterogeneous linear systems[END_REF]. For the mean-field dynamics we obtain

ẋm = (v l ⊗ I n )[(I N ⊗ A) -(L ⊗ cBF )]x = (v l ⊗ A)x = Ax m , (11)
-cf. [START_REF] Chowdhury | Synchronization networks of identical linear systems with reduced information[END_REF], [START_REF] Scardovi | Synchronization in networks of identical linear systems[END_REF]. This dynamics, which is inherent to the network, is clearly influenced by the systems dynamics.

On the other hand, the dynamics of the synchronization errors e, yield

ė = [(I N -1 N v l ) ⊗ A -(L -1 N v l L) ⊗ cBF )]x. (12)
Then, since v l is a left eigenvector of L, we have L v l = 0 N and L1 N = 0 N . Using the latter in [START_REF] Zhang | Consensus control of linear systems with optimal performance on directed topologies[END_REF] we obtain

ė = [(I N ⊗ A) -(L ⊗ cBF )][(I N -1 N v l ) ⊗ I n ]x. (13)
Hence, replacing e = [(I N -1 N v l ) ⊗ I n ]x -cf. Eq. ( 9), in [START_REF] Lv | Distributed adaptive output feedback consensus protocols for linear systems on directed graphs with a leader of bounded input[END_REF], we obtain

ė = [(I N ⊗ A) -(L ⊗ cBF )]e. (14)
In the sequel, we establish exponential stability of {e = 0} for ( 14) and provide a strict Lyapunov function for this system under the condition that c is positive and either constant or a strictly increasing function of time. Exponential stability of {e = 0} implies that all the systems synchronize and their motions tend to that of the emergent dynamics [START_REF] Maghenem | Lyapunov functions for persistentlyexcited cascaded time-varying systems: application in consensus analysis[END_REF]; note that the behavior of this system is purely determined by that of the original systems and the network topology (through the eigenvector v ), independently of the coupling strength.

B. Consensus with prescribed convergence rate

Remark 1 (notation): We use p m and p M to denote, respectively, the smallest and largest eigenvalues of P ; mutatis mutandis for Q, M and any other square matrices.

Proposition 1: Let P = P > 0 be a solution of (3) with Q L = I N and an arbitrary α > 0 and let M be the solution of (5) for any given Q 0 = Q 0 > 0. Consider N identical linear systems (1), with (A, B) stabilizable, in closed loop with (4) with c ≥ p M , F := B M , and let the coefficients a ij ≥ 0 be such that they generate, through (2), a non-symmetric Laplacian matrix L that has a simple zero eigenvalue and all others have real positive parts.

Then, dynamic consensus is achieved for the multiagent closed-loop system (8) and the synchronization errors satisfy

|e(t)| ≤ κ|e(0)|e -γt t ≥ 0, (15)
where

κ := m M p M m m p m , γ := q 0m 2m M , ( 16
)
and q 0m is the smallest eigenvalue of Q 0 -cf. Remark 1.

Proof: Consider the Lyapunov function candidate

V (e) = e [P ⊗ M ]e, (17)
which is positive definite in the synchronization errors e, as defined in ( 10), but it is not for all e ∈ R nN . Indeed,

m m p m |e| 2 ≤ V (e) ≤ p M m M |e| 2 (18)
(only) for all e = [ (I N -1 N v l ) ⊗ I n ]x -cf. Eq. ( 9). Then, we use the identity (A ⊗ B)(C ⊗ D) = AC ⊗ BD, where A, B, and D are of suitable dimensions so that one can get the matrix products AC and BD, as well as F = B M , to compute the total derivative of V along the trajectories of [START_REF] Panteley | Strict Lyapunov functions for consensus under directed connected graphs[END_REF]. We obtain

V (e) = e P ⊗ [M A + A M ] -c[P L + L P ] ⊗ M BB M e,
so, in turn, using [START_REF] Yang | Decentralized event-triggered consensus for linear multi-agent systems under general directed graphs[END_REF] with

Q L = I N , it follows that V (e) = e P ⊗ [M A + A M ] -c I N -α[P 1 N v l + v l 1 N P ] ⊗ M BB M e. ( 19
)
However, we remark that

[αP 1 N v l ⊗ I n ]e = [αP (1 N v l -1 N v l ) ⊗ I n ]x = 0 N n ,
so using the latter in [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] we obtain

V (e) = e P ⊗ [M A + A M ] -cI N ⊗ M BB M e.
Now, since P is positive definite it admits the decomposition P = CDC T where C ∈ R N ×N is orthonormal and D is the diagonal matrix having the eigenvalues of P in its diagonal. Then,

V (e) = e [C ⊗ I n ] D ⊗ [M A + A M ] -cI N ⊗ M BB M [C ⊗ I n ]e.

Next, we introduce

ẽ := [C ⊗ I n ]e = [ẽ 1 • • • ẽ N ] , so V (e) = N i=1 ẽ i λ i (D)[M A + A M ] -cM BB M ẽi V (e) ≤ p M N i=1 ẽ i M A + A M - c p M M BB M ẽi . (20)
On the other hand, by assumption, c ≥ p M . Therefore, after (5), we obtain

V (e) ≤ -p M ẽ [I N ⊗ Q 0 ]ẽ = -p M e [CC ⊗ Q 0 ]e.
Hence, since C is orthonormal, CC = I N and, consequently,

V (e) ≤ -p M e [I N ⊗ Q 0 ]e.
Global exponential stability of the manifold {e = 0} and the bound ( 15)-( 16) follow from integrating the latter inequality and using [START_REF] Ledyaev | On analytical solutions of matrix Riccati equations[END_REF].

C. Fully distributed consensus

We relax the requirement in Proposition 1 to know the maximum eigenvalue of P , which is implicit in the condition that c ≥ p M and, consequently, relies on knowledge of the Laplacian L. To that end, we redefine the coupling strength c in (7) as a "slowly" strictly increasing time-varying function.

Proposition 2: Consider the linear multiagent system (8), with (A, B) stabilizable, in closed loop with the consensus control law (4) with c : R ≥0 → R >0 strictly increasing and F and a ij as in Proposition 1. Then, the multiagent system reaches dynamic consensus exponentially.

Proof: Consider again the Lyapunov function candidate V in [START_REF] Ren | Distributed consensus in multi-vehicle cooperative control[END_REF]. Proceeding as in the proof of Proposition 1 we obtain that the total time derivative along the trajectories of ( 14) satisfies (20). Now, defining,

Q(c) := c p M M BB M -M A -A M
we see that, after ( 5) and (20),

V (e) ≤ -p M ẽ I N ⊗ [(Q(c(t)) -Q 0 ) + Q 0 ] ẽ,
so, proceeding as in the proof of Proposition 1 we obtain

V (e) ≤ -p M e [I N ⊗Q 0 ]e-p M e I N ⊗[Q(c(t))-Q 0 ] e .
Now, since t → c(t) is strictly increasing,

T := min{t ≥ 0 : c(t) ≥ p M } ∈ [0, ∞) exists. Therefore, |Q(c(t)) -Q 0 | ≤ (m M b M ) 2 := β ∀t ∈ [0, T ], while Q(c(t)) -Q 0 ≥ 0 for all t ≥ T . It follows that V (e(t)) ≤ p M β|e(t)| 2 ∀ t ∈ [0, T ] -q 0m p M |e(t)| 2 ∀ t ≥ T (21)
Then, using the inequalities in [START_REF] Ledyaev | On analytical solutions of matrix Riccati equations[END_REF] and integrating both sides of (21) we obtain

V (e(t)) ≤ e γ0t V (e(0)) ∀ t ∈ [0, T ] e -γ1(t-T ) V (e(T )), ∀ t ≥ T (22)
where γ 0 := p M β pmmm and γ 1 := q0m m M . In turn, this implies that

V (e(t)) ≤ e (γ0+γ1)t e -γ1t V (e(0)) ∀ t ∈ [0, T ] ≤ e (γ0+γ1)T V (e(0))e -γ1t , while, using V (e(T )) ≤ e γ0T V (e(0)) and the second inequality in (22), we obtain V (e(t)) ≤ e (γ0+γ1)T V (e(0))e -γ1t ∀ t ≥ T Putting the last two inequalities together and using ( 18) with e = e(t) we obtain that (15) holds with

κ := e (γ+γ0)T m M p M m m p m 1/2 , γ := q 0m 2m M . ( 23
)
Remark 2 (Convergence of the control input): For the sake of generality, in the statement of Proposition 2 there is no particular choice for the function t → c(t) which is a strictly increasing control gain. It is important to stress that the monotonicity of c does not necessarily imply that the control input grows unboundedly along the trajectories. For example, in [START_REF] Li | Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs[END_REF] a controller of the form ( 4) is also used and c is defined dynamically and grows monotonically as a function of the consensus errors. A simpler choice, stateindependent, is the slowly increasing function c(t) := ln(ε+t), which also qualifies as a suitable function for purpose of fully decentralized consensus and the control input remains bounded. Indeed, c(t) ≥ p M for all t ≥ T := e p M -ε. Hence, T ∈ [0, e p M ) exists for any ε > 0. On the other hand, the control input in [START_REF] Lv | Distributed pi control for consensus of heterogeneous multiagent systems over directed graphs[END_REF] is bounded along trajectories and, as a matter of fact, lim Then, using ( 15) with (23), we see that

|u| ≤ κ -L ⊗ B M |e(0)|c(t)e -γt , ∀ t ≥ 0.
Therefore, any strictly increasing function that does not grow faster than exponentially at the rate that the error trajectories converge γ is suitable. In particular, lim t→∞ ln(t + ε)e -γt = 0

for any ε > 0, so (2) holds.

IV. SIMULATION RESULTS

To illustrate our theoretical findings, we provide some numerical simulation results done using Matlab TM R2021a. The simulation tests are done using five harmonic oscillators modeled by Eq. ( 1), with

A = 0 1 -1 0 , B = 0 1 .
We consider two graphs with different topology. First, as in [START_REF] Xiao | Adaptive consensus in leader-following networks of heterogeneous linear systems[END_REF], [START_REF] Li | Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs[END_REF], and [START_REF] Lv | Distributed adaptive output feedback consensus protocols for linear systems on directed graphs with a leader of bounded input[END_REF], we consider a leader-follower scheme in which the graph consists in a simple rooted spanning tree. In the second case the directed graph is connected. The respective graphs and the corresponding Laplacians are provided in Figures 1 and2, below. For both cases, we use the control law in (4) with the coupling strength c defined as c(t) := ln(ε + t), with ε = 1.5. To compute the control gain F , we start by setting Q 0 = I 2 and we solve the algebraic Riccati equation (5) for M . We obtain M = 1.912 0.4142 0.4142 1.352 , which is positive definite. Then, we compute the controller gain, F = B T M , which yields F = [0.4142 1.352]. Now, for the sake of fair comparison, we use the same initial conditions for both cases:

a 43 = 1 a 53 = 1 L =       0 0 0 0 0 -1 1 0 0 0 -1 0 1 0 0 0 0 -1 1 0 0 0 -1 0 1      
a 43 = 3 a 53 = 1 a 23 = 1 a 15 = 2 L =       2 0 0 0 -2 -5 6 -1 0 0 -1 0 1 0 0 0 0 -3 3 0 0 0 -1 0 1      
x 1 (0) = [ 1 -1 1.5 2 3 ] , x 2 (0) = [ 2 -2 0.5 3 3.5 ] .
For the case of leader-follower network as shown in Figure 1, the left eigenvector v l associated with zero eigenvalue of the graph's Laplacian is given by v l = 1 0 0 0 0 . This corroborates that the first agent is the singular root node in the network. So, the emergent dynamics corresponds to that of the first agent-see Figure 3. From the evolution of individual state variables (represented by dashed lines), it can be concluded that all the trajectories converge to that of the mean-field system (represented by a solid blue line). The control input subplot in Figure 3 shows that the control input dies down to zero as the multi-agent system reaches consensus.

For the case of connected directed graph, the left eigenvector associated with the zero eigenvalue of the graph's Laplacian is given by v l = 2 7 0 1 7 0 4 7 , so the mean-field state x m := (v l ⊗ I n )x corresponds to a weighted linear combination of state variables of the agents that can transfer information to all other agents, which in this case, are the first, third, and fifth agents. This is reflected in the fact that the first, third and fifth elements of v l are non-zero. The simulation plots for this case are portrayed in Figure 4.

The states x 1 and x 2 converge to their mean-field values, showed as solid blue lines in Figure 4, so the corresponding synchronization errors converge to zero. As expected from the emergent dynamics [START_REF] Maghenem | Lyapunov functions for persistentlyexcited cascaded time-varying systems: application in consensus analysis[END_REF], the steady state oscillator is a weighted averaged oscillator influenced by the nodes in the network that can transfer information to all the remaining nodes. The control input sub-plot is shown in Figure 4 which dies down to zero as synchronization dies down to zero.

V. CONCLUSION

The collective behaviour of "general" linear systems interconnected over a directed graph may be complex. Via a change of coordinates that depends on the network topology (the eigenvectors of the Laplacian matrix) it is possible to exhibit the dichotomous character of the resulting motion. Two variables belonging to orthogonal spaces appear, one corresponding to the synchronization errors and another to a "weighted-average" dynamics. The synchronization errors converging to zero is known as dynamic consensus.

The results presented in this letter are fairly general, as they establish dynamic consensus for merely stabilizable linear systems interconnected over arbitrary connected directed graphs. The most important contribution, however, is to provide strict Lyapunov functions (in the synchronization errors space). This is significant because it may serve as basis to extend our results to other interesting scenarios. For instance, to consider linear systems with certain degree of heterogeneity or with added nonlinearities. Disposing of strict Lyapunov functions may also be useful in the study of perturbed networked systems, of robustness with respect to neglected dynamics, and networks with time-varying topology. Such topics are currently under investigation.

  t→∞ |u(e(t))| = 0. This follows from the previous proof and using the explicitly exponential bounds on the synchronization errors. More precisely, note that |u| = c(t)[-L ⊗ B M ]x = |c(t)[-L ⊗ B M ]e|, so, along the synchronization-error trajectories, |u(e(t))| ≤ c(t)| -L ⊗ B M ||e(t)|.

Fig. 1 .

 1 Fig. 1. Example 1: spanning-tree graph and corresponding Laplacian

Fig. 2 .

 2 Fig. 2. Example 2: connected graph and corresponding Laplacian

Fig. 3 .

 3 Fig. 3. Distributed consensus of x 1 and x 2 and the control input when the communication graph has a spanning tree.

Fig. 4 .

 4 Fig. 4. Distributed consensus of x 1 and x 2 and the control input when the communication graph is connected.

This work was supported by CEFIPRA under the grant number 6001-A. The work of E. Panteley and A. Loría was also supported by the French ANR via project HANDY, contract number ANR-18-CE40-0010.

M.