Emmanuel Nuño

Antonio Loría

Tonatiuh Henández

Mohamed Maghenem

Elena Panteley

Distributed Consensus-formation of Force-controlled Nonholonomic Robots with Time-varying Delays

Keywords: Nonholonomic vehicles, formation control, consensus control, systems with time-delays

We solve the leaderless consensus-formation control problem for nonholonomic robots under the influence of time-varying communication delays and via smooth feedback. The control objective is to ensure that both the Cartesian positions of the vehicles and their orientations on the plane converge to a given position in a formation pattern, relatively to an a priori unknown barycenter and a common orientation, respectively. Our controller, being of proportional-plus-damping type and smooth timevarying, is easy to implement. Furthermore, it relies on δ-persistency of excitation to overcome the well-known obstacle that for nonholonomic systems a set-point is not stabilizable by smooth time-invariant feedback. We establish asymptotic convergence of the tracking errors and we provide some simulation results that support our theoretical findings.

Introduction

For multiagent systems the well-studied consensus control problem consists in making all the systems' states converge to a common equilibrium [START_REF] Ren | Distributed consensus in multivehicle cooperative control[END_REF][START_REF] Jadbabaie | Coordination of groups of mobile autonomous agents using nearest neighbor rules[END_REF][START_REF] Olfati-Saber | Consensus problems in networks of agents with switching topology and timedelays[END_REF]. For mobile robots, because several of these can obviously not occupy the same position simultaneously, consensus must be formulated as a problem of formation control, that is, in which the robots stabilize near an equilibrium point while forming a geometric pattern [START_REF] Lin | Necessary and sufficient graphical conditions for formation control of unicycles[END_REF]. We may distinguish two consensus-formation problems: the leader-follower, in which the agreement equilibrium point is given as a desired value to a set of follower agents in the network [START_REF] Maghenem | Stability and stabilization of networked varying systems[END_REF], and the leaderless, in which such common value results from the network's topology [START_REF] Cao | Distributed Coordination of Multiagent Networks: Emergent Problems, Models, and Issues[END_REF][START_REF] Wang | Consensus of networked mechanical systems with communication delays: A unified framework[END_REF][START_REF] Hatanaka | Passivity-Based Control and Estimation in Networked Robotics[END_REF]Dong and Farrell, 2008a). Furthermore, it is important to distinguish works in which both Cartesian coordinates and orientation are controlled [START_REF] Dimarogonas | On the rendezvous problem for multiple nonholonomic agents[END_REF][START_REF] Maghenem | Stability and stabilization of networked varying systems[END_REF] from those in which orientation is neglected [START_REF] Lin | Necessary and sufficient graphical conditions for formation control of unicycles[END_REF][START_REF] Roza | A smooth distributed feedback for global rendezvous of unicycles[END_REF]. In this paper we focus on the leaderless fullconsensus problem, that is, the set-point is not given a priori and consensus is reached both in Cartesian positions and orientation angles.

Email address: emmanuel.nuno@cucei.udg.mx (Emmanuel Nuño).

Another important distinction lies in the model considered. It can be of first order, in which case either only the kinematics resulting from the nonholonomic constraints is considered [START_REF] Lin | Necessary and sufficient graphical conditions for formation control of unicycles[END_REF][START_REF] Roza | A smooth distributed feedback for global rendezvous of unicycles[END_REF][START_REF] Peng | Distributed consensus-based formation control for multiple nonholonomic mobile robots with a specified reference trajectory[END_REF]Dong and Farrell, 2008a) or it may be simplified even further to a point-mass on the plane, in which case the consensus problem boils down to that of consensus of simple integrators; a problem completely solved under various scenarios [START_REF] Ren | Distributed consensus in multivehicle cooperative control[END_REF][START_REF] Ren | On consensus algorithms for double-integrator dynamics[END_REF][START_REF] Olfati-Saber | Consensus problems in networks of agents with switching topology and timedelays[END_REF]. A more realistic model, however, is force-controlled; this model is of second order and, besides the nonholonomy and underactuation, it presents the difficulty of controlling a higher relativedegree system. This is the model that we consider in this paper; see also [START_REF] Do | A global output-feedback controller for simultaneous tracking and stabilization of unicycle-type mobile robots[END_REF][START_REF] Bautista-Castillo | Consensus-based formation control for multiple nonholonomic robots[END_REF][START_REF] Maghenem | Consensus of multi-agent systems with nonholonomic restrictions via Lyapunov's direct method[END_REF]. From a control viewpoint, the nonholonomy of the robot imposes significant difficulty to set-point stabilization; it is well known that these systems are not stabilizable via time-invariant smooth feedback [START_REF] Brockett | Asymptotic stability and feedback stabilization[END_REF]; for multiagent systems, necessary conditions for consensus are laid in [START_REF] Lin | Necessary and sufficient graphical conditions for formation control of unicycles[END_REF]. Thus, consensus of multiagent nonholonomic systems is solvable either via discontinuous time-invariant control [START_REF] Dimarogonas | On the rendezvous problem for multiple nonholonomic agents[END_REF] or time-varying smooth feedback [START_REF] Yang | Smooth time-varying formation control of multiple nonholonomic agents[END_REF][START_REF] Jin | Collision-free formation and heading consensus of nonholonomic robots as a pose regulation problem[END_REF]. From an implementation viewpoint, there are many kinds of controllers that may be designed depending on the available sensors and their capabilities. In this article we assume that the robots are equipped with global-measurement sensors and they transmit and receive measurements from "neighbor" robots that are interconnected over a network. It is most sensible to assume that communication flows in both directions (hence the network topology is undirected) and delays are time-varying [START_REF] Hatanaka | Passivity-Based Control and Estimation in Networked Robotics[END_REF][START_REF] Abdessameud | Leaderfollower synchronization of Euler-Lagrange systems with time-varying leader trajectory and constrained discretetime communication[END_REF]. Such scenario is common, for instance, in the case that a group of robots are initially too far apart for relative-measurement sensors to be of any use. When within range, however, relative measurements may result more practical than global ones in missions of formation-tracking control (after the rendez-vous has been achieved). Relative-measurement based control is considered, for instance, in [START_REF] Roza | A smooth distributed feedback for global rendezvous of unicycles[END_REF], but without communication delays and neglecting to control the orientation. Delays in the communication have been considered, e.g., in (Dong and Farrell, 2008b;[START_REF] Maghenem | Stability and stabilization of networked varying systems[END_REF]. In the former delays are considered to be constant and in the latter they are time-varying, but the control gains, as in [START_REF] Roza | A smooth distributed feedback for global rendezvous of unicycles[END_REF], must satisfy a condition that requires global connectivity information. Thus, in this paper we consider the full-consensus problem for second-order (force-controlled) nonholonomic systems interconnected over an undirected graph and with time-varying measurement delays. We propose a smooth controller of proportional-plus-damping type that achieves the control goal via a δ-persistentlyexciting (δ-PE) term -cf. [START_REF] Loría | A new persistency-ofexcitation condition for UGAS of NLTV systems: Application to stabilization of nonholonomic systems[END_REF]. Our controller is completely distributed as the control gains depend only on the actual strength of the interconnection for each agent. To the best of our knowledge, this problem, under the conditions considered here, has only been considered in [START_REF] Maghenem | Stability and stabilization of networked varying systems[END_REF]. Other important aspects, such as collision avoidance [START_REF] Jin | Collision-free formation and heading consensus of nonholonomic robots as a pose regulation problem[END_REF] and robustness with respect to external disturbances [START_REF] Ajorlou | Distributed consensus control of unicycle agents in the presence of external disturbances[END_REF][START_REF] Maghenem | A robust δpersistently exciting controller for leader-follower trackingagreement of multiple vehicles[END_REF] are out of scope in this work. The rest of the paper is organized as follows. In the next section we present the model and formulate the control problems that we address. In Section 3 we present our main theoretical results; these are illustrated via numerical simulations in Section 4. We conclude with some remarks in Section 5.

Model and problem formulation

> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > θ i 2 R 2 r x i y i
Fig. 1. Schematics of a differential wheeled mobile robot.

Consider a group of N force-controlled nonholonomic vehicles, like the one depicted in Fig. 1, each of which is modeled by the usual unicycle kinematics equations and two added integrators, that is, for each i ≤ N , we have

żi = ϕ i (θ i )v i , ϕ i (θ i ) := [cos(θ i ) sin(θ i )] (1a) vi = u vi , u vi := 1 r i m i [τ i1 + τ i2 ] (1b) θi = ω i (2a) ωi = u ωi , u ωi := 2R i I i r i [τ i1 -τ i2 ], (2b)
where z i := [x i , y i ] ∈ R 2 are the Cartesian coordinates of the robot on the plane, θ i ∈ R denotes its orientation, v i and ω i are the linear and angular velocities of the center of mass, respectively, m i is the mass, I i is the moment of inertia, r i the radius of the wheels, R i is the distance between point Q i and the wheels, u vi and u ωi are the control inputs, and τ ij are the torques at the wheels, that is, for the purpose of control implementation, we have

τ i1 τ i2 = r i 2 m i I i /2R i m i -I i /2R i u vi u ωi . (3)
The leaderless consensus-formation control problem, which we solve in this paper, involves making the robots acquire a desired formation pattern relatively to a barycenter with coordinates z c := [x c y c ] and a common orientation, denoted θ c . It is assumed that both, z c and θ c are unknown; they result from the topology of the interconnection graph and the multiagent system's initial conditions. The formation is determined by defining, for each robot, a constant vector denoted

δ i ∈ R 2 , δ i := [δ xi δ yi ]
and the relative position errors zi := z i -δ i . Then, the control goal is to steer zi → z c and θ i → θ c for all i ≤ N . We assume that each robot possesses position and velocity sensors and is able to communicate them to a number of neighbors through a relatively reliable network. More precisely, we assume that (A1) the interconnection graph is undirected, static, and connected.

Hence, the interconnection graph may be modeled by a constant Laplacian matrix,

L := [ ij ] ∈ R N ×N , where ij =    k∈Ni a ik i = k -a ik i = k, (4)
N i ⊂ Z is the set of indexes corresponding to robots transmitting information to the ith robot, a ik > 0 if k ∈ N i and a ik = 0 otherwise. By construction, L1 N = 0, where

1 N = [1 • • • 1]
and, after Assumption A1, L is symmetric, it has a unique zero-eigenvalue, and the rest of its spectrum is strictly positive. Thus, rank(L) = N -1.

On the other hand, although the network interconnections are assumed to be static, the network is logically as-sumed to be wireless. This has as a natural consequence that the communications are affected by time-varying delays [START_REF] Hatanaka | Passivity-Based Control and Estimation in Networked Robotics[END_REF][START_REF] Abdessameud | Leaderfollower synchronization of Euler-Lagrange systems with time-varying leader trajectory and constrained discretetime communication[END_REF]. Hence, we also assume that (A2) the communication from the jth to the ith robot is subject to a variable time-delay denoted T ji (t) that is bounded by a known upper-bound T ji ≥ 0 and has bounded time-derivatives.

Then, the control problem consists in designing u vi and u ωi so that lim

t→∞ v i (t) = 0, lim t→∞ ω i (t) = 0, (5)
lim t→∞ zi (t) = z c , lim t→∞ θ i (t) = θ c , (6)
for all i ≤ N .

Remark 1 Being inherently a set-point stabilization problem, leaderless consensus differs greatly from other formation control problems, notably from moving-leaderfollower consensus, which is intrinsically a tracking control problem. Firstly, in clear contrast with the case of holonomic systems, set-point stabilization cannot be considered as a particular case of tractory tracking [START_REF] Lizárraga | Obstructions to the existence of universal stabilizers for smooth control systems[END_REF]. Secondly, from a systems theory viewpoint, an important distinction lies in that nonholonomic systems are controllable about a trajectory provided that the leader velocity is nonzero [START_REF] De Luca | Control of Wheeled Mobile Robots: An Experimental Overview, volume 270 of Lecture Notes in Control and Information Sciences, chapter in "Ramsete[END_REF]. While this is relevant in tracking-control scenarii, in the present setting the main difficulty resides in that a set-point is not asymptotically stabilizable by smooth feedback, unless the latter is time-varying.

Main result

The main contribution in this paper is to provide an original solution to the problem previously stated: full leaderless consensus control of multiple second-order nonholonomic systems.

For the purpose of control design and analysis, we start by reformulating the control goal in (6) as lim t→∞ e zi (t) = 0, lim t→∞ e θi (t) = 0 (7) for all i ≤ N , where

e zi := j∈Ni a ij zi -zj (t -T ji (t)) (8)
denotes the consensus errors in terms of the Cartesian coordinates of the ith robot and the measurements received from its N i neighbours, with different time-varying delays, and

e θi := j∈Ni a ij θ i -θ j (t -T ji (t)) (9)
denotes the consensus error in terms of the orientation angles, also affected by time-varying delays.

The control approach is based on designing a controller for each part of the system's dynamics separately. One for the dynamics corresponding to the translational part, Eq. ( 1), and another one for the orientation-angles dynamics. Since the respective systems (1) and ( 2) are of second order it is natural to design controllers of proportional-derivative type -cf. [START_REF] Ren | Distributed consensus in multivehicle cooperative control[END_REF]. Nonetheless, the system being subject to nonholonomic constraints, a set-point is not stabilizable via time-invariant smooth feedback [START_REF] Brockett | Asymptotic stability and feedback stabilization[END_REF]. Furthermore, because it is also desired to reach consensus in the orientation angles, controllers tailored for pointmass models do not apply either. Thus, our controller is of proportional-derivative type and it is smooth timevarying.

First, for the translational dynamics of the ith robot we introduce the nonlinear control law -cf. [START_REF] Nuño | Consensus in networks of nonidentical Euler-Lagrange systems using P+d controllers[END_REF],

u vi = -p vi ϕ i (θ i ) e zi -d vi v i , (10
) where the parameters p vi and d vi are, respectively, the so-called proportional and derivative gains and are positive constants by design. The derivative term is simply proportional to the forward velocities v i ∈ R but, in view of the robot's kinematics, the proportional-feedback term, -p vi ϕ i (θ i ) e zi , which depends on the consensus errors e zi ∈ R 2 , involves the rank-deficient vector ϕ i (θ i ). Some readers may see here an analogy between the proportional-feedback term and the Jacobian-transpose controller used for robot manipulators, in which the Jacobian is not required to have full rank to achieve regulation. Then, the closed-loop equation, corresponding to the translational dynamics, is

Σ v : żi = ϕ i (θ i )v i vi = -d vi v i -p vi ϕ i (θ i ) e zi . (11)
In contrast to the case of an ordinary double integrator the system (11) has multiple equilibria, other than points including e zi = 0. These equilibria correspond to points in the set

E v := {z i , θ i , v i , e zi : v i = 0 ∧ ϕ i (θ i ) e zi = 0}
. This, in addition to the fact that set-points are not stabilizable for nonholonomic systems, via smooth timeinvariant feedback, considerably increases the difficulty of the consensus problem, relatively to that for linear systems or even systems with holonomic constraints. The so-needed external function of time is introduced through the second control law, at the orientation-angle dynamics. Let

u ωi = -p ωi e θi -d ωi ω i + α i (t, θ i , e zi ), (12)
where p ωi and d ωi are positive gains. Clearly, the first two terms on the right-hand side of ( 12) are also of the proportional-derivative type. The function α i , which introduces the needed time-varying terms, is defined as

α i (t, θ i , e zi ) := k αi f i (t)ϕ i (θ i ) ⊥ e zi , (13)
where

k αi > 0, ϕ i (θ i ) ⊥ = [-sin(θ i ) cos(θ i )] is the anni- hilator of ϕ i , i.e., ϕ i (θ i ) ϕ i (θ i ) ⊥ = ϕ i (θ i ) ⊥ ϕ i (θ i ) = 0,
and

f i ∈ C 2 (R ≥0 , R) satisfies, by construction, f i , ḟi , fi ∈ L ∞ , lim
t→∞ f i (t) = 0, and lim t→∞ ḟi (t) = 0. The stabilization mechanism of u ωi may be explained by observing the closed-loop equations relative to the orientation-angle dynamics,

Σ ω : θi = ω i ωi = -d ωi ω i -p ωi e θi + α i (t, θ i , e zi ). (14)
With α i ≡ 0, these equations correspond to those of a stable second-order time-invariant system with equilibria in E ω := {θ i , ω i : e θi = 0, ω i = 0}. Hence, without the term α i in the controller, complete system's trajectories may belong to E ω ∩ E v which includes, but is not restricted to, the consensus manifold. On the other hand, because ϕ i (θ i ) and ϕ i (θ i ) ⊥ are orthogonal, we have ϕ i (θ i ) ⊥ e zi = 0 for all points in E v \{v i = 0, e zi = 0}. Now, in view of the assumptions imposed on f i in (13), the function α i evaluated along complete trajectories belonging to E v cannot vanish. Thus, α i may be seen as a "controlled perturbation" that persistently impedes all the trajectories, except the trivial one, to remain in

E ω ∩ E v .
Remark 2 The excitation property of α i is reminiscent of δ-persistency of excitation [START_REF] Loría | A nested Matrosov theorem and persistency of excitation for uniform convergence in stable non-autonomous systems[END_REF]. See also [START_REF] Loría | A new persistency-ofexcitation condition for UGAS of NLTV systems: Application to stabilization of nonholonomic systems[END_REF] where the so-called δ-persistently-exciting controllers were introduced for stabilization of nonholonomic systems.

Our main result emerges from the previous rationale.

Proposition 1 Consider the system (1)-( 2), under Assumptions A1 and A2, in closed loop with (10), ( 12)-( 13) and with f i as defined above. Then, the leaderless consensus control goal is achieved, that is (5) and (6) hold, provided that

d vi > 1 2 p vi j∈Ni a ij β i + T 2 ij β j (15) d ωi > 1 2 p ωi j∈Ni a ij ε i + T 2 ij ε j (16)
for some β i , ε i > 0, for all i ≤ N .

Proof of Proposition 1

The proof relies on Barbalat's lemma and standard signal-chasing arguments. We start by showing that v i → 0 and that the solutions of Σ v are bounded. To that end, consider the Lyapunov-Krasovskii functional

V := i≤N   1 2p vi v 2 i + 1 4 j∈Ni a ij |z i -zj | 2 + 1 2β i j∈Ni a ij T ji 0 T ji t t+σ v j (η) 2 dηdσ  
whose total derivative, along the trajectories of (11),

yields V = - i≤N   d vi p vi v 2 i + ż i j∈Ni a ij (z i -zj (t -T ji (t))) - 1 2 j∈Ni a ij ( żi -żj ) (z i -zj ) - 1 2β i j∈Ni a ij T 2 ji v 2 j -T ji t t-T ji v 2 j (σ)dσ   .
Then, we use

zj -zj (t -T ji (t)) = t t-Tji(t) żj (σ)dσ,
and Lemma 6.1 from [START_REF] Cao | Distributed Coordination of Multiagent Networks: Emergent Problems, Models, and Issues[END_REF] to obtain, under Assumption A1,

1 2 i≤N j∈Ni a ij ( żi -żj ) (z i -z j ) = i≤N j∈Ni a ij ż i (z i -z j ). Therefore, V = - i≤N   d vi p vi v 2 i + j∈Ni a ij ż i t t-Tji(t) żj (σ)dσ - 1 2β i j∈Ni a ij T 2 ji v 2 j -T ji t t-T ji v 2 j (σ)dσ   .
Next, we apply Young's and Cauchy-Schwarz' inequalities on the second right-hand-term to obtain

-ż i t t-Tji żj (σ)dσ ≤ β i 2 | żi | 2 + 1 2β i t t-Tji żj (σ)dσ 2 ≤ β i 2 | żi | 2 + T ji 2β i t t-T ji | żj (σ)| 2 dσ, for any β i > 0. Now, since ϕ i ϕ i = 1, we have | żi | 2 = v 2 i and, in turn, V ≤ - i≤N   d vi p vi - β i 2 l ii v 2 i - j∈Ni a ij T 2 ji 2β i v 2 j   ,
where l ii := j∈Ni a ij is the ith element in the diagonal of the Laplacian matrix. Now, following [START_REF] Nuño | Consensus in networks of nonidentical Euler-Lagrange systems using P+d controllers[END_REF] and defining s(

v 2 i ) := [v 2 1 • • • v 2 N ] and Ψ =         dv1 pv1 -β1 2 l 11 - T 2 21 2β1 a 12 . . . - T 2 N 1 2β1 a 1N - T 2 12 2β2 a 21 dv2 pv2 -β2 2 l 22 . . . - T 2 N 2 2β2 a 2N . . . . . . . . . . . . - T 2 1N 2β N a N 1 - T 2 2N 2β N a N 2 . . . d vN p vN -β N 2 l N N         , we obtain V ≤ -1 N Ψs(v 2 i ) or, equivalently, V ≤ - i≤N   d vi p vi - j∈Ni a ij β i 2 + T 2 ij 2β j   v 2 i .
Thus, after (15), it follows that there exists λ i > 0 such that V ≤ -

i≤N λ i v 2 i .
This implies that v i ∈ L 2 and, since V is positive definite and radially unbounded with respect to v i and zi -zj , the latter belong to L ∞ and, consequently, e zi ∈ L ∞ . Now, since ϕ i is uniformly bounded, it follows from (11) that vi ∈ L ∞ and, therefore, lim

t→∞ v i (t) = 0.
So far, we have established the first limit in ( 5). Now, we analyze the behavior of the solutions of ( 14) with the aim at establishing the second limit. To that end, consider the system Σ ω with (for the time-being) α i ≡ 0 and the Lyapunov-Krasovskii functional

W := i≤N   1 2p ωi ω 2 i + 1 4 j∈Ni a ij (θ i -θ j ) 2 + 1 2ε i j∈Ni a ij T ji 0 -T ji t t+σ ω 2 j (η)dηdσ   ,
where ε i > 0. Proceeding as we did previously for the differentiation of V, we obtain

Ẇ ≤ - i≤N   d ωi p ωi - ε i 2 l ii ω 2 i - j∈Ni a ij T 2 ji 2ε i ω 2 j   , so Ẇ ≤ - i≤N   d ωi p ωi - j∈Ni a ij ε i 2 + T 2 ij 2ε j   ω 2 i .
Therefore, after (16), it follows that there exists λ i > 0 such that Ẇ ≤ -

i≤N λ i ω 2 i .
Since W is positive definite and radially unbounded with regards to ω i and θ

i -θ j , it that ω i ∈ L 2 ∩ L ∞ and θ i -θ j ∈ L ∞ . This, in turn, implies that ωi ∈ L ∞ so, after Barbalat's Lemma, lim t→∞ ω i (t) = 0.
We proceed now to establish the limit in (6). To that end, observe that, since

lim t→∞ t 0 ωi (σ)dσ = lim t→∞ ω i (t) -ω i (0) = ω i (0),
it follows, from Barbalat's lemma, that lim t→∞ ωi (t) = 0 if ωi is uniformly continuous. The latter holds since ωi = -1

I i p ωi ėθi + d ωi ωi
is bounded; indeed, this follows from Assumption A2 and the fact that ωi , ω i ∈ L ∞ . Now, from the second equation in ( 14), restricted to α i ≡ 0, it follows that lim t→∞ ωi (t) = 0 and lim t→∞ ω i (t) = 0 imply that lim t→∞ e θi (t) = 0 or, equivalently, that lim t→∞ j∈Ni

a ij θ i (t) -θ j (t) + t t-Tji(t) ω j (σ)dσ = 0.
Then, since ω i → 0 and T ji (t) is globally bounded (see A2), we have lim

t→∞ j∈Ni a ij [θ i (t) -θ j (t)] = 0 which implies that lim t→∞ Lθ(t) = 0, where θ =col(θ i ) ∈ R N . Thus, lim t→∞ θ i (t) = θ c . (17)
All the previous arguments hold provided that α i ≡ 0. If this is not the case, we remark that since, f i , e zi and ϕ ⊥ i are bounded along trajectories, so is α i -see Eq. ( 13). This and the fact that Σ ω is a marginally stable linear time-varying system with uniformly bounded time-delays ensure that ωi , ω i ∈ L ∞ , by Proposition 3 in [START_REF] Wang | Differential-cascade framework for consensus of networked Lagrangian systems[END_REF] and so is e θi .

The rest of the proof mostly consists in establishing that e zi → 0 which, since v i → 0, implies that zi → zc , thereby completing (6). To see the latter, note that e zi = j∈Ni a ij zi -zj + and z := z1 • • • zN respectively, and using żj (t) = ϕ j (t)v j (t), we obtain

e z = (L ⊗ I 2 )z + col   j∈Ni a ij t t-Tji(t) ϕ j (σ)v j (σ)dσ   .
Therefore, e zi = 0 and v i = 0 imply that (L ⊗ I 2 )z = 0 which, in view of the properties of L, implies the existence of zc ∈ R 2 such that z = 1 N ⊗ zc , or zi (t) = zc for all i ≤ n. We proceed now to show that e zi → 0. To that end, we stress that, since lim

t→∞ v i (t) = 0, lim t→∞ t 0 vi (σ)dσ = lim t→∞ v i (t) -v i (0) = v i (0).
Moreover, in view of Assumption A2 and the fact that θ i , ω i , ωi , v i , vi , and e zi ∈ L ∞ , so does vi = -

p vi m i ω i ϕ ⊥ i e zi + ϕ i ėzi + d vi p vi vi . ( 18
)
After Barbalat's lemma, it follows that lim t→∞ vi (t) = 0 and, in turn after (11), also lim t→∞ ϕ i (θ(t)) e zi (t) = 0. Note, however, that the solutions to ϕ i (θ) e zi = 0 are of the form e zi = cϕ ⊥ i . Notwithstanding, so are the solutions to ϕ i (θ) ⊥ e zi = 0 on the right-hand side of (18). Therefore, the unique solution to both equations is c = 0 or, equivalently, e zi = 0, since ϕ ⊥ i ϕ ⊥ i = 1. Thus, it is left to establish that ϕ i (θ(t)) ⊥ e zi (t) → 0. To that end, we use once more Barbalat's lemma to show that vi → 0. First, note that lim Then, to show that vi is uniformly continuous we use ( 18) and φ⊥ = -ω i ϕ i to obtain ...

v i = - p vi m i ωi ϕ ⊥ i e zi + 2ω i ϕ ⊥ i ėzi -ω 2 i ϕ i e zi + ϕ i ëzi + vi p vi vi ( 19
)
which is bounded, in view of Assumption A2 and the boundedness of all functions on the right-hand side. This implies that lim t→∞ vi (t) = 0. Similar arguments apply to establish that lim t→∞ ... v i (t) = 0 and lim

t→∞ v (4) i (t) = 0, where v (4) i = - p vi m i ωi ϕ ⊥ i e zi + 3 ωi ϕ ⊥ i ėzi -ω 3 i ϕ ⊥ i e zi + ϕ i ... e zi + 3ω i ϕ ⊥ i ëzi + ωi ϕ i e zi + ω i ϕ i ėzi + d vi p vi ... v i (20) Hence, since v i , vi , vi , ... v i , v (4)
i and ϕ i e zi converge to zero, so do ėzi , ëzi and ... e zi . It follows that all the terms in ( 18), (19), and (20) converge to zero. Thus, from (18) it follows that ω i ϕ ⊥ i e zi → 0, from (19) we have ωi ϕ ⊥ i e zi → 0, and (20) implies that ωi ϕ ⊥ i e zi → 0. Next, we use ωi = -

k αi I i p ωi k αi ėθi + d ωi k αi ωi + f i ω i ϕ i e zi -f i ϕ ⊥ i ėi -ḟi ϕ ⊥ i e zi
in the first right hand side term of (20) to conclude that the term pωi kαi ėθi ϕ ⊥ i e zi -ḟi ϕ ⊥ i e zi 2 converges to zero. Since, on one hand ω i ϕ ⊥ i e zi converges to zero and, on the other hand, e zi = cϕ ⊥ i and ϕ ⊥ i ϕ ⊥ i = 1, it follows that ω i also converges to zero. In turn, we also have ėθi ϕ ⊥ i e zi → 0. Finally, the design of f i guarantees that ḟi does not vanish, so we conclude that lim t→∞ ϕ ⊥ i (θ i (t))e zi (t) = 0. Therefore α i vanishes and the limits in (5), as well as (17), follow.

Simulations

We present now some numerical simulation results to illustrate our theoretical findings. The simulations are performed using a network of six differential-drive robots interconnected as illustrated in Fig. 2, below. For simplicity, the physical parameters of all robots are taken to be equal: 10Kg for the mass; 3Kgm 2 for the moment of inertia; 0.3m of the distance between point Q and the wheels, and 0.05m for the radius of the wheels cf. [START_REF] Shojaei | Adaptive trajectory tracking control of a differential drive wheeled mobile robot[END_REF].

robot 1 2 3 4 5 6 xi 2 1 -1 -2 -1 1 δ yi 0 2 2 0 -2 -2
Fig. 2. Communication topology and hexagonal desired formation pattern (left). The latter is defined as an hexagon, determined by the distances δxi and δyi as per in the table on the right.

The interconnection time-delays are defined as a normal Gaussian distribution with a mean of 0.2s and a variance of 0.001s hence, they are bounded by T ji = 0.33s. The functions t → f i are designed using a truncated Fourier series of a square wave, that is,

f i (t) = 5 2 + 5 k=1 4 (2k -1) π sin 2k -1 2 t ∀ i ≤ N.
The robots are assumed to be initially standing still at the coordinates

x i (

Conclusions

A solution to the leaderless consensus problem for force-controlled nonholonomic robots, in the presence of time-varying measurement delays, was presented.

To cope with the difficulty imposed by the nonholonomic constraints to achieve set-point stabilization via smooth feedback, our controller is time-varying. It relies on a condition of persistency of excitation imposed on the term α i (t, θ i , e zi ) -see Eq. ( 13). Such condition is reminiscent of δ-persistency of excitation, which for nonlinear time-varying systems is necessary for uniform asymptotic stability [START_REF] Loría | A nested Matrosov theorem and persistency of excitation for uniform convergence in stable non-autonomous systems[END_REF]. Even though this property is not established here (it is subject of current research), the previous rationale justifies the little conservativism of our assumptions.

From an implementation viewpoint, it is important to stress that the controllers have a simple proportionalplus-damping structure. Hence, the stability conditions (15) and ( 16) the obvious fact that if the proportional gains are increased, to obtain a faster response, the dissipation gain must also be increased. Surprisingly, even though the system exhibits nonholonomic restrictions, the resulting conditions are the same as that reported in [START_REF] Nuño | Consensus in networks of nonidentical Euler-Lagrange systems using P+d controllers[END_REF] for the unrestricted case.

On the other hand, the choice of f i also impacts the performance of the controller. Current research focusses on moving-leader-follower consensus, on establishing stronger stability properties, and on determining a performance-driven tunning approach of the controller parameters.

  e z := e z1 • • • e zN

  dσ = lim t→∞ vi (t) -vi (0) = -vi (0).

.Fig. 3 .Fig. 4 .

 34 Fig. 3. Orientations of the six robots against time

Acknowledgment

This article is supported by the Mexican CONACyT Basic Scientific Research grant CB-282807, by the University of Paris Saclay, by the French ANR via project HANDY, number ANR-18-CE40-0010, by CEFIPRA under the grant number 6001-A, and by the Government of Russian Federation grant 074-U01.


Carmen Villegas

Evangelos Krokos

Pierre-Antoine Bouit

Juan Luis Delgado

Dirk M Guldi
email: dirk.guldi@chemie.uni-erlangen.de

Nazario Martín

Efficient Light harvesting Anionic Heptamethine Cyanine-[60] and [70]Fullerene Hybrids

New anionic fullerene-heptamethine conjugates absorbing as far as 900 nm have been synthesized and fully characterized by physicochemical means. In terms of electron transfer, we have succeded in realizing the rarely occurring scenario of a charge transfer commencing with a higher excited state. Kinetic studies reveal that charge separation and charge recombination dynamics occur in the normal and inverted Marcus region, respectively. .

Introduction

Artificial systems that mimic the elegant operations of the photosynthetic process, namely light-harvesting, light-funnelling, charge separation and charge-shift, converting light into useful electrochemical potentials, have been at the forefront of extensive research in recent years, predominantly in the area of optoelectronics. [START_REF]For some recent special issues, see[END_REF] Fullerenes, three-dimensional carbon allotropes, exhibit attributes that are particularly well suited for electron transfer processes. Videlicet, their notable electron accepting properties 2 and exceptionally low reorganization energies in electron transfer processes 3 render them excellent building blocks for electron donor-acceptor conjugates. In fact, fullerenes are known to accelerate charge separation, midst retarding charge recombination. The affluent chemistry that fullerenes feature 4 enables their covalent linking to a diverse collection of chromophorsincluding porphyrins, 4 phthalocyanines 5 or perylenes 6 that function as light harvesting antennas and electron donors.

The aforementioned systems have been thoroughly studied despite the fact that their absorptions rarely extend beyond 700 nm. Nevertheless, electron donor-acceptor systems containing near-infrared absorbing dyes are of pivotal importance because of the amount of near infrared radiation in the solar spectrum. Charge transfer efficiency, however, may be at peril when chromophores with light-harvesting abilities that span beyond 700 nm are being employed. To this end, the interactions between structure and charge transfer reactivity are expected to be as crucial as the parameters met in conventional systems, essentially the energy gap between the higher (S2) and the lowest (S1) singlet excited state as well as the lifetime of S2. This prompted us to study the electronic interactions between fullerenes and heptamethine cyanine dyes. Cyanines manifest remarkable optical properties in the infrared section of the solar spectrum and noteworthy chemical stability. These traits, which render cyanines more than adequately suited to be utilized for exploring the prospect of near-infrared photoinduced charge transfer reactions, are due to a delocalized charge on a -conjugated bridge featuring an odd number of carbons. 7 We have previously reported on molecular electron donor-acceptor conjugates containing cationic cyanines and fullerenes (Figure 1). In that study, time resolved transient absorption measurements revealed that photoexcitation is followed by charge transfer 55 evolving from higher singlet excited states of the heptamethine cyanine as electron donor to an either covalently linked C60 or C70 as electron acceptor. 8 This prominent photophysical event 9 introduced new possibilities in electron transfer reactions that could be beneficial for photovoltaics. In order to ascertain the 60 feasibility of implementing near-infrared absorbing heptamethine dyes as light harvesters for photovoltaics, molecular PV cells were prepared, in which a mixture of hepamethine dyes and PCBM as fullerene derivative was used as the only active layer. 10 Notwithstanding other strongly absorbing cyanines endowed 65 with a negative charge, we decided to focus on an anionic polymethine cyanine absorbing as far as 900 nm. These compounds, displaying maxima in the NIR region of the solar spectrum, intrigued us to probe them in conjunction with C60 (6, scheme 1) as well as C70 (7, scheme 1) as photo-and redox-active 70 building blocks and to compare their properties with their previously studied cationic analogs.

Herein we describe the preparation of new electron donor acceptor conjugates using an anionic cyanine and fullerenes. The slow internal conversion within the light harvesting polymethine 75 and the strong electronic coupling between the individual constituents are the basis for powering a charge transfer from higher excited state of the polymethine.

N N O R 1 R 1 N N O R 1 R 1 1.R 1 =C 6 H 13 ;R 2 =C 8 H 17 2.R 1 =C 6 H 13 ;R 2 =C 8 H 17 I I N R 2 N R 2

Results and Discussion

Synthesis and Characterization 10

The synthesis of the new polymethine cyanine-fullerene electron donor-acceptor conjugates 6 and 7 was carried out in two steps, starting from previously described heptamethine cyanine (3) 11 as it is depicted in Scheme 1. 8 The introduction of the phydroxybenzaldehyde unit to the chloro-heptamethine cyanine (3) was achieved from 3 and 4, by substitution of the meso-chlorine atom to form 5 (Scheme 1). Thus, reaction of formyl-cyanine 5 with either C60 or C70 in the presence of N-octylglycine and chlolorobenzene as solvent, 12 afforded 6 and 7 as deep green solids in 70 % and 40 % yields, respectively. As expected, 7 was obtained and studied as a mixture of isomers due to the multiple possibilities of addition of azomethine ylides to C70. The structures of 6 and 7 were confirmed by 1 H and 13 C-NMR. The presence of the conjugated cyano groups was confirmed by the intense band at 2207 cm -1 in the FT-IR spectra.

Optical properties

The absorption properties of compounds 5, 6, and 7 were studied in dichloromethane as solvent. All the compounds display 35 an intense absorption band in the near-infrared (ɛ  300000 L/mol/cm) centered at 896 nm, due to the presence of the anionic polymethine cyanine. 6 and 7 display also absorption bands in the UV-visible, which are the typical absorption features for both C60 and C70, respectively (Figure 2). As expected, 6 and 40 7 display a longer absorption maxima shifted towards longer wavelength, when compared to the cationic analogues 1 and 2, as already observed in previous studies on simpler related systems. 11

Electrochemistry

The electrochemical properties of 5, 6, and 7 have been studied at room temperature by means of cyclic voltammetry. As described for similar anionic cyanine heptamethines, 10 5 reveals 50 an amphoteric redox character with reversible one-electron reduction processes at -0.94 and -1.76 V. One-electron oxidation processes, on the other hand, are noted at +0.18 Vfully reversibleand 0.87 Virreversible. 6 displays additionally three reversible one-electron reductions, due to the presence of 55 C60, at -0.95, -1.32, and -1.88 V (Figure 3). In 7, the one-electron reductions for C70 were detected at -0.94, -1.31, and -1.78 V. The characteristic reductions of the anionic cyanine moiety (-0.94 and -1.76 V) are superimposed to the first and third fullerene reduction, thus increasing the intensity of these steps in comparison with the second reduction (-1.32 or -1.31 V) (Figure 3). On the oxidation side, 6 and 7 display the characteristic oxidations of the cyanine moiety (6: +0.16, +0.88 V; 7: +0.16, +0.86 V). 6 displays similar electrochemical properties to its cationic analogue 1, recorded under the same experimental conditions (see table S1). In both cases, the first reduction potential of the cyanine is close to the first reduction potential of the fullerenopyrrolidine. These data suggest that the LUMOs of both electro-active units are energetically close. The electrochemical properties of 5, 6, and 7 were also studied in benzonitrile as solvent. With these data in hands we have derived radical ion state pair energies of 1.11 eV ( 6) and 1.13 eV (7) in benzonitrile. Photophysical Studies Laser flash photolysis experiments were carried out, employing excitation wavelengths of 387 and 775 nm to photoactivate C60 or C70 and the cyanine moiety, respectively. Excitation of C60 or C70 leads to the generation of their singlet excited states with characteristic absorption maxima at 610 / 920 and 610 / 1360 nm, respectively. Intersystem crossing to the corresponding triplet manifolds is the predominant deactivation pathways of these singlet excited states. In fact, the singlet lifetimes are quite short and range from 0.7 to 1.4 ns.

Characteristics of the triplet excited states are maxima at 700 nm for C60 and 970 nm for C70. At first glance, the transient absorption spectra upon 387 nm 30 excitation of 5 in dichloromethane or benzonitrile are dominated by strong ground state bleaching centered around 920 nm (Figure 4). A closer analysis reveals that the instantaneously formed higher singlet excited state of 5 features maxima at 490, 600, and 990 nm, minima at 565, 630, 735, and 1400 nm, and broad bleach 35 between 820 and 955 nm. With a lifetime of 4.4 ps these features transform rapidly into those of the lowest singlet excited state with distinct maxima at 520, 585, 625, and 1240 nm plus distinct minima at 730 and 920 nm. Next, the decay of the singlet excited state follows within approximately 2000 ps. Population of the 40 higher lying singlet excited states was bypassed upon 775 nm excitation, which leads to the rapid formation (> 0.5 ps) of the lowest singlet excited state. The lifetime of the correspondingly formed singlet excited state as determined from a multiwavelenght analysis is 1.9 ns (5.2 x 10 8 s -1 ). In accordance with the absorption spectra that reveal a partition of 5 and C60 or C70 absorption of 1:1 for 6 and 3:1 for 7 at 387 nm see Figure 2 we note upon photoexcitation of 6 and 7 at this wavelength nearly exclusively the transients of the higher singlet excited state features of 5vide supra. Owing to the slow 20 internal conversion and vibrational relaxation in 5 the latter start to transform in benzonitrile into main maxima at 520, 545, 600, 655, 1000, 1015, 1085, and 1235 nm ( 6) as well as 520, 545, 600, 655, 1000, 1085, 1235, and 1300 nm (7)the radical ion pair statesbefore the population of the lowest singlet excited state 25 (Figures 5 andS10).

Spectroelectrochemically and pulse radiolytically generated one-electron oxidized form of 5 further support the notion of radical ion pair state formation (Figure S11). In fact, these are in perfect agreement with the transient maxima observed at 520, 600, and 1240 nm. The transient 30 features, on the other hand, at around 1015 and 1300 nm are in accordance with previous work by pulse radiolysis assigned to the one-electron reduced forms of C60 and C70, respectively. From multi-wavelength analyses we have derived charge separation (i.e., kCS1) and charge recombination dynamics (i.e., kCR) of 3 ps 35 (3.3 x 10 11 s -1 ) and 400 ps (2.5 x 10 9 s -1 ) in 6 as well as 4 ps (2.5 x 10 11 s -1 ) and 430 ps (2.3 x 10 11 s -1 ) in 7. Remarkable is the competition between charge transfer (i.e., 3 ps) and internal conversion / vibrational relaxation (i.e., 2.5 ps). In light of this kinetic competition, the radical ion pair state and the lowest 40 singlet excited state are formed with 45.5 and 54.5 %, respectively. In fact, during the early times of our femtosecond experiments the accordingly formed singlet excited states of 5 are discernablea shoulder at 520 and the 1250 nm maximum. The latter reveals shortened lifetimes of approximately 12 ps for 6 and 45 18 ps for 7 in benzonitrile, when compared to the intrinsic lifetime of 5 to generate the same radical ion pair state that originated from the higher singlet excited state. In turn, we have deduced from the latter charge separation kinetics (i.e., kCS2) of 8.3 x 10 10 s -1 for 6 and 5.5 x 10 10 s -1 for 7. Such lifetimes suggest 50 charge separation (i.e., kCS1, kCS2) and charge recombination dynamics (i.e., kCR) that are in the normal and inverted region of the Marcus parabola, respectively. Relating these charge separation kinetics to the intrinsic decay of the photoexcited cyanine moiety (5.2 x 10 8 s -1 ) we conclude charge transfer 55 efficiencies close to unity, that is, nearly 100 %. In line with reference experiments, exciting 6 and 7 at 775 nm the directly and solely formed singlet excited state features emerge at the conclusion of the photoexcitation. In particular, maxima at 520, 585, 625, and 1240 nm plus distinct minima at 730 and 920 20 nm are discernable. In contrast, to the slow singlet excited state deactivation that was seen for 5, these features are short-lived in 6 and 7. In fact, they decay (kCS2) in benzonitrile for 6 with 15 ps (6.7 x 10 10 s -1 ), while 13 ps (7.7 x 10 10 s -1 ) was derived for 7, to yield the radical ion pair state features. The latter include for 6 the 25 spectroscopic markers at 600, 1015, and 1100 nm, while 7 exhibits the markers at 600, 1100, and 1300 nm.

Conclusions

In summary, we have carried out the synthesis of a new type of fullerene derivatives bearing covalently connected an anionic 35 heptamethine cyanine moiety. The new molecule is a powerful light harvesting system absorbing at 900 nm with very high extinction coefficients. Remarkably, photoexciting at 387 nm leads to the formation of higher singlet excited states in compounds 6 and 7, which is followed by a kinetic competition 40 between the charge separation and the internal deactivation to the lowest singlet excited state. The measured values reveal that charge separation and charge recombination dynamics occur in the normal and inverted region of the Marcus parabola, respectively. Nevertheless, the charge recombination is fast a fact that is due to the presumably large reorganization energy that is imposed by the anionic heptamethine cyanine moiety.

Madrid (MADRISOLAR-2, S2009/PPQ-1533). J.L.D. thanks the MICINN of Spain for a Ramón y Cajal Fellowship

Notes and references

a Departamento de Química Orgánica,Facultad de Ciencias Químicas, Universidad Complutense de Madrid, Ciudad Universitaria s/n, 28040 ,

Figure 1 .

 1 Figure 1. Previously synthesized cyanine-C60 conjugate 1 and cyanine-C70 2.

5 Scheme 1 .

 51 Scheme 1. Synthesis of fullerene-heptamethine conjugates 6 and 7.

Figure 2 .

 2 Figure 2. Absorption spectra of 5 (blue), 6 (green), and 7 (red) 45

Figure 3 .

 3 Figure 3. Cyclic voltammograms of 6; top (reduction), bottom (oxidation) -V vs Ag/AgNO3; GCE as the working electrode; 0.1 M TBAP; o-DCB/MeCN (4:1); scan rate 100 mV/s.

Figure 4 :

 4 Figure 4: Upper partdifferential absorption spectra (visible and near-infrared) obtained upon femtosecond flash photolysis (387 nm) of 5 in benzonitrile with several time delays between 0 and 5

Figure 5 :

 5 Figure 5: Upper partdifferential absorption spectra (visible and near-infrared) obtained upon femtosecond flash photolysis (387 nm) of 6 in benzonitrile with several time delays between 0 and 22 ps at room temperaturesee Figure legend for details. Upper central partdifferential absorption spectra (visible and nearinfrared) obtained upon femtosecond flash photolysis (775 nm) of 6 in benzonitrile with several time delays between 0 and 22 ps at 10

Figure 6 :

 6 Figure 6: Energy diagram illustrating the different excitation, charge separation (kCS1, kCS2, kCS3) and charge recombination (kCR) 30

  The 1 H-NMR 25 spectrum of 6 shows, in addition to the aromatic protons, the expected protons of the pyrrolidine ring, and the presence of the vinylic protons relating to the cyanine moiety.

		Furthermore, high
		resolution MS measurements confirmed the formation of both
	30	dyads, 6 ([M] -= 1465,35757 (calcd. C106H46N7O3: 1465,36999)
		and 7 [M] -= 1585,36895 (calcd. C116H46N7O3: 1585,37000) see
		SI for details.

Acknowledgments. Forschungsgemeinschaft (Grant SFB 583), Cluster of Excellence "Engineering of Advanced Materials" is gratefully acknowledged. This work has been supported by the European Science Foundation (SOHYD, MAT2006-28170-E), the MEC of Spain (CT2008-00795/BQU, and Consolider-Ingenio 2010C-07-25200, Nanociencia Molecular) and Comunidad de


Pierre-Antoine Bouit

Carmen Villegas

Juan Luis Delgado

Pedro M Viruela

Rosendo Pou-Amérigo

Enrique Ortí
email: enrique.orti@uv.es

Nazario Martín

ExTTF-Based Dyes Absorbing over the Whole Visible Spectrum

The development of new materials able to transform sunlight into chemical energy or electricity efficiently is currently a major challenge in science. In this regard, organic dyes are an appealing choice as light harvesters provided their synthetic availability, high extinction coefficients and low contamination due to the lack of metal atoms. The recent development of organic and hybrid solar cells, that are able to reach 7% and 11% of power conversion efficiency, respectively, has pointed out the crucial need for new organic materials to build up efficient photovoltaic (PV) devices. 1 For such

applications, the targeted molecules have to fulfill several chemical and electronic parameters to be integrated into a device. Among them, chemical and thermal stability, processability, and a favorable band offset between the donor and the acceptor (either a fullerene derivative for organic solar cells or TiO2 for dye-sensitized solar cells (DSSCs)) are critical requirements. Furthermore, the capability of the molecule to efficiently collect light in a large part of the solar spectrum, especially in the visible and the near infrared (NIR) regions where the majority of the sunlight intensity is displayed, is an essential issue for these light-harvesting applications. In this context, the synthesis of push-pull chromophores, extensively studied in the field of nonlinear optics (NLO), 2 recently appeared 2 a) Choa, M. J.; Choia, D. H.; Sullivan, P. A.; Akelaitis, A. J. P.; Dalton, L. R. Prog. Polym. Sci. 2008, 33, 1013- )

 (nm)
as a promising way to build up highly efficient "all organic" and hybrid PV devices. 3 In this communication, we report the synthesis and the optical and electrochemical properties of new push-pull molecules based on 2- [9-(1,3-dithiol-2-ylidene)anthracen-10(9H)-ylidene]-1,3-dithiole (exTTF, see Scheme 1), whose absorption covers the whole visible spectrum and enters in the near-infrared region. Theoretical density functional theory (DFT) calculations were performed to get more insights into the electronic properties of these systems that appear as promising molecules for lightharvesting applications. ExTTF, a -extended analog of tetrathiafulvalene (TTF), is an organic molecule which possesses strong donor ability and singular butterfly shape. 4 During the last decade, its electronic and geometric features have been used in supramolecular chemistry 5 as well as in materials science (DSSCs, 6 charge transfer complexes, 7 molecular wires, 8 NLO, 9 etc). In order to improve the optical absorption properties of the exTTF unit in the visible and NIR range, we have designed new push-pull systems (3a and 3b, Scheme 1) in which the strongly electrondeficient tricyanofuran (TCF) group 10 has been covalently connected through a -conjugated bridge to the exTTF moiety. The attachment of SMe groups to the dithiole rings allows for a fine tuning of the electronic properties of the donor unit.

S S S S O CN CN CN R R R R S S S S O O CN CN CN R R R R 1a R = H; 1b R = SMe 3a R = H; 3b R = SMe 2
Scheme 1. Synthesis and structure of the new dyes 3a and 3b.

The synthesis of the two exTTF-based aldehydes 1a and 1b (Scheme 1) was achieved according to an already published multistep synthesis, starting from commercially available 2-(hydroxymethyl)anthraquinone (see the Supporting Information for full synthetic details). 11 Then, the TCF 12 moiety 2 was introduced by means of microwave-assisted Knoevenagel condensation to afford the targeted molecules 3a and 3b in moderate yields (46 and 41 % respectively) as dark green solids. The compounds displayed good solubility in the main organic solvent. In the 1 H NMR spectra (300 MHz, CDCl3) of 3a and 3b, the coupling constant of the vinyl bond ( 3 JHH = 16 Hz, see SI) proved the trans character of the oligoenic backbone.

Figure 1 displays the minimum-energy molecular geometry computed for 3a at the B3LYP/6-31G* level (see the SI for computational details). As expected, the exTTF fragment possesses the typical concave saddle-like shape, 13 in which the central ring of the anthracene unit folds up in a boat conformation and the dithiole rings are tilted down. As shown in Figure 1, the TCF moiety is coplanar with the ethylene bridge and the aromatic ring of the anthracene skeleton to which it is attached. This result suggests that an efficient -conjugation can operate between the donor and acceptor units in these molecules. Figure 2 shows the atomic orbital (AO) composition of the highest-occupied (HOMO-2 to HOMO) and lowestunoccupied (LUMO and LUMO+1) molecular orbitals of 3a. The HOMO (-5.17 eV) and HOMO-1 (-5.76 eV) are localized on the electron-donor exTTF unit and are calculated at lower energies than the HOMO (-4.64 eV) and HOMO-1 (-5.22 eV) of the exTTF molecule. In contrast, the LUMO (-3.22 eV) spreads over the electronacceptor TCF unit with a large participation of the ethylene bridge and is obtained at lower energies than the LUMO of TCF (-2.86 eV). Therefore, compound 3a presents a small HOMO-LUMO energy gap of 1.95 eV and low-energy charge-transfer (CT) absorption bands are to be expected in the electronic spectrum. The electronic communication between the donor and acceptor units is evidenced by the shift to lower energies of the HOMOs and by the charge transfer of 0.20e that takes place from the exTTF unit to the TCF moiety. The redox properties of 3a and 3b were examined by cyclic voltammetry (CV) and compared with those of exTTF and TCF (Figure 3, Table 1, and Table S1). 3a undergoes a quasi-reversible, two-electron oxidation at E°ox = 0.25 V vs. Ag/AgNO3, which is slightly influenced by the SMe substituents on the dithiole rings (3b: 0.35 V). The oxidation process is assigned to the oxidation of the exTTF fragment, by comparison with the reference exTTF compound (Eox 1/2 = 0.24 V). No appreciable impact on the oxidation of the exTTF unit is therefore detected despite the presence of the electron-acceptor TCF unit and in contrast with the lower energy predicted for the HOMO of 3a. Using Koopmans′ theorem, a lower-energy HOMO would lead to a more positive oxidation potential. The reason for this apparent discrepancy between experiment and theory is that Koopmans′ theorem is a one-electron approach and does not apply to the oxidation process of 3, which involves two electrons. 3a and 3b also undergo a non-reversible one electron reduction at Ered = 0.55 V, which is strongly shifted compared to the reduction of 2 (Ered = -1.17 V). The anodic shift is in agreement with the lower energy predicted for the LUMO of 3a and is due to the great conjugation introduced by the ethylene bridge (Figure 2). The absorption properties of the dyes were measured in diluted dichloromethane solution (Table 1 and Figure 4). 3a and 3b present a broad absorption covering the whole visible and entering the NIR, which can be divided into three main absorption bands. In first approximation, the lowest-energy band with absorption maxima at 635 (3a) and 628 nm (3b) was attributed to a charge transfer (CT) band, as this band appears with the insertion of the TCF acceptor unit and it displays the typical shape of a CT transition (broad low intensity). This band, situated in the red part of the visible, indicates a small HOMO-LUMO energy gap in these dyes. The optical gap (Eg opt ) obtained from the absorption onset recorded about 800 nm is 1.55 eV. The second band around 500 nm is also associated to the insertion of the TCF unit and presents a small red shift in passing from 3a to 3b. Surprisingly for a polar compound, these transitions are weakly solvatochromic without clear trend depending on the solvent polarity (see Figure S7 andS8). 14 One reason might be that the two transitions are overlapping and might have opposite solvatochromic behavior. It is important to remind that the absence of intermolecular interactions in exTTF derivatives 6 ruled out the possibility of solvent induced aggregation. Finally, a third and more intense band is observed at 420 nm. This band is in principle attributed to an electronic transition within the exTTF unit which presents an intense absorption at 428 nm. 15 These absorption properties, with exctinction coefficient superior to 5000 L.mol -1 .cm -1 from 300 to 700 nm, make compounds 3a and 3b very interesting as materials for light-harvesting applications. recorded in dichloromethane at room temperature.

To investigate the nature of the electronic transitions that give rise to the absorption bands observed in the electronic spectra, the lowest-energy singlet excited states (Sn) were calculated for 3a using the time-dependent DFT (TDDFT). 16 PBE0/6-31G** calculations predict that the absorption bands observed for 3a at 635 nm (1.95 eV) and 499 nm (2.48 eV) are due to electronic transitions to the first two excited singlets, S1 and S2, calculated at 1.75 and 2.33 eV. These states originate in the HOMOLUMO and HOMO-1LUMO monoexcitations, respectively, and imply an electron density transfer from the exTTF moiety, where the HOMO and HOMO-1 reside, to the TCF unit, where the LUMO is mainly located (see Figure 2). Calculations therefore confirm the CT nature of the two low-energy absorption bands of compounds 3. The medium intensities observed for these bands are in agreement with the oscillator strengths calculated for S1 and S2 (f = 0.17 and 0.32, respectively), and are due to the planar structure of the conjugated ethylene bridge that allows for a significant overlap between the HOMO/HOMO-1 and the LUMO.

The intense absorption band observed at 418 nm is assigned to electronic transitions to the S3 and S4 states that are computed close in energy at 2.99 eV (415 nm) 16 Vertical transitions to the lowest-energy electronic states were calculated at both the B3LYP/6-31G** and PBE0/6-31G** levels. Both calculation levels lead to identical descriptions of the electronic transitions in terms of monoelectronic excitations, but the PBE0 functional provides slightly higher excitation energies (see Table S2). and 3.12 eV (398 nm), respectively. The S3 state results from the HOMO-2LUMO excitation, that mainly implies the TCF moiety and the ethylene bridge (Figure 2), and is calculated to be very intense (f = 0.92). In contrast, the S4 state originates in the HOMOLUMO+1 excitation centered on the exTTF moiety and has a lower intensity (f = 0.28). Therefore, the intense band at 418 nm mainly results from the excitation of the TCF moiety. Excitation of the exTTF fragment contributes in a lower degree to this absorption band.

In summary, two new push-pull molecules associating -extended TTF as electron donor and tricyanofuran as electron acceptor have been synthesized. The new chromophores possess a broad optical absorption covering the visible spectrum from 300 to 800 nm. The broad absorption is due to the efficient electronic connection between the donor and acceptor units through the conjugated ethylene bridge, which determines the appearance of medium-intensity charge-transfer bands extending into the near-infrared region. These low bandgap molecules may be used as new molecular materials in the field of photovoltaics. 17 Molecular engineering is currently in progress in the laboratory in order to take advantage of the outstanding optical properties of these compounds in molecular bulk heterojunction devices 18 or as photosensitizers grafted on TiO2 in DSSCs.

  1058. b) Beverina, L.; Fu, J.; Leclercq, A.; Zojer, E.; Pacher, P.; Barlow, S.; Van Stryland, E. W.; Hagan, D. J.; Brédas, J.-L.; Marder, S. R. J. Am. Chem. Soc. 2005, New push-pull dyes featuring -extended tetrathiafulvalene (exTTF) as donor group and tricyanofuran (TCF) as acceptor group were synthesized and characterized. Their broad absorption covers the entire visible spectral range and enters the near infrared region. Electrochemistry and theoretical calculations provided an understanding of these these singular electronic properties. The new dyes are appealing candidates as light-harvesters in photovoltaic devices.

Figure 1 .

 1 Figure 1. Minimum-energy B3LYP/6-31G** structure of 3a.

Figure 2 .

 2 Figure 2. Electron density contours (0.03 e bohr -3 ) and orbital energies calculated for the HOMOs and LUMOs of 3a at the B3LYP/6-31G** level. H and L denote HOMO and LUMO, respectively.

Figure 3 .

 3 Figure 3. CV trace obtained for 3a (CH3CN, 0.1 M Bu4NClO4, v = 100 mV.s -1 , V vs. Ag/AgNO3).

Figure 4 .

 4 Figure 4. Absorption spectra of 3a (red) and 3b (blue)

Table 1 .

 1 Electrochemical and optical data for 3a and 3b

			compound	Eox 1/2 (V) a	Ered (V) a	Eg cv (eV) b	max (nm) c (L.mol -1 .cm -1 ) c
			3a		0.25		-0.56	0.81	635	8800
			3b		0.35		-0.55	0.90	628	9400
	a Measured by CV (CH3CN, 0.1 M Bu4ClO4, v = 100 mV s -1 , V vs. Ag/AgNO3). Measured in
	DCM, 10 -5 mol L -1 .
		35000
		30000
		25000
	)
	-1
	.cm	20000
	-1
	(L.mol	15000
	
		10000
		5000
		0
		300	400	500	600	700	800
				 (nm)

b

Electrochemical energy gap determined as Eox -Ered.

c

C. Reichardt Chem. Rev. 1994, 94, 2319-2358.

Guldi, D. M.; Sánchez, L.; Martín, N. J. Phys. Chem. B 2001, 105, 7139-7144.

Acknowledgments. This work has been supported by the European Science Foundation (SOHYD, MAT2006-28170-E), the MICINN of Spain (CT2008-00795, CTQ2009-08790, and Consolider-Ingenio CSD2007-00010 on Molecular Nanoscience), and Comunidad de Madrid (MADRISOLAR-2, S2009/PPQ-1533). P.A.B. thanks IMDEA-Nanociencia for a postdoctoral research grant. J.L.D. thanks the MICINN of Spain for a Ramón y Cajal Fellowship co-financed by the EU Social Funds. Supporting Information Available Synthetic procedure, complete characterizations, and computational details. This material is available free of charge via the Internet at http://pubs.acs.org.


Emmanuel Nuño

Ioannis Sarras

Antonio Loría

Mohamed Maghenem

Emmanuel Cruz-Zavala

Elena Panteley

Strict Lyapunov-Krasovskiȋ Functionals for Undirected Networks of Euler-Lagrange Systems with Time-varying Delays

Keywords: Euler-Lagrange Dynamics, Multi-Agent Systems, Time-Delays, Strict Lyapunov Functions

For an undirected network of nonidentical interconnected Euler-Lagrange systems whose communication is affected by varying time-delays that may not be differentiable, we consider the problem of establishing leaderless and leader-follower consensus via the simplest Proportional plus damping decentralized controller. The main contribution of this work is to prove that the agents' positions and velocities converge uniformly, globally, and asymptotically to a common non-specified position in the leaderless case, and to a given reference in the leader-follower case. The main results are established via Lyapunov's direct method; a Strict Lyapunov-Krasovskiȋ Functional is constructed, to the best of our knowledge, for the first time in the literature. It is shown that the resulting closed-loop system is Input-to-State Stable with regards to external additive inputs (perturbations). In turn, the separation principle applies to a certainty-equivalence controller, implemented with any globally convergent velocity estimator, such as the Immersion & Invariance observer.

Introduction

It is said that Euler-Lagrange (EL) systems achieve consensus if all their generalized positions converge to a common value. We identify two particular consensus problems: leader-follower, in which case an agreement trajectory is given as a common reference [START_REF] Abdessameud | Leader-follower synchronization of Euler-Lagrange systems with time-varying leader trajectory and constrained discrete-time communication[END_REF] and the leaderless, in which case the agents' positions converge to a common non-specified value [START_REF] Olfati-Saber | Consensus problems in networks of agents with switching topology and time-delays[END_REF][START_REF] Cao | Distributed Coordination of Multi-agent Networks: Emergent Problems, Models, and Issues[END_REF]. The practical applications of consensus of EL-agents are diverse and can be found in different fields of mechanical, electrical and electromechanical systems such as, for example, in robotics [START_REF] Hatanaka | Passivity-Based Control and Estimation in Networked Robotics[END_REF] or in control of aerial vehicles [START_REF] Abdessameud | Formation control of VTOL unmanned aerial vehicles with communication delays[END_REF]. Consensus problems may be and have been solved via decentralized feedback of the agents' neighbors' positions and, possibly, their velocities. Pioneer works addressing the consensus problem for EL systems include [START_REF] Rodriguez-Angeles | Mutual synchronization of robots via estimated state feedback: A cooperative approach[END_REF][START_REF] Chopra | Advances in Robot Control: From Everyday Physics to Human-Like Movements[END_REF][START_REF] Chung | Cooperative robot control and synchronization of Lagrangian systems[END_REF][START_REF] Chung | Cooperative robot control and concurrent synchronization of Lagrangian systems[END_REF][START_REF] Ren | Distributed leaderless consensus algorithms for networked Euler-Lagrange systems[END_REF]. Notably, the simplest consensus controller is the Proportional plus damping (P+d) scheme reported in [START_REF] Ren | Distributed leaderless consensus algorithms for networked Euler-Lagrange systems[END_REF]; moreover, it is showed in [START_REF] Ye | Distributed modelindependent consensus of Euler-Lagrange agents on directed networks[END_REF] that this controller works also with directed interconnection topologies.

Consensus control for networked-interconnected systems is, however, typically stymied by the effect of interconnection delays. This problem has been addressed, for instance, in [START_REF] Nuño | Synchronization of networks of nonidentical Euler-Lagrange systems with uncertain parameters and communication delays[END_REF][START_REF] Liu | Controlled synchronization of heterogeneous robotic manipulators in the task space[END_REF][START_REF] Liu | Consensus for multiple heterogeneous Euler-Lagrange systems with time-delay and jointly connected topologies[END_REF][START_REF] Wang | Consensus of networked mechanical systems with communication delays: A unified framework[END_REF][START_REF] Nuño | Consensus of Euler-Lagrange systems using only position measurements[END_REF][START_REF] Nuño | Achieving consensus of Euler-Lagrange agents with interconnecting delays and without velocity measurements via passivity-based control[END_REF][START_REF] Klotz | Synchronization of uncertain Euler-Lagrange systems with uncertain timevarying communication delays[END_REF]. See also [START_REF] Nuño | Consensus in networks of nonidentical Euler-Lagrange systems with variable time-delays[END_REF][START_REF] Nuño | Consensus in networks of nonidentical Euler-Lagrange systems using P+d controllers[END_REF] where the results of [START_REF] Ren | Distributed leaderless consensus algorithms for networked Euler-Lagrange systems[END_REF] are extended by considering the presence of interconnection Email address: emmanuel.nuno@cucei.udg.mx (Emmanuel Nuño) delays. A common drawback of all these works, nonetheless, is that they rely on the assumption that the timevarying delays are differentiable and that their derivative is bounded. Clearly, this imposes a practical limitation because several communication media (such as Internet) do not ensure this hypothesis. Remarkably, in [START_REF] Abdessameud | Leader-follower synchronization of Euler-Lagrange systems with time-varying leader trajectory and constrained discrete-time communication[END_REF][START_REF] Abdessameud | Synchronization of Lagrangian systems with irregular communication delays[END_REF] the leader-follower (in the former) and the leaderless (in the latter) synchronization problems are solved under the assumption that delays may be non-differentiable. Moreover, in both references it is assumed that the systems' parameters are unknown hence, the convergence of the consensus errors is ensured via adaptive control.

The main contribution of this paper is the design of an original strict Lyapunov-Krasovskiȋ functional to guarantee uniform global asymptotic consensus for a network of EL-systems controlled by decentralized P+d controllers in the presence of, possibly non-differentiable, time-varying delays and under an undirected-graph interconnection topology. With this proposal we are able to reveal new properties of the simplest available control laws (P+d) for consensus of EL-systems, despite the fact that these have been exhaustively studied and implemented in the literature (see [START_REF] Ren | Distributed leaderless consensus algorithms for networked Euler-Lagrange systems[END_REF][START_REF] Nuño | Consensus in networks of nonidentical Euler-Lagrange systems with variable time-delays[END_REF][START_REF] Nuño | Consensus in networks of nonidentical Euler-Lagrange systems using P+d controllers[END_REF][START_REF] Ye | Distributed modelindependent consensus of Euler-Lagrange agents on directed networks[END_REF]).

The importance of having a strict Lyapunov-Krasovskiȋ functional cannot be overestimated; it allows to prove uniform global asymptotic consensus. Furthermore, we establish input-to-state stability for the closed-loop system. This is in clear contrast with previous works in which only convergence is established. Indeed, the latter is a weaker property that does not guarantee robustness with respect to bounded disturbances.

We also stress that establishing input-to-state stability is well beyond pure academic interest. As we show, in addition, a byproduct of our main statements is an extension to the case in which velocity measurements are replaced by estimates obtained from a converging velocity observer that guarantees uniform exponential convergence of the observation errors, such as the Immersion & Invariance (I&I) velocity observer [START_REF] Astolfi | Nonlinear and adaptive control design and applications[END_REF], to render the delaydependent stability condition decentralized -cf. [START_REF] Nuño | Consensus in networks of nonidentical Euler-Lagrange systems with variable time-delays[END_REF][START_REF] Nuño | Consensus in networks of nonidentical Euler-Lagrange systems using P+d controllers[END_REF], and to prove Input-to-State Stability (ISS) with respect to external additive inputs (perturbations). See also [START_REF] Abdessameud | Synchronization of networked Lagrangian systems with input constraints[END_REF] where a partial-state feedback controller under constant delays is proposed.

Although we are not aware of other articles in which the consensus problem is treated under the exact same setting and where input-to-state stability is established, a clear drawback of our main results, that contrasts with [START_REF] Abdessameud | Leader-follower synchronization of Euler-Lagrange systems with time-varying leader trajectory and constrained discrete-time communication[END_REF] and [START_REF] Abdessameud | Synchronization of Lagrangian systems with irregular communication delays[END_REF], is that we assume that part of the robots' parameters are known and, more importantly, that the interconnection graph is undirected.

The rest of the paper is organized as follows. In the following section we present some background material on the systems' models and lay our main hypotheses. In Sections 3 and 4 we present our main results: on leaderless, leaderfollower, and output-feedback consensus respectively. We conclude with some remarks in Section 5.

Background

Throughout the paper, the following notation is employed. R := (-∞, ∞), R >0 := (0, ∞), R ≥0 := [0, ∞) and N := {1, 2, 3, . . . }. I n ∈ R n×n denotes the n × n identity matrix, 1 n ∈ R n defines the vector of n elements equal to one and 0 n ∈ R n is the all-zeros vector. For any x ∈ R n , ∇ x := [∂ x1 , . . . , ∂ xn ] stands for the gradient operator of a scalar function and |x| for the Euclidean norm. λ m {A} and λ M {A} are the minimum and the maximum eigenvalues of the symmetric matrix

A ∈ R n×n . N := {1, 2, ..., N } for N ∈ N. A continuous function γ : R ≥0 → R ≥0 is of class K if γ(0) = 0 and it is strictly increasing. Further, it is of class K ∞ if it is of class K and unbounded. For an absolutely continuous function φ : [-h, 0] → R n , h > 0, we define φ := max -h≤θ≤0 |φ(θ)| (1)
and we denote by C[-h, 0] the space of such functions. The norm defined in (1) is used in this paper to formulate conditions for uniform global asymptotic stability of the null solution of functional differential equations

ẋ = f (t, x t ), f (t, 0) ≡ 0, (2)
where x t is short-hand notation for x t (θ) := x(t + θ) for all θ ∈ [-h, 0], f : R ≥0 × R n × C[-h, 0] → R n is continuous and locally Lipschitz in the second argument.

Remark 1. We observe that in the first works on stability of solutions of systems (2) by N. N. Krasovskiȋ, some of which are compiled in [START_REF] Krasovskiȋ | Stability of motion[END_REF], the L 2 norm of φ(θ) is used instead of the max(•).As in [START_REF] Gu | Stability of Time-Delay Systems[END_REF], we use [START_REF] Rodriguez-Angeles | Mutual synchronization of robots via estimated state feedback: A cooperative approach[END_REF], which results convenient to formulate the following statement.

Theorem 1. -cf. [26, Theorem 1.3] 1 Let β i : R ≥0 → R ≥0 be class K ∞ functions. If there exists a continuous differentiable functional V : R ≥0 × ×C[-h, 0] → R ≥0 such that β 1 ( x t (0) ) ≤ V(t, x t ) ≤ β 2 ( x t ), (3)
and

V(t, x t ) ≤ -β 3 ( x t (0) ), (4)
then the trivial solution of ( 1) is uniformly globally asymptotically stable.

Remark 2. Functionals V satisfying the above are commonly referred to as Strict Lyapunov Krasovskiȋ functionals.

Euler-Lagrange Agents

We consider a network composed of N fully-actuated and conservative EL-agents, with n-Degrees-of-Freedom (DoF). The dynamics of the ith-agent is given by

d dt ∇ qi L i (q i , qi ) -∇ qi L i (q i , qi ) = τ i , i ∈ N , (5)
where L i (q i , qi ) is the Lagrangian function that is defined as L i (q i , qi ) := K i (q i , qi ) -U i (q i ), with K i (q i , qi ) := 1 2 q i M i (q i ) qi the kinetic energy and U i (q i ) the potential energy. q i , qi ∈ R n are the generalized position and velocity, respectively, M i (q i ) ∈ R n×n is the generalized inertia matrix, which is symmetric positive definite, and τ i ∈ R n is the control input.

The EL-equations of motion ( 5) can be written in a compact form as

M i (q i )q i + C i (q i , qi ) qi + ∇ qi U i (q i ) = τ i , (6)
where C i (q i , qi ) ∈ R n×n is the Coriolis and centrifugal forces matrix, defined via the Christoffel symbols of the first kind. We restrict to EL-agents (6) that satisfy the following assumption:

Assumption A1: There exist m 1i ∈ R >0 and m 2i ∈ R >0 such that, for all q i ∈ R n , m 1i I n ≤ M i (q i ) ≤ m 2i I n .
This assumption is ubiquitous in robot control but it also holds for other different physical systems, see [START_REF] Ortega | Passivity-based Control of Euler-Lagrange Systems: Mechanical, Electrical and Electromechanical Applications[END_REF][START_REF] Ghorbel | On the uniform boundedness of the inertia matrix of serial robot manipulators[END_REF]. Further, model (6) has the following properties [START_REF] Kelly | Control of robot manipulators in joint space[END_REF]:

Property P1: Matrix Ṁi (q i ) -2C i (q i , qi ) is skew- symmetric. Further, Ṁi (q i ) = C i (q i , qi ) + C i (q i , qi ).
Property P2: There exists k ci ∈ R >0 such that, for all

q i ∈ R n , C i (q i , qi ) qi ≤ k ci qi 2 .

Interconnection Topology

As it is customary, we use graphs to represent the interconnection topology among the N EL-agents. In particular, we employ the graph Laplacian matrix

L := {L ij } ∈ R N ×N that is defined as L ii = j∈Ni a ij and L ij = -a ij ,
where a ij > 0 if j ∈ N i and a ij = 0 otherwise. The set N i contains all the neighbors of the ith-EL-agent. Note that, by construction, L has a zero row sum. Therefore L1 N = 0 n .

We assume that the agents exchange information according to the following assumption. Assumption A2: The EL-agents interconnection graph is undirected, static and connected.

From A2, the Laplacian L is symmetric; positive semidefinite; it has a single zero-eigenvalue, with the associated eigenvector 1 N , and all of the other eigenvalues are strictly positive; and rank(L) = N -1. Further, ker(L) = α1 N , ∀α ∈ R.

We also consider the fact that the information exchange between agents is subjected to time-varying delays; for these interconnection delays we assume the following. Assumption A3: The communications, from the j-th agent to the i-th agent, is subjected to a variable timedelay T ji (t) with a known upper-bound * T ji , that is,

0 ≤ T ji (t) ≤ * T ji < ∞. (7)
Remark 3. One of the main contributions of this work is that the time-varying delay T ji (t) is not required to be differentiable.

In addition to A2, we make the following assumption for the leader-follower interconnection. Assumption A4: There is a nonempty set of followers that have a direct access to the desired constant position q .

Next, let us define a diagonal matrix A := diag{a i } ∈ R N ×N to model the leader-follower interconnections, such that a i > 0 if node i receives the leader desired position and a i = 0, otherwise. Define N as the set of all follower agents that receive the leader desired position. The following lemma, which is a special case of Lemma 3 in [START_REF] Hong | Tracking control for multi-agent consensus with an active leader and variable topology[END_REF] and Lemma 1.6 of [START_REF] Cao | Distributed Coordination of Multi-agent Networks: Emergent Problems, Models, and Issues[END_REF], provides a fundamental property of the composed Laplacian matrix L := L + A . Lemma 1. Consider the matrix A := diag{a i } ≥ 0 ∈ R N ×N and suppose that N is non-empty. Assume that A2 holds, then the matrix L is symmetric, positive definite and of full rank.

Note that A4 ensures that there is at least one agent receiving information from its neighbors.

Leaderless Consensus

We speak of leaderless consensus if there exists

q c ∈ R n such that lim t→∞ | qi (t)| = 0, lim t→∞ q i (t) = q c (8)
for all i ∈ [0, N ]. In [START_REF] Nuño | Consensus in networks of nonidentical Euler-Lagrange systems using P+d controllers[END_REF] it has been shown that the P+d controller

τ i = ∇ qi U i (q i )-d i qi -p i j∈Ni a ij q i -q j (t-T ji (t)) , (9)
where p i , d i ∈ R >0 are the proportional and the damping gains, respectively, guarantees leaderless consensus provided that the control gains satisfy

2d i > p i j∈Ni a ij α i + * T 2 ji α j , (10)
for some

α := [α 1 , . . . , α N ] ∈ R N such that α i > 0.
Nonetheless, the main statement in [START_REF] Nuño | Consensus in networks of nonidentical Euler-Lagrange systems using P+d controllers[END_REF] has two significant limitations. Firstly, that the mere convergence properties [START_REF] Hatanaka | Passivity-Based Control and Estimation in Networked Robotics[END_REF] do not guarantee robustness with respect to bounded disturbances, a property that, in contrast, is guaranteed by uniform asymptotic stability [START_REF] Krasovskiȋ | Stability of motion[END_REF]. Secondly, and more importantly, it is assumed in the latter reference that the time-delays are differentiable. As it appears, both limitations may be lifted, by using Lyapunov's direct method. Indeed, our main contribution is to provide a strict Lyapunov-Krasovskiȋ functional to establish the following statement. Proposition 1. For each i ∈ N , consider the system (6) in closed loop with [START_REF] Nuño | Synchronization of networks of nonidentical Euler-Lagrange systems with uncertain parameters and communication delays[END_REF] and let (10) hold. Then, under Assumptions A1-A3, there exists q c ∈ R n such that the equilibrium of the closed-loop system ( q, q) = (0, 1 N ⊗q c ), where q := [q 1 • • • q N ] , is uniformly globally asymptotically stable. Furthermore, the closed-loop system is Input-to-State stable with respect to additive inputs u :=

[u 1 • • • u N ] .
Proof. We start by deriving the closed-loop equations; after ( 6) and ( 9) we have

qi = -M -1 i (q i ) [C i (q i , qi ) qi + d i qi + p i e i ] -p i M -1 i (q i ) j∈Ni a ij t t-Tji(t) qj (σ)dσ, ( 11
)
where we defined the relative position errors

e i := j∈Ni a ij (q i -q j ), i ≤ N (12)
and we used the identity t t-Tji(t) qj (σ)dσ = q j -q j (t -T ji (t))

Next, we proceed to construct a Lyapunov-Krassovskiȋ functional. First, as in [START_REF] Nuño | Consensus in networks of nonidentical Euler-Lagrange systems using P+d controllers[END_REF], we introduce the energy-like function V i : R 2n → R ≥0 , defined as

V i (q i , qi ) = 1 2p i q i M i (q i ) qi + 1 4 j∈Ni a ij |q i -q j | 2
Also, using [START_REF] Nuño | Consensus of Euler-Lagrange systems using only position measurements[END_REF] we find that the derivative of V i along the trajectories of ( 11) yields Vi = -

d i p i | qi | 2 - j∈Ni a ij q i t t-Tji(t) qj (σ)dσ - 1 2 j∈Ni a ij ( qi + qj ) (q i -q j ).
Furthermore, using successively Young's and Cauchy-Schwartz inequalities we obtain that, for any α i > 0,

q i t t-Tji(t) qj (σ)dσ ≤ α i 2 | qi | 2 + 1 2α i t t-Tji(t) qj (σ)dσ 2 ≤ α i 2 | qi | 2 + * T ji 2α i t t- * Tji | qj (σ)| 2 dσ, hence, Vi ≤ j∈Ni a ij α i 2 | qi | 2 + * T ji 2α i t t- * Tji | qj (σ)| 2 dσ - d i p i | qi | 2 - 1 2 j∈Ni a ij ( qi + qj ) (q i -q j ).
Next, for each i ≤ N let the given constants on the right hand side of (10) generate d i satisfying this condition. Then, there exists δ i > 0 such that

2d i > p i j∈Ni a ij α i + * T 2 ji α j + * T ij δ i 2 . ( 14
)
Consider next the functional

W i : C[-h, 0] n → R ≥0 de- fined as W i ( qt ) := * T ji 2α i + δ i j∈Ni a ij 0 - * Tji t t+σ | qj (θ)| 2 dθdσ, (15)
where qt (θ) := q(t + θ). We note that W i ( qt ) ≥ 0 and there exists c i > 0 such that W i ( qt ) ≤ c i qt 2 -see (1). The total derivative of Ẇi yields Ẇi = * T ji

2α i + δ i j∈Ni a ij * T ji | qj (t)| 2 - t t- * Tji | qj (σ)| 2 dσ .
Next, defining E i (q i , qi , qt ) := V i (q i , qi ) + W i ( qt ) -cf.

[29, 30], we obtain Ėi ≤ -

d i p i | qi | 2 + j∈Ni a ij α i 2 | qi | 2 + * T 2 ji 2α i + δ i * T ji | qj | 2 -δ i j∈Ni a ij t t- * Tji | qj (σ)| 2 dσ - 1 2 j∈Ni a ij ( qi + qj ) (q i -q j )
and, defining E(q, q, qt ) :=

i∈

N E i (q i , qi , qt ), we obtain Ė ≤ - i∈ N j∈Ni a ij d i p i L ii - α i 2 | qi | 2 - * T 2 ji 2α i + δ i * T ji | qj | 2 - i∈ N δ i j∈Ni a ij t t- * Tji | qj (σ)| 2 dσ
where L ii is the diagonal element of the Laplacian, defined as L ii := j∈Ni a ij , and we used the fact that the Laplacian matrix is symmetric, so

i∈ N j∈Ni a ij ( qi + qj ) (q i -q j ) = 0.
Now, as in [START_REF] Nuño | Consensus in networks of nonidentical Euler-Lagrange systems using P+d controllers[END_REF], to compact the notation, we introduce

Q := | q1 | 2 • • • | qN | 2 and Ψ =      α 1 2 -d 1 p 1 L 11 . . . * T N 1 2α 1 + δ1 * TN1a1N . . . . . . . . . * T 1N 2α N + δN * T1N aN1 . . . α N 2 -d N p N L N N      , so Ė ≤ 1 N ΨQ - i∈ N δ i j∈Ni a ij t t- * Tji | qj (σ)| 2 dσ.
In view of [START_REF] Liu | Controlled synchronization of heterogeneous robotic manipulators in the task space[END_REF], for each i ≤ N , the numbers

λ i := d i p i L ii - α i 2 - j∈Ni a ij * T ij 2α j + δ i * T ij are positive and, consequently, -Ψ is diagonal dominant. Therefore, Ė ≤ - i∈ N   λ i | qi | 2 + δ i j∈Ni a ij t t- * Tji | qj (σ)| 2 dσ   (16)
that is, Ė is negative semi-definite. In order to render it negative definite we introduce the cross term U (e, q) := h(e) P -1 M(q) q (17)

where We are now ready to present our Lyapunov-Krasovskiȋ functional candidate. Let V : R ≥0 × C[- * T, 0] → R ≥0 , with * T := min{ * T ij }, be defined as

P :=diag[p i ] ⊗ I n ∈ R N n×N n ; M(q) :=blockdiag M i (q i )
V(t, x t ) = E(q(t), q(t), qt ) + U (e(t), q(t)) (18)
where x t := [e(t) q(t) q t ] and > 0. We show that V verifies the conditions of Theorem 1. To that end, we stress that U (e, q) can be also written as

U (e, q) = i∈ N 1 p i h i (e) M i (q i ) qi , with h i (e) = 1 1+|e| e i . Therefore, E(q, q, qt ) + U (e, q) = i∈ N W i + 1 4 i∈ N j∈Ni a ij |q i -q j | 2 + i∈ N 1 2p i [ qi + h i ] M i [ qi + h i ] -2 h i M i h i and the term h i M i h i = 1 (1+|e|) 2 e i M i e i admits the bound 1 p i h i M i h i ≤ m 2i p i j∈Ni a ij (q i -q j ) 2 ≤ m L j∈Ni a ij |q i -q j | 2 ,
where m := max

i∈ N m2i pi and L := max i∈ N {L ii }. Therefore, since c i qt 2 ≥ W i ≥ 0 for all i ≤ N , by setting < 1 2 m L , we find that V satisfies (3) with β 1 (s), β 2 (s) ∝ s 2 .
We compute next the total derivative of E + U . To that end, we write U = ḣ P -1 M q + h P -1 Ṁ q + Mq and we observe that

ḣ P -1 M q ≤ 2 mλ M {L}| q| 2 .
Then, using P1 and (11), the last terms of U can be written as

h P -1 Ṁ q + Mq = i∈ N 1 p i h i C i qi -d i qi -p i e i - i∈ N h i j∈Ni a ij t t-Tji(t) qj (σ)dσ.
On the other hand, in view of P2 and the boundedness of h, we have

h i C i (q i , qi ) qi ≤ k ci | qi | 2 .
In addition, invoking Young's inequality for any µ > 0, we obtain

-h i qi ≤ µ 2 |h i | 2 + 1 2µ | qi | 2 ,
and

h i j∈Ni a ij t t-Tji(t) qj (σ)dσ ≤ µ 2 |h i | 2 + 1 2µ j∈Ni a ij t t-Tji(t) qj (σ)dσ 2 , ≤ µ 2 |h i | 2 + L ii 2µ j∈Ni a ij * T ji t t- * Tji | qj (σ)| 2 dσ,
where the last inequality follows from the fact that ( ab) 2 ≤ a 2 b 2 and the Cauchy-Schwartz inequality. Moreover,

|h i | 2 ≤ 1 1 + |e| |e i | 2 (19) so U satisfies U ≤ i∈ N 2 mλ M {L} + k ci p i + d i 2µp i | qi | 2 + i∈ N µd i 2p i + µ 2 -1 1 1 + |e| |e i | 2 + i∈ N   L ii 2µ j∈Ni a ij * T ji t t- * Tji | qj (σ)| 2 dσ   .
and, in turn, it holds that for any µ satisfying µ < 2 d+1 , ensures that V satisfies (4). The first statement follows invoking Theorem 1.

V ≤ - i∈ N λ i - 2 mλ M {L} + k ci p i + d i 2µp i | qi (t)| 2 - i∈ N 1 - µd i 2p i - µ 2 1 1 + |e| |e i (t)| 2 - i∈ N   j∈Ni a ij δ i - L ii * T ji
{ * T ji } we obtain V ≤ -µ 1 | q(t)| 2 - µ 2 1 + |e(t)| |e(t)| 2 -µ 3 i∈ N j∈Ni
Next, let u i be an additive input to the system (6), which defines a passive map u i + τ i . Therefore, after the previous computations we obtain

V ≤ -µ 1 | q(t)| 2 - µ 2 1 + |e(t)| |e(t)| 2 + i∈ N 1 p i qi (t) + h i (e i (t)) u i (t).
So, using the triangle inequality and ( 19) we obtain

V ≤ - µ 1 2 | q(t)| 2 - 1 2 µ 2 1 + |e(t)| |e(t)| 2 + 1 2p 2 1 µ 1 + µ 2 |u(t)| 2 (21)
where p := min{p i }. Therefore, the system is Input-to-State stable with respect to u.

Leader-Follower Consensus

We speak of leader-follower consensus if, given an agreement position q ∈ R n which is accessible to at least one follower (but not necessarily to all of them)

lim t→∞ | qi (t)| = 0, lim t→∞ q i (t) = q (22)
for all i ∈ [0, N ].

Proposition 2. Consider the system (6) in closed-loop with

τ i = ∇ qi U i (q i ) -d i qi -p i a i (q i -q ) -p i j∈Ni a ij q i -q j (t -T ji (t)) (23)
for each i ≤ N and let [START_REF] Liu | Controlled synchronization of heterogeneous robotic manipulators in the task space[END_REF] hold and under Assumptions A1-A3. Then, the equilibrium ( q, q) = (0, 1 N ⊗ q ) is UGAS provided that the damping gain satisfies [START_REF] Liu | Controlled synchronization of heterogeneous robotic manipulators in the task space[END_REF]. Furthermore, the the closed-loop system [START_REF] Liu | Consensus for multiple heterogeneous Euler-Lagrange systems with time-delay and jointly connected topologies[END_REF] is ISS with respect to additive inputs u i ∈ R n .

Proof. The proof follows along the same lines as that of Proposition 1. Let N i := N i ∪ {j = }. We (re)define the relative position errors [START_REF] Wang | Consensus of networked mechanical systems with communication delays: A unified framework[END_REF] as

e i := a i (q i -q ) + j∈Ni a ij (q i -q j ) = j∈N i a ij (q i -q j ). (24)
Then, the closed-loop equations ( 6), ( 23) are given by [START_REF] Liu | Consensus for multiple heterogeneous Euler-Lagrange systems with time-delay and jointly connected topologies[END_REF] with N i replaced by N i . Next, we introduce

H(q, q, qt ) = 1 4 i∈ N 2 p i q i M i qi + a i |q i -q | 2 + i∈ N   1 4 j∈N i a ij |q i -q j | 2 + W i ( qt )   ,
where W i ( qt ) is redefined upon (15) by replacing N i with N i . Akin to Ė, im view of ( 14), the total derivative of H along the closed-loop trajectories satisfies

Ḣ ≤ - i∈ N   λ i | qi | 2 + δ i j∈N i a ij t t- * Tji | qj (σ)| 2 dσ   .
Thus, we introduce the Lyapunov-Krasovskiȋ functional V(t, x t ) = H(q(t), q(t), qt ) + U (e(t), q(t)),

where U (e, q) is defined in [START_REF] Nuño | Consensus in networks of nonidentical Euler-Lagrange systems using P+d controllers[END_REF]. Note that H(q, q, qt ) + U (e, q) = i∈ N

1 2p i [ qi + h i ] M i [ qi + h i ] + a i 4 |q i -q | 2 + i∈ N j∈N i a ij * T ji 2α i + δ i 0 - * Tji t t+σ | qj (θ)| 2 dθdσ + 1 4 i∈ N   j∈N i a ij |q i -q j | 2 - 2 2 p i h i M i h i   . Since 1 p i h i M i (q i )h i ≤ m L j∈N i a ij |q i -q j | 2 ,
where L := max i∈ N {L ii + a i }, setting < 1 m L ensures that H(q, q, qt ) + U (e, q) is positive definite and radially unbounded with regards to e and q.

Proceeding as in the Leaderless consensus case, we can show that V(t, x t ) = H(q(t), q(t), qt ) + U (e(t), q(t))

satisfies V ≤ -φ 1 | q(t)| 2 - β 2 1 + |e(t)| |e(t)| 2 -φ 3 i∈ N j∈N i a ij t t- * Tji | qj (σ)| 2 dσ,
where φ 1 := λ -2 mλ M {L } + kc + d 2µ ; and φ

3 := δ - L * T 2µ . Thus, setting < min λ 2 mλ M {L } + kc + d 2µ , 2µδ L * T , 1 2 m L , (25)
for any µ satisfying µ < 2 d+1 , ensures that V is negative definite so, after Theorem 1, the result follows. The statement on Input-to-State stability follows mutatis mutandis as for Proposition 1.

Remark 5. Suppose that velocity measurements are not available, but that a velocity estimate qi is available from a velocity observer. In this case, the certainty-equivalent controllers ( 9) and ( 23) become τ i = ∇ qi U i (q i ) -d i qi -p i j∈Ni a ij q i -q j (t -T ji (t)) and τ i = ∇ qi U i (q i ) -d i qi -p i a i (q i -q )

-p i j∈Ni a ij q i -q j (t -T ji (t)) ,

respectively. Correspondingly, the closed-loop system (11) becomes qi = -M -1 i (q i ) [C i (q i , qi ) qi + d i qi + p i e i ]

-p i M -1 i (q i ) j∈Ni a ij t t-Tji(t) qj (σ)dσ

+ d i M -1 i (q i ) qi ,
where qi := qiqi is the velocity estimation error; similarly for the closed-loop system with the controller [START_REF] Gu | Stability of Time-Delay Systems[END_REF] in which case N i is replaced with N i . In either case, (21) holds with u(t) = q(t) so, under the hypothesis that the latter converges asymptotically to zero, so do q(t) and ė(t).

If furthermore, the origin for estimation errors dynamics is uniformly globally asymptotically stable, so is the null solution for the overall closed-loop dynamics.

Conclusions

We propose a novel Strict Lyapunov-Krasovskiȋ Functional for the leaderless and leader-follower consensus in networks of EL-agents controlled by simple decentralized P+d schemes for which the communications impose variable time-delays that may not be differentiable. We prove that the agents positions and velocities globally, uniformly and asymptotically converge to a consensus position and to zero, respectively. Moreover, by directly replacing the velocity measurements by the velocity estimates obtained from a velocity observer that accepts a SLF, as the I&I velocity observer, the same UGAS property holds. Finally, we also establish the fact that the resulting closed-loop systems are ISS with regards to external inputs (perturbations).

One future research avenue concerns the inclusion of non-holonomic restrictions in the agents dynamics.

Remark 4 .

 4 e, e := [e 1 • • • e N ] . For further development, we observe that h satisfies the following: i) |h(e)| ≤ 1, ii) |h(e)| ≤ |e|, and iii) | ḣ(e)| ≤ 2| ė|. Furthermore, from the fact that ė = (L ⊗ I n ) q, it holds that | ḣ| ≤ 2λ M {L}| q|.

  Tji | qj (σ)| 2 dσ   and, defining λ := min i∈ N {λ i }, δ := min i∈ N {δ i }, kc := max i∈ N { kci pi }, d := max i∈ N { di pi }, and * T := max i∈ N , j∈Ni

|

  qj (σ)| 2 dσ, where µ 1 := λ -2 mλ M {L} + kc + d 2µ , µ 2 := 1µ 2 d + 1 , and µ 3 := δ -

Preprint published in Syst. & Control Letters, vol 135, pp. 104579, 2020

In its original form, in[START_REF] Krasovskiȋ | Stability of motion[END_REF], this theorem is formulated for asymptotic stability.

Acknowledgements

This work is partially supported by the Mexican CONA-CyT Basic Scientific Research grant CB-282807; by the Dept. STITS of University of Paris Saclay; by the Government of Russian Federation (grant 08-08); and by the ECOS-Nord contract M14M02.


Sahar El

Olivier Harle

Jean-Roch Mouret

Alain Rapaport

Thibault Nidelet

Design and mathematical optimization of multi-strain ecosystems in winery

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Design and mathematical optimization of multi-strain ecosystems in winery

Sahar El'Aida 1 , Olivier Harle 2 , Jean-Roch Mouret 1 , Alain Rapaport 3 , Thibault Nidelet 1 1 : SPO, Univ Montpellier, INRA, Montpellier SupAgro, Montpellier, France 2 : STLO, INRA, Agrocampus Ouest, Rennes, France 3: MISTEA, Univ Montpellier, INRA, Montpellier SupAgro, Montpellier, France * Corresponding Author: thibault.nidelet@inra.fr For food processing, fermentation by microorganisms allows food preservation and production of several fermented products, among the most consumed are bread, cheese, beer and wine. In all these anthropic environments, microbes form communities where individual microorganisms interact in various ways such as competition, predation, commensalism or mutualism that may have a tremendous effect on the final product quality. Exploiting these interactions by ecological engineering (by choosing the good partners in the good proportions) appears to be a promising strategy for improving fermented food. However, it demands a profound understanding of the interaction's mechanisms and an exponential number of experiments to assess all possible microbes combination. To meet these challenges in winery, we choose to guide our ecological designs by a mathematical model simulating the microbial and metabolic dynamics of the oenological fermentation. We perform a deep phenotyping of fermentation driven by different yeast-yeast combination that gives us the necessary information to build a mathematical model (of ordinaries differential equations) that simulate the dynamic of mixed cultures from isolated cultures data. With this model, we test different inoculation protocols (which strain/species, in what proportion and at which time) and evaluate their performance for maximizing different functional traits.

Oenology, fermentation, yeast-yeast interactions, population dynamics, non-saccharomyces species, ecological engineering, microbiology, wine.